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On Faraday s III. By Lines of Force. Clerk Maxwkll, B.A. Fellow of J. Trinity College, Cambridge. [Read Dec. The 11, 1856.] of electricity on the surface of conductors have been duction of galvanism and No science. between electrical theory electricity at into fallen with relation can now be put forth, unless rest current electricity, but and inductive effects of electricity in both states. laws, the mathematical form of which is it have been reduced the other parts of the shews the connexion not between the attractions and Such a theory must accurately those satisfy known, and must afford the means of calculating the where the known formulae are inapplicable. effects in the limiting cases esta- the theory of the con- mutual attraction of conductors that of the but have not formulDe, data are wanting; the experimental while in other parts to matliematical The analytically some parts of the mathematical theory of magnetism are deduced from experiment; only and Feh. present state of electrical science seems peculiarly unfavourable to speculation. laws of the distribution blished, 10, 1855, appreciate the requirements of the science, the student must In order therefore to make himself materially interferes with further progress. The first with a familiar considerable body of most intricate mathematics, the mere retention of which in the memory process therefore in the effectual study of the science, must be one of simplification and reduction of the results of previous investigation to a form in which the mind can grasp them. The results of this simplification the form of a purely mathematical formula or of a physical hypothesis. entirely lose sight of the phenomena to be explained; and though we In the may may first trace take case out we the consequences of given laws, we can never obtain more extended views of the connexions of the subject. If, on the other hand, we adopt a physical hypothesis, we see the phenomena only through a medium, and are liable to that blindness to facts a partial explanation encourages. which allows the mind committed to We it is in assumption which must therefore discover some method of investigation at every step to lay hold of a clear physical conception, without any theory founded borrowed, so that and rashness neither drawn being on the physical science from which that conception is aside from the subject in pursuit of analytical subtleties, nor carried beyond the truth by a favourite hypothesis. 4—2 Mr maxwell, ON FARADAY'S LINES OF FORCE. 28 In order to obtain physical ideas without adopting a physical theory we must make oura physical analogy I Thus illustrate the other. the mathematical sciences are founded on relations between all physical laws and laws of numbers, so that the aim of exact science is to reduce the of nature to the determination of quantities by operations with numbers. most universal of all analogies to a very partial one, we mathematical form between two different phenomena giving The changes of direction which light undergoes which intense forces problems Passing from the same resemblance rise to a physical theory of light. from one medium to another, to the direction, and not to the and velocity of motion, was long believed to be the true explanation of the refraction of light; we still find it useful in the solution of certain in moving through a narrow space This analogy, which extends only act. the find in passing are identical witii the deviations of the path of a particle in in mean that between the laws of one science and those of another which makes each of partial similarity them By the existence of physical analogies. selves familiar with problems, in which we employ it without danger, The other analogy, between light and the vibrations of an elastic but, though its importance and fruitfulness cannot be overfarther, much medium, extends estimated, we must recollect that it is founded only on a resemblance inform between the laws as an artificial method. of lio-ht By and those of vibrations. stripping observation, but probably deficient both method. I of its physical dress and reducing we might obtain a system of truth a theory of " transverse alternations," its it in the vividness of its strictly it to founded on conceptions and the fertility of have said thus much on the disputed questions of Optics, as a preparation for the discussion of the almost universally admitted theory of attraction at a distance. We have all acquired the mathematical conception of these attractions. about them and determine distinct mathematical significance, phenomena. .the There is appropriate forms their and no formula their results are in applied found uniform media appear at first those relating to attractions. heat, conductivity. matical laws of tlie The sight The a at distance. among viorA force is be in accordance with natural in the the most different in foreign to the subject. in minds of men than that of The laws of the conduction of heat quantities which enter into uniform motion of heat can reason mathematics more consistent with nature than formula of attractions, and no theory better established the action of bodies on one another to We These formula have a or formula;. tiieir in physical relations from them are temperature, flow of Yet we find that the mathe- Iiomogeneous media are identical in form with those of attractions varying inversely as the square of the distance. We have only to substitute source of heat for centre of attraction, flow of heat for accelerating effect of attraction at any point, and temperature for jwtential, and the solution of a problem in attractions is transformed into that of a problem in heat. This analogy between the formulae of heat and attraction was, I believe, first pointed out by Professor William Thomson in the Cambridge Math. Journal, Vol. III. Now the conduction of heat is supposed to proceed by an action between contiguous parts of a medium, while the force of attraction is a relation between distant bodies, and yet, if we knew nothing more than is expressed in the mathematical formulfe, be nothing to distinguish between the one set of phenomena and the other. tiiere would Mr. It that if true, is maxwell, ON FARADAY'S LINES OF FORCE. we introduce otlier considerations subjects will assume very different their laws may remain, and will aspects, but and observe additional the two facts, mathematical resemblance of some of tiie be made useful still 29 exciting appropriate mathematical in ideas. It is by the use of analogies of mind, in a convenient and manageable form, those mathematical ideas which to the study of the phenomena of kind that I this The methods electricity. by the processes of reasoning which are found though they have have attempted in to the researches of Faraday *, and which, been interpreted mathematically by Prof. those employed by the professed mathematicians. in which it I evident that I am Thomson and By strict I adopt, I iiope not attempting to establish any pliysical theory of a science my design is to application of the ideas and methods of Faraday, the connexion of the very different orders of the niatlietnatical mind. others, are very when compared with method which the have hardly made a single experiment, and that the limit of shew how, by a are necessary are generally those suggested generally supposed to be of an indefinite and unmathematical character, to render bring before the phenomena which he has discovered may be I shall therefore avoid as much clearly placed before as I can the introduction of anything which does not serve as a direct illustration of Faraday's methods, or of the mathematical deductions which I shall use may be made from them. In treating the Faraday's mathematical methods as well as his ideas. subject requires it, I shall use analytical notation, simpler parts of the subject When the complexity of the confining myself to the development still of ideas originated by the same philosopher. I have When and placed in the first place to explain and illustrate a body in is electrified in any given position, If the small body be now the idea of "lines of force." any manner, a small body charged with positive urging experience a force will negatively electrified, it will in it electricity, a certain direction. be urged by an equal force in a the north or south poles of a direction exactly opposite. The same small magnet. relations hold between If the north pole a magnetic urged is in body and one direction, the south pole is urged in the opposite direction. In this way we might find a line passing through any point of space, such that it represents the direction of the force acting on a positively electrified particle, or on an elementary north pole, and the reverse direction of the force on a negatively south pole. at electrified Since at every point of space such a direction any point and draw a line so that, we go along as it, its account a line of force. filled all We might by passes, this their direction curve if any point XXXVIII. of tlie shall will indicate the that of lines tlie of force, force at point. • See especially Series we commence and might be called on that same way draw other in the space with curves indicating it found, direction at always coincide with that of the resultant force at that point, direction of that force for every point through which particle or an elementary may be Experimental Researches, and Phil. Mag. 1852. till we had any assigned so maxwell, ON FARADAY'S LINES OF FORCE. Mb. We should thus obtain a geometrical model of the physical plienomcna, us the direction of tell but as any point. according to any given law, by regulating tubes. This method a tube is and velocity vary way we might in this by the motion of the of representing the intensity of a force by the fluid in these velocity of an imaginary applicable to any conceivable system of forces, but the case in simplification in direction its the ve- since tiien, we may make the section of the tube, tlie represent the intensity of the force as well as fluid in these curves not as mere lines, an incompressible fluid, inversely as the section of the tube, is some method of indicating require still we consider If tubes of variable section carrying fine of the fluid locity but we should force, tlie the intensify of the force at which would it is capable of great which the forces are such as can be explained by the hypothesis of attractions varying inversely as the square of the distance, such as those observed in electrical and magnetic phenomena. will generally In the case of a perfectly arbitrary system of forces, there be interstices between the tubes; but in the case of electric The possible to arrange the tubes so as to leave no interstices. it is motion of a surfaces, directing the commence fluid filling up the whole the investigation of the laws of these forces by and magnetic forces tubes will then be mere It has been usual to space. assuming that the pJienomena at once between certain points. are due to attractive or repulsive forces acting We may however obtain a different view of the subject, and one more suited to our more difficult inquiries, by adopting for the definition of the forces of which we treat, that they magnitude and direction by the uniform motion of an incompi-essible propose, I then, to first can be clearly conceived describe a may be represented in fluid. method by which the motion of such a fluid secondly to trace the consequences of assuming certain conditions ; of motion, and to point out the application of the method to some of the less complicated phenomena of electricity, magnetism, and galvanism of these methods, and the introduction attractions By fluid, shew how by an extension lastly to to Faraday, the laws of the as to physical nature of the electricity, or adding anything to has been already proved by experiment. referring everything to the purely geometrical idea of the motion of an imaginary I hope premature mere and and inductive actions of magnets and currents may be clearly conceived, without making any assumptions that which ; of another idea due to attain generality and precision, theory professing to explain speculation which I have collected and the cause of are found to avoid the dangers arising from a the to phenomena. be of the If any use to results of experimental philosophers, in arranging and interpreting their results, they will have served their purpose, and a mature theory, in which physical facts will be physically explained, will be formed by those who by interrogating Nature herself can obtain the only true questions which the mathematical theory I. (1) The solution of the suggests. Theory of the Motion of an incompressible Fhiid, substance here treated of must not be assumed to possess any of the properties of ordinary fluids except those of freedom of motion and resistance to compression. It is not Mr maxwell, ON FARADAY'S LINES OF FORCE. even a liypothctical fluid which is 31 introduced to explain actual phenomena. It is merely a may he employed for cstahlisliing certain theorems in a way more intclligihle to many minds and more applicahle to physical which algebraic symbols alone are used. The use of the word " Fluid" collection of imaginary properties wliich pure mathematics in problems than that in not lead us into error, will if we remember that it denotes a purely imaginary substance with the following property The portion ofjluid ivhich at any instant occupied a given volume, will at any succeed- ing instant occupy an equal volume. This law expresses the incompressibility of the measure of quantity, namely its volume. direction of motion of tlie fluid its The and furnishes us with a convenient fluid, unit of quantity of the fluid will therefore be the unit of volume. The (2) the space which may occupies, but since the direction be different at diS'erent points of we determinate for every such point, is conceive a line to begin at any point and to be continued so that every element of the line indicates a it will in general manner by its direction the direction of motion at that point of space. Lines drawn in such that their direction always indicates the direction of fluid motion are called lines of fluid motion. If the motion of the fluid be what velocity of the motion at any fixed is called steady motion, that is, if the direction and point be independent of the time, these curves will repre- sent the paths of individual particles of the fluid, but if the motion be variable this will not generally be the case. The come under our cases of motion which will notice will be those of steady motion. (3) and if If upon any surface which cuts the lines of fluid motion from every point of this curve we draw a generate a tubular surface which we may call line we draw a closed curve, of motion, these lines of motion will a tube of fluid motion. Since this surface generated by lines in the direction of fluid motion no part of the fluid can flow across that this imaginary surface (4) The is as impermeable quantity of fluid which in unit of time crosses any fixed section of the tube and no part runs tlirough the the second section is sides of the tube, therefore equal to that which enters through the For tlie fluid is the quantity is incompressible, which escapes from first. If the tube be such that unit of volume passes through any section in unit of time called a unit tube so to the fluid as a real tube. the same at whatever part of the tube the section be taken. (5) it, is it is offluid motion. In what follows, various units will be referred to, and a finite number of surfaces will be drawn, representing in terms of those units the motion of the fluid. in order to define the motion in every part of the to be drawn at indefinitely small intervals would involve continual reference ; fluid, but since lines or Now an infinite number of lines would liave tlie to the theory of limits, description of such a system of lines it has been thought better to suppose Mk maxwell, on FARADAY'S LINES OF FORCE. 32 the lines drawn as small as we To (fi) at intervals depending on the assumed surface which cuts tlie all curves intersecting the The From These motion be drawn. Similar impermeable surfaces curves of which, by their intersection with the unity of time, this any it draw a second system of every point in a curve of the lines may first system form a surface through which no will be drawn for all the curves of the first let fluid system. system, will form quadrilateral tubes, which will be first Since each quadrilateral of the cutting surface transmits unity of fluid transmit unity of fluid through any of its* every tube in the svstem will The motion sections in unit of time. determined by the same surface second system will give rise to a second system of impermeable surfaces, tlie tubes of fluid motion. in a system of unit tubes. quadrilaterals formed by the intersection of the two systems tlie of curves shall be unity in unit of time. a line of fluid unit. system, and so arranged that the quantity of fluid which crosses first the surface within each of assume the unit to of fluid motion, and draw upon lines On system of curves not intersecting one another. passes. by means of define the motion of the whole fluid fixed and afterwards by taking a small submultiple of the standard please Take any unit, at every part of the of the fluid system of unit tubes for ; the direction of motion through the point in question, and the velocity is space is it occupies that of the is tube reciprocal of the area of the section the of the unit tube at that point. We (7) have now obtained a geometrical construction wliich completely defines the motion of the next to shew motion of A these fluid by dividing the space how by means it occupies into a system of unit tubes. of these tubes we have may ascertain various points relating to the may begin and end at different points, and tlie fluid. unit tube may be may either return into itself, or either in the In the that space. first boundary of the space case there is in which we investigate the motion, or within continual circulation of fluid in the tube, in a second the fluid enters at one end and flows out at the other. may be supposed are in the bounding surface, the fluid from an unknown source, and to flow out origin of We tiie tube or its to tiie If the extremities of the tube be continually supplied from without at the other into an termination be within the space unknown reservoir; but if the under consideration, then we must conceive the fluid to be supplied by a source within that space, capable of creating and emitting unity of fluid in unity of time, and to be afterwards swallowed receiving and destroying the same There created, it and reduced up by a sink capable of continually. nothing self-contradictory in the conception of these sources where the fluid is and sinks where have made amount it is annihilated. incompressible, and to nothing at others. The properties of the fluid are at our disposal, now we suppose The it produced from notiiiiig at certain points places of production will be called sources, and their numerical value will be the number of units of fluid which they produce in unit of time. places of reduction will, for want of a better name, be called sinks, and will be estimated number of units of fluid absorbed is we in unit of time. sources, a source being understood to be a sink when Both places its sign is will negative. The by the sometimes be called Mn MAXWELL, ON FARADAY'S LINES OF FORCE. It is evident that tlie (8) the by number of that of motion in the tubes. A carries in. the other, in as and the direction in which the it, measured by determined and must carry therefore much as fluid out of the surface as it tube which begins within the surface and ends without it will carry it out unity will carry in the same In order therefore to estimate the amount of fluid which flows out of the closed quantity. we must subtract the number of tubes which end within the surface, is fluid passes is If the surface be a closed one, tiien any tube whose ter- and one which enters the surface and terminates within of fluid; of tubes which begin there. we If which passes any fixed surface fluid on the same side of the surface must cross the surface as many times in the one lie direction unit tubes which cut its minations amount of 33 If the result is the quantity of fluid produced within the surface minus the number of unit sinks, and number negative the fluid will on the whole flow inwards. beginning of a unit tube a unit source, and call the surface from the its termination a unit sink, then estimated by the is number of unit sources must flow out of the surface on account of the this inconipressibility of the fluid. In speaking of these unit tubes, sources and sinks, we must remember what was stated (5) to the as magnitude of the number we may If (9) and distribute we know we conceive if them according sum of to a third case in which and velocity of the the direction the two former cases at fluid at any surface crosses fluid any point corresponding points, in the third then is the case will be the algebraic the quantities which pass the same surface in the two former cases. which the their any law however complicated. which passes a given fixed surface fluid and increasing size in the direction and velocity of the fluid at any point in two different cases, the resultant of the velocities in amount of and how by diminishing their unit, For the rate at the resolved part of the velocity normal to the surface, is and the resolved part of the resultant is equal to the sum of the resolved parts of the com- ponents. Hence the number of unit tubes which cross the surface outwards in the third case be the algebraical sum of the numbers which cross it sum as we of sources within any closed surface will be the may be taken Since the closed surface of sources and sinks in the II. two former Theory of as small in the third case arises The which the the fluid fluid is will in general uniform motion we may it is and the number evident that the distribution of an I. medium. its motion is opposed by the medium through This resistance depends on the nature of the medium, depend on the direction Part imponderable incompressible Jluid through a conceive to be due to the resistance of a supposed to flow. in which the fluid moves, as well as restrict ourselves to the case of a all directions. Vol. X. please, here supposed to have no inertia, and is For the present we may the same in cases, cases. action of a force which and two former of the numbers in the two former cases. from the simple superposition of the distributions resisting (10) in the must The law which we assume is on its velocity. uniform medium, whose resistance as follows. 5 is Mr maxwell, ON FARADAY'S LINES OF FORCE. 34 Any retarding force proportional to volume of the unit of velocity fluid in by may neutralise the effect of may make as small as we please, two of its faces. Then the resistance difference of pressures is the resistance. Conceive a cubical unit of fluid (which we (5)), and let move it so that it, tlie in a direction perpendicular to be kv, and therefore the difference of pressures on the will the pressure diminishes in the direction measured along the motion line of ference of pressure at its is supposed the surface (p) of equal pressure. call corresponding to the pressures pressure belonging to unit of pressure is we measure a length equal vary continuously in the to equal to a given pressure p will is is kv, so that of motion at the rate of kv for every unit of length so that if ; and second faces first to k units, the dif- extremities will be kvh. Since the pressure (11) which the pressure In order, therefore, that must be a greater pressure behind any portion of the there fluid than there bv be a force equal to kv acting on v, then the resistance will a direction contrary to that of motion. may be kept up, in front of directhj opposed by a is its velocity. If the velocity be represented the medium portion of the fluid moving thrmirih the resisting it, lie at may If a series of these surfaces be constructed in the fluid then the 0, 1, 2, 3 &c., number and the surface niav be referred that pressure which fluid, all the points on a certain surface which we is of the surface will indicate the The to as the surface 0, 1, 2 or 3. produced by unit of force acting on unit of surface. In order therefore to diminish the unit of pressure as in (5) we must diminish the unit of force same proportion. in the (12) ' It is easy to see that of fluid motion lines ; for if tliese surfaces of equal pressure the fluid were to move resistance to its motion which could not be balanced remember to be considered.) it, by any difference of pressures. so that the resistances There are therefore two to the any other direction, there would be a in that the fluid here considered has no inertia or mass, only which are formally assigned to must be perpendicular sets of surfaces and tliat its (We must properties are those and pressures are the only things which by their intersection form the system of unit tubes, and the system of surfaces of equal pressure cuts both the others at Let h be the distance between two consecutive surfaces of equal pressure mea- right angles. sured along a line of motion, then since the difference of pressures kvh = which determines the relation of v Let s to /;, 1, so that one can be found when the other is known. definition of a unit tube vs find by the last (13) The and section s. = \, equation s = kh. surfaces of equal pressure cut the unit tubes into portions whose length These elementary portions of unit tubes each of them unity of volume of fluid passes from unit 1, be the sectional area of a unit tube measured on a surface of equal pressure, then since bv the we = of time, and will a pressure be /) called to therefore overcomes unity of resistance in that time. overcoming resistance is therefore unity in every cell in unit a pressure The work every unit of time. is cells. (p-1) h In in spent in 35 Mr maxwell, ON FARADAY'S LINES OF FORCE. If (14) may be tubes known, the direction and magnitude of pressure are may be of the fluid at any point the velocity unit of equal surfaces tlie found, after and the beginnings and endings of these tubes ascertained constructed, and marked out as the sources whence the fluid is derived, In order to prove the converse of that may be pressure at every point if we down lay two at every point in the fluid in the pressure at any point the two former it disappears. certain preliminary propositions. sum the is different cases, and of the pressures at then the velocity at any point in the third cases, the distribution of sources of the velocities in the other two, and the resultant is and the sinks where the distribution of sources be given, the if we must found, take a third case in which corresponding points in case this, we know the pressures If (15) the complete system of wliicli is that due to the simple superposition of the sources in the two former cases. For the in that any direction velocity in direction ; that so if proportional to is new system resolved the in the the same in The two original systems. It in by follows from this, new the system, the (<)), sum It all is evident in tliat will therefore are simply combined which in the pressure two given systems the distribution of sources will is old ones, and that difference the of pressure be got by changing the sign of first. If the pressure at every point of a closed surface be the same and equal to p^ (16) if tlie form the third. to the sources in the second system and adding them to those in the and resolved parts be the resultant former systems. of the corresponding quantities in system the the two rates, the velocity sum of the the new system tlie in combined the that the quantity of fluid which crosses any fixed surface the sources of the two original systems in the sum of direction will be velocity in of the velocities at corresponding points is, rate of decrease of the pressure two systems of pressures be added together, since the rate of decrease of pressure along any line will be the in tlie there be no sources or sinks within the surface, then there will be no motion and the pressure within fluid within the surface, For if there be motion of fluid tlie it will of the be uniform and equal to p. within the there will surface be tubes of fluid motion, and these tubes must either return into themselves or be terminated either within the surface or at its Now boundary. pressure to places of less pressure, it since the fluid always flows from places of greater cannot flow in a I'e-entering curve; there are since no sources or sinks within the surface, the tubes cannot begin or end except on the surface and since the pressure in tubes havinjr both surface, at points all extremities on If the distribution pressure at surface. if of sources within two is the same, there can be no Hence there is p, the pressure at any point within every point of a pressures can exist within the For tiie is ; motion no motion within the and therefore no difference of pressure which would cause motion, and since the pressure at the bounding surface (17) of the surface the surface given closed be also known, it is surface be also p. known, and the then only one distribution of surface. different distributions of pressures satisfying these conditions could be found, a third distribution could be formed in which the pressure at any point should be the 5—2 Mr MAXT\'ELL, ON FARADAY'S LINES OF FORCE. 36 difference of the pressures in tlie at the in surface at tlie Then bv but this and there cases are the same, centre of which in the pressures is in a unit source the source being supposed The the sources within all the must be zero; distribution third the two former cases, and therefore these at any point of an placed, the pressure is body of infinite an at infinite fluid distance from be zero. to flow out will fluid and only one distribution of pressure possible. Let us next determine the pressure (18) zero, every point in the at the difference of the is would be the pressure distributions, (15). pressure the (l6) this case, since the pressures are the same in both it distribution third by surface would vanish, In two former distributions. the surface and the sources within from and since unity of volume symmetrically, centre the flows out of every spherical surface surrounding the point in unit of time, the velocity at a distance r from the source will be 1 = V . 47rr^ k The when r of decrease of pressure rate the actual pressure at infinite, is therefore kv or is any point , he will and since the pressure = p= . 47rr The pressure It evident that is therefore inversely is proportional to the pressure due to a the distance from the source. unit sink will be negative and equal to k 47rr source formed by the coalition of If we have a pressure will be ]) = , so 5 unit sources, then the resulting that the pressure at a given distance varies as the resistance iTTl' and number of sources conjointly. If a (19) number of sources and sinks coexist in the fluid, then in order to determine add the pressures which each source or the resultant pressure we have only For by be a solution of the problem, and by (17) By this this will (lo) method we method of (I-i) of pressures is We (20) space and in we can determine the if we will be the only one. distribution of sources to which a given distribution due. have next to shew that if we conceive any imaginary surface the lines of motion of the this surface a distribution of sources any way the motion of the For it sink produces. can determine the pressures due to any distribution of sources, as by the intersecting on one side of to describe the fluid fluid, we may as substitute for upon the surface itself fixed in the fluid without altering on the other side of the surface. system of unit tubes which defines the motion of the fluid, and wherever a tube enters through the surface place a unit source, and wherever a tube goes out through the surface place a unit sink, and at the same time render the surface impermeable to the fluid, the motion of the fluid in the tubes will go on as before. Mr maxwell, ON FARADAY'S LINES OF FORCE. 37 If the system of pressures and the distribution of sources which produce them (21) be known in a medium whose same system of pressures resistance then in order to produce the k, medium whose resistance is unity, the rate of production For the pressure at any point due to a given k. a in measured by is at each source must be multiplied by source varies as the rate of production and the resistance conjointly; therefore must vary inversely be constant, the rate of production On (22) the conditions to be fulfilled at a surface coe^cients of resistance are k and medium both media, The resistance. which separates two media whose the that quantity medium The to the other. velocity normal to the surface and therefore the rate of diminution of pressure of the tubes direction of motion the same and the surfaces of equal pressure will is this refraction will be, that is it the normal to the the tangent of the angle of to as k' is to k. Let the space within a given closed surface be and with a filled the pressures at any point of this medium that given distribution of sources within and without the surface be given exterior to it, let required it is ; the same system of pressures to determine a distribution of sources which would produce medium whose difi'erent compound system due from to a a out the and that the tangent of the angle of incidence (23) in is proportional the direction of incidence and takes place in the plane passing through refraction flows to be altered after passing through the surface, and the law of surface, which of fluid at any point flows into the other, and that the pressure varies con- tinuously from one in the pressure k'. These are found from the consideration, of the one if as the resistance. coefficient of resistance is unity. Construct the tubes of fluid motion, and wherever a unit tube enters either medium place a unit source, and wherever it leaves it Then unit sink. place a if we make the surface impermeable all will go on as before. Let the resistance of the exterior medium be measured by by k'. Then (including if those we multiply in and that of the interior the rate of production of all the sources in the exterior medium the surface), by k, and make the pressures will remain as before, and the same multiply all the sources in it by k', will k, coefficient of resistance unity, be true of the interior including those in the surface, and medium make if the we resistance its unity. Since the pressures on permeable We if we both sides of the surface are now equal, we suppose it please. have now the original system of pressures produced combination of three systems of sources. multiplied by k, the second is The medium had an equal but now the source is first of these is multiplied medium on by k and the sink by external unit source on the surface, a source sources the original system of pressures = (k the by medium by a given external system k', and the third is the In the original case every source in the itself. sink in the internal a uniform in the given internal system multiplied system of sources and sinks on the surface external may — k'). may be produced k', the other side of the surface, so that the result By means in a is for every of these three systems of medium for which k= 1. Mr maxwell, ON FARADAY'S LINES OF FORCE. 38 Let there be no resistance (24) let = k' 0, in addition to the given and sinks within the surface supposing the medium p uniform pressure which external and internal sources, and by we can render the pressure pressures so found for the external medium, together with the tlie medium, in the internal will be the true and only distribution of pressures possible. is For is, uniform and equal to p, and the is the same within and without the surface, the surface uniform, that surface, closed If by assuming any distribution of pairs of sources also p. is the witliin then the pressure within the closed surface pressure at the surface itself at medium the in if two such distributions could be found by taking different imaginary distributions of pairs of sources and sinks within the medium, then by taking the difference of the two we should have the pressure of the bounding for a third distribution, the new system and many as sources as sinks within and therefore whatever it, medium without that The sources surrounding the internal medium. the surface at the fluid must be outwards from every point of the surface But inwards towards the surface. because if any fluid is by pressure if it neitlier positive, zero, the motion of be negative, must flow it of these cases is possible, at zero in the third system. is all points in the boundary of therefore zero, and there are no sources, tlie medium internal and therefore the pressure in is the third case everywhere zero, (16). The is has been shewn that it ; is infinite is enters the surface an equal quantity must escape, and therefore the pressure at the surface The pressure at infinity If the pressure at the surface constant. is flows point. and we have an sources destroy one another, the all fluid some other in at any point of the surface, an equal quantity must flow out at In the external medium surface constant in no pressure resistance, the bounding surface of the internal in therefore it is difference of pressures in the zero tliroughout two given one distribution of pressure which is ; medium is and there also zero, but the pressure in the third case and there cases, therefore these are equal, is is the only possible, namely, that due to the imaginary distribution of sources and sinks. When (25) through of fluid impermeable If the resistance or into it to the fluid, infinite in is The bounding it. and the tubes of fluid surface medium, there can be no passage may be considered as therefore motion will run along it without cutting by assuming any arbitrary distribution of sources within the surface the given sources velocities as in in the the outer case true one. For since no medium, we can fluid passes from what it also small, would be if the condition of there being fulfil through the surface, the tubes may be If the extent of the internal in the two media be addition to system of pressures in the outer medium will be the the independent of those outside, and (26) in it. medium, and by calculating the resulting pressures and of a uniform no velocity across the surface, altered the internal then in the interior are taken away without altering the external motion. medium be small, the position the external and of the medium if the difference of resistance unit tubes filled will not the whole space. be much Mr maxwell, ON FARADAY'S LINES OF FORCE. On this supposition of the internal medium we can easily calculate the 39 kind of alteration which the introduction produce; for wherever a unit tube enters the surface we must will conceive a source producing fluid at a rate —— - -k —--— , and wherever a tube leaves we must it k' place a sink annihilating fluid at the rate that the resistance in botii media , then calculating pressures on the supposition k the same as is in medium, we shall obtain we may get a better result by the external the true distribution of pressures very approximately, and repeating the process on the system of pressures thus obtained. abrupt change from one If instead of an (27) coefficient of resistance to another take a case in which the resistance varies continuously from point to point, medium the as if it were composed of thin shells each of we may may treat the case as if the resistance were equal to unity throughout, as in sources will then be distributed whenever the motion positive when is less to places the same in The (23). and be will of greater resistance, and negative whatever direction the motion of the fluid takes place a case in which the resistance will not in resistance at is diff'erent in different directions. velocity at any point, and a, ft, the same point, these quantities will be connected linear equations, ; but we medium to be may conceive In such cases the lines of general be perpendicular to the surfaces of equal pressure. be the components of the y the If a, components of h, the and will be referred to conductivity. In order to that name. b c In these equations there are simplify the equations, let nine = P^a + Q3/3 + ^,7, = P,ft + Q,7 + R,a, = P37 + Q.,a + R,ft. independent coefficients + i?j = 2^,, Q^-R^ = &c 4:T'={Q,- BJ' + where I, m, The n of us put Q, {Q, - are direction cosines of a certain 2lT, &c. R.f + (Q, - R^, fixed line in space. equations then become = P^a-^ S,ft + S,y + (n/3 - m 7) T, b = P.,ft+S,y + S,a + (ly -na)T, c^ P^y + S,a + S,ft - {ma - lft)T. transformation of coordinates we may get rid of a By The c by the following system of which may be called " equations of conduction" a and whole medium, Hitherto we have supposed the resistance at a given point of the (28) by tlie we shells, the contrary direction. in motion continuously throughout from places of By which has uniform resistance. properly assuming a distribution of sources over the surfaces of separation of the we treat the ordinary equations then become the coefficients marked .S*. Mr maxwell, ON FARADAYS LINES OF FORCE. 40 P^a+ a = = P,'/3 + (I'y - na)T, = P^y +{m'a - l'(i)T, b C where If t, m, n («'/3- 7n'y)T, new are the direction cosines of the fixed line with reference to the axes. we make — dx' u ^ - , — = ana z = -— dz , dy' the equation of continuity da db — + — dx dy dc ^ = 0, dz becomes p;^+p;"j^ " ' and x = ^/Pi^, we make if y the ordinary equation of the coefficient T. o, ^ = v rib- = V-P^V^ drp d'p dp rfp dtr d(^- ^3^1 0, of conduction. It appears therefore that the distribution this ' ^ +-—+-_ = then = + p;"^ dy' - ' dx'- of pressures Thomson has shewn how Professor is to not altered conceive a substance in which determines a property having reference to an axis, coefficient of Pj, P,, P, by the existence which unlike the dipolar. is For further information on the equations of conduction, see Professor Thomson on On Stokes Conduction of Heat in Crystals (Cambridge and Dublin Math. Jo?crn.), and the Professor Dynamical Theory of Heat, Part V. (Transactions of Royal Society of the Edinburgh, Vol. axes XXI. Part It is evident that all I.) that has been proved in (u), (15), (16), (17), with respect to the superposition of different distributions of pressure, and there being only one distribution of pressures corresponding to a given distribution of sources, will be true also in the case in which the resistance varies from point to point, and the resistance at the same point different in to different For directions. if we examine the proof we shall find is applicable it such cases as well as to that of a uniform medium. (29) We now are prepared to prove certain general propositions which are true in most general case of a medium whose resistance is different in different directions the and varies from point to point. We may by tlie method of (28), when the distribution of pressures surfaces of equal pressure, the tubes of fluid motion, that since in each cell into which a unit tube unity of fluid passes from pressure done by the fluid in each cell in The number p to pressure overcoming of cells in eacli unit tube pressure through which the endp', then the it passes. number of is is and the sources and known, construct the sinks. It evident — (}> 1) in unit of time, unity of work is resistance. is determined by the number of surfaces of equal If the pressure at the beginning of the tube be cells in it will is divided by the surfaces of equal pressure be p - jy . Now if jy and at the tube had extended from the Mr maxwell, ON FARADAY'S LINES OF FORCE. source to a place where the pressure the tube had come from the is the difference of these. is the Therefore we if number of numhcr of zero, the due to we must cut we denote the source of S S may be written the - S', sinks if always system will cells in the (Sp). The same (30) cell. Now work done each Now from the number previously obtained. off S'p' cells being reckoned negative, and the general expression for the number of be if number the true from which S tubes proceed to be p, Sp the source S; but if S' of the tubes terminate in a sink at tubes by S, the sink of S' tubes .S* would have been p, and cells number would have been p, and sink to zero, the find the pressure at a source cells a pressure j), then is 41 in by conclusion may be each source S, the fluid S in by observing arrived at that unity of work done on is units of fluid are expelled against a pressure p, so that overcoming resistance is At .S'p. each sink which in S' tubes terminate, S' units of fluid sink into nothing under pressure p'; the work done upon the fluid by the pressure is The whole work done by therefore S'p. the fluid may therefore be expressed by W=I.Sp-1S'p', or more, concisely, considering sinks as negative sources, W=1{Sp). Let (31) S represent the rate of production of a source in any medium, and the pressure at any given point due to that source. Then we superpose on if equal source, every pressure will be doubled, and thus by successive superposition a source nS would produce more generally the pressure a pressure np, or a given source varies as the rate of production of the source. at let p be this another we find that any point due to This may be expressed by the equation p==RS, where R is a coefficient depending on the nature of the source and the given point. hS R medium and on In a uniform medium whose resistance is the positions of the measured by k, k „ may be called the coefficient of resistance of the medium between By combining any number of sources we have generally p = ^{RS). the source and the given point. (32) In a uniform medium the pressure due to a source p= At another source S' at a distance r we if p' be the pressure at S due to S". 2(5"), and if the pressures due to the S — 47r r . have shall o. —k S k S^ 47r r If therefore there be two systems of sources first be ^{S'p) For every term S'p has a term Sp' equal to Vol. X. Paet I. p and to the = -EiSp). second p, 1{S) and then it. 6 Mr maxwell, ON FARADAY'S LINES OF FORCE. 42 Suppose tbat (33) to in a uniform medium tlie one plane, then the surfaces of equal pressure motion of the fluid will is everywhere parallel be perpendicular to tliis If plane." we take two parallel planes at a distance equal to k from each other, we can divide the space between these planes into unit tubes by means of cylindric surfaces perpendicular to the planes, and these together with the surfaces of equal pressure will divide the space into cells of which the leno-th is equal to the breadth. For h be the distance between consecutive surfaces of if equal pressure and s the section of the unit tube, we have by (l3) s But is the product of the breadth and depth s is but the depth ; = is kh. k, therefore the breadth h and equal to the length. If two systems of plane curves cut each other at right angles so as to divide the plane into little areas of distance k from the of which the length and breadth are equal, then by taking another plane at first and erecting cylindric surfaces on the plane curves be formed which cells will as bases, a system whether we suppose the will satisfy the conditions fluid to run alonsr the first set of cutting lines or the second *. Application of the Idea of Lines of Force. I have now to shew how the idea of motion as described above lines of fluid may be modified so as to be applicable to the sciences of statical electricity, permanent magnetism, magnetism of induction, and uniform galvanic currents, reserving the laws of electro-magnetism for special consideration. I shall assume that the phenomena of the mutual action of two opposite kinds of matter. electricity have been already explained by statical electricity we consider one of If and the other as negative, then any two particles of a force which is electricity repel these as positive one another with measured by the product of the masses of the particles divided by the square of their distance. Now we found in (18) that the velocity of r varies inversely as r". Let us see what every particle of positive electricity. The our imaginary fluid due to a source will be the velocity effect therefore tlie resultant velocity be proportional to the resultant attraction of find the resultant pressure at we may aX a. distance due to each source would be proportional to the attraction due to the corresponding particle, and sources would S of substituting such a source for all the particles. any point by adding the pressures due due to all Now we may to tlie given sources, find the resultant velocity in a given direction from the and the rate of decrease of pressure in that direction, and this will be proportional to the resultant attraction of the particles resolved in that direction. Since the resultant attraction in the electrical problem pressure in the imaginary problem, and since the imaginary problem, we may assume we may is select proportional to the decrease of any values rically equal to the decrease of pressure in that direction, or A. — -—— dai • for the constants in that the resultant attraction in any direction See Cambridge and Dublin Malhemalical Journal, Vol. III. p. 28li. is nume- Mn MAXWELL, ON FARADAY'S LINES OF FORCE. By this assumption we find that if Vhe dV= Xdx V= or since at an infinite distance 43 the potential, + Ydy + Zdz = - dp, p = 0, V = —p. and In the electrical problem we have In the fluid p = 2 ( — -) ; '. —k dm. S= 4nr If k be supposed very great, the amount of fluid produced by each source in order to keep up the pressures will be very small. The potential of any system of electricity on itself will be 2(pd«)= ^,npS) = ^W. iir If ^(dm), 2 (dm') be two systems of respectively, then by electrical particles — 2ipS') = , 47r the potential of the on the and pp' the potentials due to them (32) S(pcZm')= or 47r first ~, S (p'^ = 2 (p'rfm), 47r system on the second equal to that of the second system is first. So that problems the analogy in the ordinary electrical in fluid motion is of this kind : F=-p, X=-'/ dx dm — = ku, —k S, 47r whole potential of a system = coming — "LVdm = — W, where W is the work done by the fluid in over- resistance. The by those lines of force are the unit tubes of fluid motion, and they may be estimated numerically tubes. Theory of Dielectncs. The electrical induction exercised on a body at a distance depends not only on the distri- bution of electricity in the inductric, and the form and position of the inducteous body, but on the nature of the interposed medium, or dielectric. " Series Faraday * expresses this by the conception XI. 6—2 Mr maxwell, ON FARADAY'S LINES OF FORCE. 44 of one substance having a greater inductive capacity, or conducting the lines of inductive more action medium than another. freely the resistance is we suppose that If in our analogy of a fluid in a resisting different in different media, then by making the resistance less we obtain the analogue to a dielectric which more easily conducts Faraday's lines. It is electricity evident from (23) that in this case there will always be an apparent distribution of dielectric, there on the surface of the and positive electricity being negative electricity where the lines enter In the case of the fluid there are no real sources where they emerge. on the surface, but we use them merely for purposes of calculation. In the dielectric there be no real charge of electricity, but only an apparent electric action due to the surface. may medium, we should If the dielectric had been of less conductivity than the surrounding have had precisely opposite namely, positive electricity where lines enter, and negative effects, where they emerge. If the conduction of the dielectric tricity perfect or nearly so for the small quantities of elec- is with which we have to do, then we have the case of considered as a conductor, its surface is The (fii). dielectric is then a surface of equal potential, and the resultant attrac- tion near the surface itself is perpendicular to it. Theory of Permanent Magneti. A its maofnet is conceived to be own north and south o-overned made up of elementary magnetized the action poles, by laws mathematically motion can be employed to illustrate it may be useful to examine be represented by the unit may magnet by one made to this subject, the way cells in in to Hence the same is applica- and the same analogy of In each unit cell face, so that the first face sink and the second a unit source with respect to the rest of the compared north and south poles fluid which the polarity of the elements of a fluid motion. and flows out by the opposite face which has it. But enters of which upon other identical with those of electricity. of the idea of lines of force can be tion particles, each of fluid. It unity of fluid becomes a unit may therefore be an elementary magnet, having an equal quantity of north and south magnetic matter distributed over two of its faces. system, the fluid flowing out of one If we now consider the cell as forming part of a cell will flow into the next, and so on, so that the source will be transferred from the end of the cell to the end of the unit tube. If all the unit tubes the bounding surface, the sources and sinks will be distributed entirely begin and end on on that surface, and in the case of a tubular distribution of magnetism, all magnet which has what has been called a solenoidal or the imaginary magnetic matter will be on the surface*. Theory of Paramagnetic and Diatnagnetic Induction. Faraday "f has shewn that the field may be • effects of paramagnetic and diamagnetic bodies in the magnetic explained by supposing paramagnetic bodies to conduct the lines of force better. See Professor Thomson On + Experimental the Mathematical Theory of Magnetism, Chapters III. Researches (3292). &. V. Phil. Trans. 1851. 45 Mr maxwell, ON FARADAY'S LINES OF FORCE. By and diamagnetic bodies worse, than the surrounding medium. referring to (23) and (26), and supposing sources to represent north magnetic matter, and sinks south magnetic matter, then a paramagnetic body be in the neighbourhood of a north pole, the lines of force on if entering it will produce south magnetic matter, and on leaving Since the quantities of magnetic matter on amount of north magnetic matter. equal, but the southern matter is they will produce an equal it on the other hand the body be diamagnetic, or a worse conductor of surrounding medium, there where the will the whole are If nearest to the north pole, the result will be attraction. of force than the lines be an imaginary distribution of northern magnetic matter lines pass into the worse conductor, and of southern where they pass out, so that on the whole there will be repulsion. We system may is obtain a more general law from the consideration that the potential of the whole proportional to the amount of work done by the fluid in overcoming resistance. medium introduction of a second ance is increases or diminishes the greater or less than that of the tion will first medium. vary as the square of the velocity of the Now, by the work done according The amount as the resist- of this increase or diminu- fluid. theory of potentials, the moving force in any direction is measured by the rate of decrease of the potential of the system in passing along that direction, therefore the resistance within the second k', rounding medium, there where it is less, is medium, is greater than Ic, the resistance in a force tending from places where the resultant force v is when the sur- greater to to less values of the resultant body moves from greater so that a diamagnetic The force *. In paramagnetic bodies k' is less than k, so that the force points of greater resultant magnetic force. of k and k' , may it is now from points of less to Since these results depend only on the relative values evident that by changing the surrounding medium, the behaviour of a body be changed from paramagnetic to diamagnetic at pleasure. It is evident that we should obtain the same mathematical the magnetic force had a power of exciting a polarity as the lines in paramagnetic bodies, fact is we have not as yet these three theories, induced polarity. come to and in bodies we had supposed that results if which is in the same direction in the reverse direction in diamagnetic bodies -f. In any facts which would lead us to choose any one out of that of lines of force, that of imaginary magnetic matter, and that of As the theory of lines of force time least theoretic statement, we shall allow it admits of the most precise, and at the same to stand for the present. Theory of Magnecrystallic Induction. The may be * son, 1847. theory of Faraday thus stated. \. with respect to the behaviour of crystals in the magnetic field In certain crystals and other substances the lines of magnetic force are Experimental Researches (2797), (2798). See ThomCambridge and Dublin Mathematical Journal, May, t Exp. Res. Ixxxvii. p. 145. + (2429), (3320). See Weber, Poggendorft', Prof. Tyndall, Phil. Trans. 185G, p. 237. E.vp. Res. (283G), &c. Mr maxwell, ON FARADAY'S LINES OF FORCE. 46 conducted with different The body when suspended directions. different in facility in a uniform magnetic field will turn or tend to turn into such a position that the lines of force shall pass through with least resistance. it It to express the laws of this kind of action, The formula2. of reduce them in certain cases to numerical to and of imaginary magnetic matter are here principles of induced polarity use; but the theory of lines of force little class of not difficult by means of the principles in (28) is and even is capable of the most perfect adaptation to this phenomena. Theory of the Conduction of Current It the calculation of the laws of constant electric currents that the theory of fluid in is Electricity. motion which we have laid down admits of the most direct application. researches of of M. Ohm on this subject, we have those of M. Kirchhoff, In addition to the Aim. de Quincke, xlvii. 203, on the Conduction of Electric Currents in CJiim. xli. 496, and According to Plates. the received opinions we have here a current of fluid moving uniformly in conducting circuits, which oppose a resistance to the current which has an electro-motive force at some part of the circuit. be overcome by the application of to On account of this resistance to the motion This pressure, of the fluid the pressure must be different at different points in the circuit. which is commonly called electrical tension, in statical electricity, If we knew what amount of we assume electricity as electricity, measured finite between to be physically identical with the potential statically, passes phenomena. sets of along that current which This has as yet been done only approximately, but we *. (b be certain that the conducting powers of different substances degree, and that the difference between but found our unit of current, then the connexion of electricity of tension with current would be completed know enough is and thus we have the means of connecting the two glass quantity in glass, and a small but electricity statical and fluid and metal finite motion turns is, that the differ resistance Thus quantity in metal. is only in a great the analogy more perfect than we might have out supposed, for there the induction goes on by conduction just as in current electricity, but the quantity conducted is insensible On When must act owing to the great resistance of the dielectrics Electro-motive Forces. a uniform current exists in a closed circuit on the pressures, then it fluid besides the pressures. For if it is evident that some other forces the current were would flow from the point of greatest pressure point of least pressure, whereas in • See Exp. Res. (371). -j-. reality it circulates in f Exp. Res. in due to difference of both directions to the one direction constantly. Vol III. p. 513. We Mr maxwell, ON FARADAY'S LINES OF FORCE. must therefore admit the existence of Of a closed circuit. in A action. certain forces capable of keeping most remarkable tlie that which is is up a constant current produced by chemical of a voltaic battery, or rather the surface of separation of the fluid of the cell and the cell these 4? zinc, the is an seat of electro-motive force opposition to the resistance of the circuit. of electric currents, the positive current we adopt If from the is circuit, and the zinc, back to the fluid again. can maintain a current whicii through the platinum, the conducting fluid If the electro-motive force act only in the surface of separation of the fluid and zinc, then the tension of electricity in the fluid exceed that by the zinc in a quantity and on the strength of the current in If F In order to keep up this difference the conductor. be the electro-motive force, of electrical units delivered in must depending on the nature and length of the circuit of pressure there must be an electro-motive force whose intensity of pressure. in the usual convention in speaking / measured by that difference is the quantity of the current or the unit of time, and Ka number quantity depending on the length and resistance of the conducting circuit, then F = IK = p where p is the electric the part which remains attached F p — p\ or p'. affords a quantity and intensity more than in this : — The and unit of time, The intensity or IK, where The same The quantity number of K is and p the in zinc. to the platinum will be p, very useful and of the that the is amount of which electricity measured by / the number of unit currents which its is power of overcoming in resistance, and it transmits it contains. measured by is may be applied to the case of magnetism-j". any section of a magnetic body magnetic force which pass through it. The is measured by the intensity of magnetization depends on the resisting power of the section, as well as on the number of the section, and / it. the If k be the number F of the whole circuit. quantity and intensity of magnetization other will This distinction of but must be distinctly understood to mean nothing '", quantity of a current is no current the tension of is measure of the intensity of the current. the resistance idea of which pass through intensity is of a current lines of in the section the fluid in be broken at any point, then since there If the circuit be tension - p, resisting power of the material, and of force of lines which pass through it, .S* then lines the area of the whole throughout the section —IWhen p for magnetization the magnetic is tension produced by at Exp. Hes. Vol. influence of other magnets only, any point, then F= • the for the whole magnetic solenoid I f^dj!= IK = p-p'. III. p. 519. t we may put ^^P- ^es. (2870), (3293). Mr maxwell, ON FARADAY'S LINES OF FORCE. 48 When depend If magnetized circuit returns into a solenoidal be the quantity of the magnetization i magnetization does the itself, some magnetizing force of which the intensity on difference of tensions only, but on any point, or the number of at force passing through unit of area in the section of the solenoid, then the of niao-nctization in the circuit / = "^idyds, where dydx is number of the is which pass lines total through the element of the section, and the summation is not is F. lines of quantity any section performed over the whole section. The mao-netization, is of magnetization intensity measured by is measured by the sum of the ki any point, at =f, and the or force required the round the keep up the of magnetization in the circuit total intensity local intensities all to circuit, F=^ ifdx), where dx the is the clement of length in the circuit, and the summation is extended round entire circuit. In the same circuit we have always and depends on circuit, On The mathematical investigated From its the total is it resistance of the composed. is Curretits at a Distance. laws of the attractions and repulsions of conductors have been most ably have stood the his results assumption, element of another current the line joining the K IK, where form and the matter of which the Action of closed by Ampere, and the single F= is test of subsequent experiments. the action of an element of one that an attractive or repulsive force acting two elements, he determined has by current in simplest the upon an the direction of experiments the mathematical form of the law of attraction, and has put this law into several most elegant and useful forms. We must however that no experiments have been made on recollect these elements of currents except under the form of closed currents either in rigid conductors or in fluids, and that the laws of closed currents only can be deduced from such experiments. Hence if Ampere's formulae applied currents give true results, their truth to closed that the action between proved for elements of currents unless we assume must be along the line Although which joins them. philosophical in the present state of science, gation if ultimate we endeavour to do without it, it will and to this assumption most warrantable and be more conducive to freedom of investi- assume the laws of closed currents as the that when currents are combined according to parallelogram of forces, the force due to the resultant current to the has is the law of the the resultant of the forces component currents, and that equal and opposite currents generate equal and opposite forces, He not datum of experiment. Ampere has shewn due is is two such elements and when combined neutralize each other. also shewn that a closed circuit of any form has no tendency to turn a moveable circular conductor about a fixed axis through the centre of the circle perpendicular to its plane, and that therefore the forces a complete differential. in the case of a closed circuit render Xdx+ Ydy+Zdz Mr maxwell, OK FARADAV'S LINES OF FORCE. has shewn Finally, he systems of circuits similar and similarly corresponding conductors beino- same, the whatever be the absolute dimensions of the systems, which resultant forces are equal, the there be two if quantity of electrical current in the situated, that 49 proves that the forces are, cceferis paribus, inversely as the square of the distance. From these results very small is it follows that the mutual action of two closed currents whose areas are the same as that of two elementary magnetic bars magnetized perpendicularly to the plane of the currents. The direction of magnetization of the equivalent that a current travelling round magnet may be predicted by remembering the earth from east to west as the sun appears to do, would be equivalent to that magnetization which the earth actually possesses, and therefore in the when pointing reverse direction to that of a magnetic needle If a in number of closed unit currents in contact exist on a surface, then at which two currents are will produce no freely. but effect, points all contact there will be two equal and opposite currents which in round the boundary of all tlie surface occupied by the currents there will be a residual current not neutralized by any other; and therefore the result will be the same as that of a single unit current round the boundary of From this appears that it perpendicular to the attractions of external those due surface are the same as its each of the elementary currents in the a round to a current the same former case has the currents. all uniformly magnetized shell edge, its as an effect for element of the magnetic shell. If we examine the lines of magnetic force produced by a closed current, we shall find that they form closed curves passing round the current and embracing intensity of the magnetizing force all along the closed line of force tity of the The number current only. electric closed current depends on embrace and that the The it. unit of magnetic quantity is unit cells is in number measured simply by the number of unit case this are portions produced by unity of magnetizing cell is therefore inversely as the intensity of the total of unit lines * of magnetic force due to a the form as well as the quantity of the current, but the of unit cells f in each complete line of force currents which it, depends on the quan- force. magnetizing force, and its of space in which The length section is of a inversely as the quantity of magnetic induction at that point. The whole number of cells due to a given current therefore proportional to the strength is of the current multiplied by the number of lines of force which pass through any change of the form of the conductors the number of be a force tending to produce that change, so that there is transverse to the lines of magnetic force, so as to cause cells If by it. can be increased, there will always a force urging a conductor more lines of force to pass through the closed circuit of which the conductor forms a part. The number of cells due to two given currents respectively. sum of its own Now lines • Vol. X. is got by multiplying the number of magnetic action which pass through each by the quantity of the currents lines of inductive by (9) the number of lines which pass through the first current and those of the second current which would pass through the Erp. Res. (3122). Pakt I. See Art. (fi) of this paper. + Art. (13). is the first if the Mr maxwell, ON FARADAY'S LINES OF FORCE. 50 Hence the whole number of second current alone were in action. by any motion which causes more durino- the motion will be measured by the number of new On All the actions produced. cells may be deduced from each other conductors on of closed pass through either circuit, and therefore tend to produce such a motion, and the work done by this force force will the resultant lines of force to be increased cells will this principle. Electric Currents produced by Induction. Faraday has shewn * that when a conductor moves transversely to the lines of magnetic If the force, an electro-motive force arises in the conductor, tending to produce a current in it. conductor is closed, there is a continuous current, if open, tension conductor move transversely to the lines of magnetic induction, then, which pass through circuit will be in If a closed the result. is if tlie number of lines does not change during the motion, the electro-motive forces it equilibrium, and there will be no current. depend on the number of lines Hence in the the electro-motive forces which are cut by the conductor during the motion. If the motion be such that a greater number of lines pass through the circuit formed by the conductor after than before the motion, then the electro-motive force will be the number of and lines, When the additional lines. circuit the is That is tlie increase of generate a current the reverse of that which would have produced number of the lines of inductive increased, the induced current will tend to diminish the number that, in will measured by magnetic action through the number of the lines, and when diminished the induced current will tend to increase them. this is the true expression for the whatever way the number of be increased, the electro-motive effect of the conductor itself, lines is law of induced currents is shewn from the fact of magnetic induction passing through the circuit the same, whether the increase take place by the motion or of other conductors, or of magnets, or by the change of intensity of other currents, or by the magnetization or demagnetization of neighbouring magnetic bodies, or by the change of intensity of the current itself. In all these cases the electro-motive force depends on the change lastly in the number of lines of inductive magnetic action which pass through the circuit f • Erp. Res. (3077), &c. + The electro-magnetic forces, of the material conductor, which tend must be to produce motion carefully distinguished from the electro-motive forces, which tend to produce electric currents. Let an electric current be passed through a mass of metal The distribution of the currents within the metal In this case we have electro-magnetic forces acting on the material conductor, without any electro-motive forces tending to modify the current which it carries. Let us take as another example the case of a linear conductor, not forming a closed circuit, and let it be made to traverse the lines of magnetic force, either by its own motion, An electro-motive force of any form. or by changes in the magnetic field. Now let a by the laws of conduction. constant electric current be passed through another conductor near the first. If the two currents are in the same direction the two conductors will be attracted towards each other, and would come nearer if not held in their positions. But though will act in the direction of the conductor, and, as will be determined electric tension at it cannot pro- no circuit, it will produce the extremities. There will be no electro- duce a current, because there is the material conductors are attracted, the currents (which are any course within the metal) will not alter their magnetic attractionon the material conductor, for this attraction depends on the existence of the current within it, and this is prevented by the circuit not being closed. Here then we have the opposite case of an eleclro-motive original distributibn, or incline towards each other. force acting on the electricity in the conductor, but free to choose For, since no change lakes place in the system, there will be no electromotive forces to modify the original distribution of currents. on its material particles. no attraction Mr MAXVVELI., on FARADAY'S It 51 natural to suppose that a force of this kind, which depends on a change in is number of A OF FORCE. I,INES lines, is due change of state which to a may closed conductor in a magnetic field As long the magnetic action. measured by the number of these is be supposed to be state arising from effect takes place, but, when in a certain unchanged no as this state remains tlie lines. the state changes, electro-motive forces arise, depending as to their intensity and direction on change of this state. cannot do better here than quote a passage from the I first series of Faraday's Experimental Researches, Art. (60). "While to the wire be in a peculiar common in its is subject to either volta-electric or magneto-electric induction state, for formation of an electrical current in resists the it condition, such a current would be produced and when ; it ; appears it whereas, uninfluenced left it if has the power of originating a current, a power which the wire does not possess under ordinary This electrical condition of matter has not hitherto been recognised, but circumstances. many probably exerts a very important influence in by currents of advising with Finding that tonic state, all the this learned friends, ventured new immediately appear (71) will to his second still series rejected designate as it as the I have, after electro-tonic state." reference to the electro- not necessary may be some think that there to it it not most of the phenomena produced phenomena could be otherwise explained without Faraday in researches* he seems about For reasons which electricity. several if ; but in his recent physical truth in his conjecture state of bodies. Tlie conjecture of a philosopher so familiar with nature may sometimes be more pregnant with truth than the best establisiied experimental law discovered by empirical inquirers, and though not bound to admit it as a physical truth, our mathematical conceptions may be rendered In this outline of view, I can do no electrical phenomena can be and not as best comprehended and reduced my mind clearly explained without reference to operations. By fluids, I state hope adapted use symbols new idea by which to calculation, The freely, in my ideas, aim has lines or nor readily adapt idea of the electro-tonic state, however, has such a form that take for to and embodied form, as systems of its nature and properties mere symbols, and therefore and I believe that granted I the propose may be in the following ordinary mathematical a careful study of the laws of elastic solids and of the motions of viscous to discover a to general method of forming a mechanical conception of reasoning Part or in an mere symbols, which neither convey the same not yet presented itself to to as a mathematical methods by which state the themselves to the phenomena to be explained. investigation it Faraday's electrical theories, as they appear from a mathematical point of more than simply been to present the mathematical ideas to the mind surfaces, we may accept clearer. II. this electro-tonic •}-. On Faraday's "Electro-tonic State." When a conductor moves in tlie neighbourhood of a current of electricity, or of a magnet, when a current or magnet near the conductor is moved, or altered in intensity, then a force " (3172) (3269). tion of Electric, Magnetic and Galvanic Forces. Camb. and I t See Prof. W. Thomson On a Mechanical Representa- \ Dub. Math. Jour. Jan. 1847. 7—2 Mr maxwell, ON FARADAY'S LINES OF FORCE. 52 the conductor and produces electric tension, or a continuous current, according as the acts on circuit This current open or closed. is is produced only by changes of the electric or phenomena surrounding the conductor, and as long as these are constant there effect on conductor. the conductor the Still magnet, and when away from its is in influence, since the different states is when near magnetic no observed a current or removal or destruction of the current or magnet occasions a current, which would not have existed if magnet or current had not the been previously in action. Professor Faraday to connect with his discovery of the Considerations of this kind led all bodies are thrown by the itself by any known phenomena in this state is indicated by a current or tendency induction of electric currents, the conception of a state into which presence of magnets and currents. as long as it is This state does not manifest undisturbed, but any change towards a current. To he gave the name of the " Electro-tonic State," and although this state he afterwards succeeded explaining the phenomena which in suggested by means of it hypothetical conceptions, he has on several occasions hinted at the probability nomena might be discovered which would Faraday had been he has well established, and which he abandoned only for direct proof of the Perhaps investigation. phenomena unknown are not state, by the study of laws which led want of experimental data for the have not, I think, been made the subject of mathematical may be thought it that the quantitative determinations of the various rigorous to be sufficiently some phe- render the electro-tonic state an object of legitimate Tliese speculations, into which induction. tliat less made the of a mathematical theory basis Faraday, however, has not contented himself with simply stating the numerical results of his experiments and leaving the law to be discovered by calculation. he has at once stated it, in is by experiment, he has merely assisted ; and if the other laws capable of being it own the physicist in arranging his ideas, which confessedly a necessary step in scientific induction. In the following investigation, therefore, the laws established by as he has perceived a law terms as unambiguous as those of pure mathematics mathematician, receiving this as a physical truth, deduces from tested Where true, and it be shewn will that laws can be deduced from them. If to include one set of phenomena, may class, these by following out it ; will be assumed and more general should then appear that these laws, originally devised be generalized so as to extend to phenomena of a different mathematical connexions physical connexions Faraday his speculations other may suggest to physicists the means and thus mere speculation may be turned to of establishing account in experimental science. On Quantity and It is found that certain different sections of the different the material eifect effects of The circuit they are estimated. Intensity as Properties of Electric same an electric current are equal at whatever part of the quantities of water or of any other electrolyte circuit, are always found to be and form of the circuit may be of a conducting wire is Currents. at decomposed at two equal or equivalent, however the two sections. The magnetic also found to be independent of the form or material of the wire Mr IMAXWELL, ON FARADAY'S LINES OF FORCE. same in the circuit. There therefore an electrical effect which is equal at every section of the is If we conceive of the conductor as the channel aloncj which a fluid circuit. move, then the quantity of 53 is constrained to transmitted by each section will be the same, and we fluid may define the quantity of an electric current to be the quantity of electricity which passes across a complete section of the current in We unit of time. of electricity by the quantity of water which In order to express mathematically the may for the present would decompose it electrical currents in in unit measure quantity of time. any conductor, we must have a definition, not only of the entire flow across a complete section, but also of the flow at a given point in a given direction. Def. The quantity of a current at a given point and in a given direction when uniform, by is measured, the quantity of electricity which flows across unit of area taken at that point perpendicular to the given direction, and when variable by the quantity which across this area, supposing the flow uniformly the same would flow as at the given point. In the following investigation, the quantity of electric current at the point {nyz) estimated in the directions of the axes The quantity of .r, electricity y, x respectively which flows = dS will be in imit of {la., + mb.2 where Imn are the direction-cosines of the normal to + denoted by External These may be electro-motive magnets, or from changes forces 5^ c-i- time through the elementary area dS ncS), dS. This flow of electricity at any point of a conductor which act at that point. a.^ due is to the electro-motive forces either external or internal. either arise in their intensity, or from the relative motion from other causes acting of currents and at a distance. Internal electro-motive forces arise principally from difference of electric tension at points of the conductor in the immediate neighbourhood of the point in question. The other causes are variations of cnemical composition or of temperature in contiguous parts of the conductor. all Let P2 represent the electric tension at any point, and J^2 J'a -^2 the sums of the parts of the electro-motive forces arising from other causes resolved parallel to the co-ordinate axes, then if a.j fi, y-i be the effective electro-motive forces 02 = ^2 - dp„ -r-" do! (A) 7, = Z2 Now If the resistance of the Qj if dz the quantity of the current depends on the electro-motive force and on the resistance of the medium. but - = AJjOo, /3., medium be uniform = h.hi, 72 = in all directions hfii, These quantities tlie a> /Bo 72 directions of wyz. may be to k.^, (B) the resistance be difl^erent in different directions, the law will be action in and equal more complicated. considered as representing the intensity of the electric Mr maxwell, OX FARADAY'S LINES OF FORCE. 54 The measured along an element intensity e where Imn = + la a curve dcr of m[i + ny, are the direction-cosines of the tanjjent. The integral feda taken with respect to a given portion of a curve intensity along that line. If the curve a closed one, is line, represents the total represents the total intensity of the it electro-motive force in the closed curve. Substituting the values of 0/87 from equations (A) fedff If, therefore (J^dx will vanish, and + Ydy + Zdz) in all closed = f(Xdx + Ydy + -p + C. a complete differential, the value of is jetZcr for a closed curve curves /erfo- the integration Zdz) = f{Xdx + Tdy + Zdz), being effected along the curve, so that in a closed of the effective electro-motive force is equal to the total intensity curve the total intensity of the impressed electro- motive force. The total quayitity of conduction through any surface is expressed by fedS, where e = + mh + la nc, Imn being the direction-cosines of the normal, fedS = ffadydz + ffbdzdx + .*. the integrations being effected over the given surface. we may If find ffcd.vdy, When the surface is a closed one, then by integration by parts we make da db dc dx dy dz fedS = 4 IT fjfpdxdydz, where the integration on the right side of the equation within the surface. is effected over every part In a large class of phenomena, including all of space cases of uniform currents, the quantity p disappears. Magnetic Quantity and Intensity. From in the his study of the lines of magnetic force, tubular surface* formed by a system of such across any section of the tube in passing from one is constant, and that the substance to another, capacity in the two substances, which electric Faraday has been led is is to lines, to the conclusion that the quantity of magnetic induction alteration of the character of these lines be explained by a difference of inductive analogous to conductive currents. Exp. Res. 3271, de6nition of " Sphondyloid.' power in the theory of Mr maxwell, ON FARADAY'S LINES OF FORCE. In following investigation we shall have occasion to treat of magnetic quantity and tlie intensity in connexion with electric. by the and suffix 1, In such cases the magnetic symbols will be distinguished by the the electric suffix 2. The equations connecting a, p, and p, are the same in form as those which we have just given, a, b, c k, denotes the resistance to magnetic induction with respect to quantity ; and may be a, j8, "y, are the effective nected different in different directions; a, witli h, by equations (B) c, afterwards explained by equations a, b, c all ; p, is (C). As all b, c, k, a, y3, y, are the symbols of magnetic induction, magnetizing forces, con- the magnetic tension or potential which will be p denotes the density of real magnetic matter and ; the details of magnetic calculations after the exposition of the connexion of say that 55 magnetism with will connected with be more intelligible be electricity, it will is sufficient here to the definitions of total quantity, with respect to a surface, and total intensity with respect to a curve, apply to the case of magnetism as well as to that of electricity. Electro-magnetism. Ampere has proved following laws the of the and attractions repulsions of electric currents Equal and opposite currents generate equal and opposite I. A II. crooked current is forces. equivalent to a straight one, provided the two currents nearly coincide throughout their whole length. Equal currents traversing III. and similarly situated closed curves act with similar equal forces, whatever be the linear dimensions of the circuits. IV. A closed to be observed, that the currents with which Ampere worked were constant and exerts current turn a circular conductor about no force tending to its centre. It is therefore currents, All re-entering. and assumption results are therefore deduced from experiments on this action is exerted in the direction of the line joining those elements. no doubt warranted by the universal consent of men of science is attractive forces considered as due to the mutual action of particles; in treating of phenomena, the currents alone, but also in the surrounding medium. The The as first and second laws shew that currents are to be combined like velocities or tliird law is the expression of a property of all attractions which may be shews that the electro-magnetic forces may always be reduced forces. conceived of depending on the inverse square of the distance from a fixed system of points fourth This but at present we are proceeding on a different principle, and searching for the explanation of the not in closed expressions for the mutual action of the elements of a current involve the his assumption that his ; and the to the attractions and repulsions of imaginary matter properly distributed. In fact, identical given with the action of a very small electric circuit that of a small on a point magnetic element on a point outside in its it. neighbourhood If is we divide any portion of a surface into elementary areas, and cause equal currents to flow in the same direction round all these little areas, the effect on a point not in the surface will be the Mr maxwell, ON FARADAY'S LINES OF FORCE. 56 same as that of a surface. shell But by the coinciding with the surface, and uniformly magnetized normal to first law the currents forming the all little circuits another, and leave a single current running round the bounding effect of a uniformly magnetized shell edge of the the sun, shell. line. its destroy one will So that the magnetic equivalent to that of an electric current round the is If the direction of the current coincide witli that of the apparent motion of the imaginary shell will be magnetization of then the direction of the same as that of the real magnetization of the earth*. The total intensity of the closed current current. As is magnetizing force may constant, and this intensity is in a closed therefore be made a measure of the quantity of the independent of the form of the closed curve and depends only on the quantity of the current which passes through tlie curve passing through and embracing it, we may consider the elementary case of current which flows through the elementary area dydx. Let the axis of x point towards the west, be the position of a point round the four quantity we Let wys sides of the element is Total intensity The towards the south, and y upwards. the middle of the area dydz, then the total intensity measured in + therefore if sr = 7>. dy (j^- - of electricity conducted 2 J -jAdy ds. through the elementary area dydx is adydz, and define the measure of an electric current to be the total intensity of magnetizing force in a closed curve embracing These equations enable us we know the values of a, /3, it, we shall have dz dy dy. dai d.v dz da, rfj3. dy d.v deduce the distribution of the currents of to y, the magnetic intensities. a function of xyz with respect to • See Experimental Researches (3265) embracing curves. .r, If a, j8, y electricity be exact differentials of y and z respectively, then the values of as 62 for the relations between the electrical whenever and magnetic circuit, c> disappear considered as mutually Mr maxwell, OX FARADAY'S and we know that the magnetism which we are investigating. OF FORCE. 57 not produced by electric currents in that part of the is due either It is LLNTES to the presence of field permanent magnetism within the field, or to magnetizing forces due to external causes. We which may is observe that the above equations give by differentiation da^ db, dc.> o.r ai/ d~ Our investigations are therefore for we know little of the magnetic effects of any the equation of continuity for closed currents. the present limited to closed currents; and in fact currents which are not closed. Before entering on the advantageous to state calculation certain of general these electric theorems, and truth tlie magnetic states may be of which may be it established analytically. Theorem The equation d'V V and p (where are functions of distance,) can be satisfied by The value of d-V <PV xyx never infinite, Theorem F which will satisfy the — Qu x' \' + y — is found by integrating the expression y' \' + s — z' r)' to include every point of space where p may be found proofs of these theorems in electricity, and See Arts. to Electricity, See also Gauss, on Attractions, translated in Taylor's Scientific Memoirs. Theorem U and V is finite. any work on attractions or Green's Essay on the Application of Mathematics 19 of this Paper. Let See Art. (17) above. pdrdydss where the limits of xyz are such as 18, for all points at an infinite II. above conditions rrr ''''•' in particular in and vanishing one, and only one, value of V. ^ The I. be two functions of .ryz, III. then dUdV dUdV\, JJj^ [d^'-'df^ d^r^^''' =-JjJ[-d^-d:v^^d^^ Tzi^r^y'^'' rr Trr I d~V cPV rf r\ , , , rrr/dUdV where the integrations are supposed to extend over differing from (Green, p. This theorem shews that equal and opposite. all the space in which Part if there And by making I. , U and V have values 10.) be two attracting systems the actions between them are ^7"= F we find that the potential of a proportional to the integral of the square of the resultant attraction Vol. X. , system on through all itself is space 8 ; a Mr maxwell, OX FARADAY'S LINES OF FORCE. 58 result deducible from Art. (Arts. 12, the velocity !.•?), (.'50), volume of each since the number of and therefore the cell is in cells inversely as the square of a given space is directly as the square of the velocity. Theorem Let and a, /3, 7, p be quantities k be wiven let then the equation in for through a certain space and vanishing in the space beyond, parts of space as a continuous or discontinuous function of .vyz, p d and only has one, all finite IV. d l/_ dp\ 1/ one solution, in dp\ which is p d dp\ If always finite and vanishes at an infinite distance. The Dublin Math. If W. Thomson, may proof of this theorem, by Prof. be found in the Cambridge and Journal, Jan. ISiS. a/37 be ance, then the above equation and the theorem p the electro-motive forces, is the electric tension, and k the coefficient of resist- identical with the equation of continuity da^ db, dc-i ax dy dz ' shews that when the electro-motive forces and the rate of production of electricity at every part of space are given, the value of the electric tension is determinate. Since the mathematical laws of magnetism are identical with those of electricity, as far as we now consider them, we may regard 0/^7 as magnetizing forces, p as mafftietic tension, and p as real magnetic density, k being the coefficient of resistance to magnetic induction. The where proof of this theorem rests on the determination of the V is and p has minimum got from the equation to d'V d'V d-7 dor dy^ dz- be determined. The meaning of this integral in electrical sence of the media in which k "quantity" resolved in language may be thus brought out. If the pre- has various values did not affect the distribution of forces, then the x would be simply dV — dV and the intensity -] and a - and intensity are -fa — , and the parts due alone are therefore 1 , dp\ <iP\ -(a -3dx) k dp dV dV__^_ --— anda--T^ dx k—- But the actual quan- cue ufi*' tity value of ax - dV k-—. , ax to the distribution of media Mn MAXWELL, ON FARADAY'S LINES OF FORCE. Now the product of these represents the work done on account of the distribution of sources being determined, and talcing in the media, Q we get the expression point which is due done by for the total worlc to the distribution of tliat 59 tliis distribution of terms in y and z part of the whole effect at any conducting media, and not directly to the presence of the sources. This quantity which satisfies Q is minimum by one and rendered a Theorem If it is a, b, c only one value of p, namely, that the original equation. be three functions of .r, V. y, s satisfying the da db dc ax dy dz a, /3, 7 always possible to find three functions equation wliicli shall satisfy the equations d^_dy_ ^ Let of y, A dz dy dy da dx dz da dd dy dx =fcdy, where the integration treating x and « B = fadz, C = Jhdx, Let as constants. performed upon c considered as a function to be is = jbdz, A' i?' = fcdx, C integrated in the same way. Then a^A-A'^f, ax dy dz will satisfy the given equations d/3 dy dz dy ; = for rda I — dz , rdc — - r , dx - J dz J dy •> db — dx dy rda — dy, + J dy and = rda / J dji dy dz dy — = — dx + J dx — dx + r(^^ , dy — dx + Jrda dy dy rda / J dx = , , ~y~ -^ r^c — rda + , dx; J dz , -—dz J dz a. 8—2 = fady, Mr maxwell, ON FARADAY'S LINES OF FORCE. 60 In the same way it may be sliewn that the values of a, The function \|/ may be considered at present The method here given is taken from /3, 7 satisfy the other given equations. as perfectly indeterminate. W. Prof. Thomson's memoir on Magnetism (Phil. Trans. 1851, p. 283). As we cannot perform the required integrations when r, y, z, the following more method, which perfectly general C discontinuous functions of be determined from the equations by the methods of Theorems I. drA d'A d'A dx' ay dz- d'B d-B d'B dx' dy'' d%^ d'C d^C d'C dx^ dy^ dz^ and II., so that A, B, C are is infinite. Also let _dB^_dC dz then a, b, c are though more complicated, may indicate clearly the truth of the proposition. Let A, B, or 2 is dy dj, dx never infinite, and vanish when .r, y, Mr maxwell, ON FARADAY'S LINES OF FORCE. The function \|/ is to be determined from the condition da dfi dx'^ dy if "^ dy _ /d^ ds ~ [dx' Theorem a, b, c be any three functions of >f, y, z, d^\ d'= ^ "*" the left-hand side of this equation be always zero, Let x^^ "*" d^'J it is possible to find three functions a, dB dy ans dy dx a=-dd dy dV ,_dy da dV dx dx dy c Let and let V be found from the equation then satisfy the condition - also. VI. da d% ^ must be zero a fourth V, so that and 61 ^ r + dy dx ' j3, 7 and Mb maxwell, ON FARADAY'S LINES OF FORCE. 62 where ai6,c„ aifti 7, are any functions whatsoever, Q in = + - ffJl^Trppi + (ao«2 capable of transformation into + fioh •y^c.i)}dMydx, which the quantities are found from the equations au^o7o^' are determined from da, db, dc, dx dy dz da, rf/3, d.v dy by the a, &, Cj ^7i ^ found from 62 C2 are a, 71 by /3i last is J- &c., dy found from the equation For, we put if a, in the d^p d-p rf'p da?* %• dz' and treat h, bering that dx dy we have by integration by parts through infinity, remem- the functions vanish at the limits, all Q=- similarly, then Cj ' ' form dz and dw equations tlie dz p „ / theorem, so that dy a,= and , ax; ds a-j is KS ^ I' ^ '^) Q= + or "" (f - '^) ^ ^' {^ - '^) ^ - (^ - ^) I-*-. /// |(47r Fp') - (a^a. + jSo&2 + 7oC;i) } dcfdyd;?, and by Theorem III. ffjVp'dxdydz = JJfppdxdydz, - + ^A = so that finally Q= If Oi a., functions (ao«2 7t>f'0 i dxdydx. represent the components of magnetic quantity, and a, 6, Ci intensity, then tension, fffl'iTrpp h, p c.-, will represent will the real magnetic density, be the components of quantity of deduced from a,h,c,, which Faraday's Electro-tonic will electric /3i 71 those of magnetic and p the magnetic potential or currents, and ao/3o 7o ^^^^ be three be found to be the mathematical expression for state. Let us now consider the bearing of these analytical theorems on the theory of magnetism. Whenever we suffix ( , ). deal with quantities relating to magnetism, Thus a, 61 c, are the components resolved in we shall distinguish them by the the directions of x, y, z of the Mr maxwell, ON FARADAY'S LINES OF FORCE. 63 quantity of magnetic induction acting tlirough a given point, and aSiJi are the resolved intenof magnetization at the same point, or, what sities is tlie same tiling, tlie components of the which would be exerted on a unit south pole of a magnet placed at that point without force disturbing the distribution of magnetism. The found from the magnetic intensities by the equations electric currents are Oo — = When &c. dy dz there are no electric currents, then + y^dz = a^dx + ^^dy a perfect differential of a function of On y, z. .?, f/;;„ the principle of analogy we may call p^ the magnetic tension. The forces which act on a mass m of south magnetism at any point are dpi — m-— , m — is , and — m dp^ , dz and therefore the whole work done during any displacement of a in the direction of the axes, magnetic system — dp^ dy d.v equal to the decrement of the integral Q= UlpiPxdxdydz throughout the system. Let us now of Q and will call Q the total potential of the system on itself. measure the work lost or will therefore enable us to By Theorem III. Q may which a, /3i or decrease be put under the form — ffj(C\(^\ + ^ifti + Ci7i)rf,rrfyd«, 7, are the differential coefficients of pi with respect to we now assume If increase determine the forces acting on that part of the system. Q= + in The gained by any displacement of any part of the system, that this expression for Q is ,r, y, « respectively. true whatever be the values of oi /3i 71, we pass from the consideration of the magnetism of permanent magnets to that of the magnetic effects of electric currents, and ^^Ifj So that 1^'^' ~ ^ ["""- in the case of electric currents, the by the functions a^fi^yo respectively, system gives us the part of We we have then by Theorem VII. have now obtained Q due "*" ^°^^ "'' ^"^''j r'^'^^y'^^ components of the currents have and the summations of all in the functions a^figyf, the such products throughout the means of avoiding the consideration of Instead of this method we have the natural one of considering the current with reference same space with the current be multiplied to those currents. the quantity of magnetic induction which passes through the circuit. in the to itself. To or components of the Electro-tonic intensity. these I give the name of artificial to quantities existing Electro-tonic functions, Mr maxwell, ON FARADAY'S LINES OF FORCE. 64 Let us now consider the conditions of the conduction of the electric currents within the medium diirin<f changes in the electro-tonic state. The method which we shall adopt is an by Helmholtz application of that given mass currents whose quantity Then the amount of work due measured by a^ is + and their intensity by + b.fi.^ a.> fi-z y... is cry.i)dxdydz form of resistance overcome, and dt d 47r dt in b>c., which would generate in the con- to this cause in the time dt dtjjj{a.fii in the Conservation of Force*. tlie source of electric currents Let there be some external ductintr memoir on in his the form of fjfia^ao + ^So + cr/^dxdydz work done mechanically by the electro-magnetic action of these currents. If there be no external cause producing currents, then the quantity representing the whole work done by the external cause must vanish, and we have dt fjf{aMi+ + 6./3. c.,y.2)d.vdi/dz+ —— where the integrals are taken through any arbitrary space. a.,a.2 + b,l3-, + C,y., = and We ^^^ ^°^ ^^ a.b^c^. variations of Oo/SoYo' We must c,yo)d,vdydz, tlierefore have ^'sVo) at must be remembered it Mo + — - («.ao + ^So + 47r for every point of space; + ///(«2ao Q that the variation of must therefore is supposed due to treat a.^b.^c, as constants, and the equation becomes a, a.>+ •V' ^1 47r forces must due = ; A /3., ~] + ^TT +c., 7. + 'V- dt) may be independent In order that this equation efficients +6, dt) /^ =0dt 4Tr of the values of a., ) h, and therefore we have the following expressions to the action of magnets and currents c-,, for each of these cothe electro-motive at a distance in terms of the electro-tonic functions, ""* It ^" ' 47r dt '^' ' 47r dt appears from experiment that the expression 47r —° refers to dt the change of electro-tonic state of a given particle of the conductor, whether due to change in the electro-tonic functions themselves or to the motion of the particle. If Oj, be expressed as a function of article, ,r, j/, z, and t, and if ,?', y, -z be the co-ordinates of a moving then the electro-motive force measured in the direction of Idar, dx da^ dy dua dz da^\ 47r \da; dt dy dt dz dt dt New Scientific 1 "'~ x * Translated in Taylor's Memoirs, Part is J II. Mr maxwell, ON FARADAY'S LINES OF FORCE. The 65 The expressions for the electro-motive forces in y and % are similar. distribution of currents due to these forces depends on the form and arrangement of the conducting media and on the resultant The paper tical discussion of these functions would involve us in mathematical formulae, of which this already too is It is only on full. By their present form. help of those who which a more patient consideration of their relations, and with the are engaged in physical inquiries both in obviously connected with in account of their physical importance as the mathema- expression of one of Faraday's conjectures that I have been induced to exhibit them at in all any point. electric tension at all its relations it, I hope may be subject and in others not this a form to exhibit the theory of the electro-tonic state in distinctly conceived without reference to analytical calcula- tions. Summary of We in may the Theory of the Electro-tonic State. conceive of the electro-tonic state at any point of space as a quantity determinate magnitude and direction, and we may represent the electro-tonic condition of a portion of space by any mechanical system which has at every point some quantity, which may be a velocity, a displacement, or a force, whose direction and magnitude correspond to those of the supposed electro-tonic of state. artificial notation. electro-tonic state, This representation involves no physical theory, In analytical investigations and call them we make use of the electro-tonic functions. electro-tonic intensity at every point of a closed curve, call and tliree it is only a kind components of the We take the resolved part of the find by integration what we may the entire electro-tonic intensity round the curve. Prop. divided I. itifo If on any surface a closed curve he drawn, and if the surface within it be small areas, then the entire intensity round the closed ctirve is equal to the sum of the intensities round each of the small areas, all estimated in the same direction. For, in going round the small areas, every boundary line between two of them along twice Every effect in opposite directions, effect is Law that I. and the intensity gained of passing along the interior divisions due The to the exterior closed is in the one case is lost in Prop. I. it entire electro-tonic intensify appears that what is therefore neutralized, and the whole round the boundary of an element of surface true of elementary surfaces finite magnitude, and therefore any two surfaces which are will passed curve. measures the quantity of magnetic induction which passes through that surface, words, the number of lines of magnetic force which pass through that surface. By is the other. is or, in other true also of surfaces of bounded by the same closed curve have the same quantity of magnetic induction through them. Law II. The magnetic intensity at any point is connected with the quantity of magnetic induction by a set of linear equations, called the equations of conduction*. • See Art. (28 Vol. X. Part I. mk maxwell, on faradays lines of force. 66 Law The III. entire magnetic inteyisifii round the boundary of any surface measures the quantity of electric current which passes through that surface. Law The quantity and IV. intensity of electric currents are connected by a system of equations of conduction. By the values of the electro-tonic functions. I be deduced from have not discussed the values of the units, as that We be better done with reference to actual experiments. will may these four laws the magnetic and electric quantity and intensity come next to the attraction of conductors of currents, and to the induction of currents within conductors. Law The V. total electro-magnetic potential of a closed current is measured by the product of the quaiitity of the current multiplied by the entire electro-tonic intensity estimated in the same direction circuit. displacement of the conductors which would cause an increase in the potential will be Any by assisted round the force a measured by the rate of increase of the potential, so that the mechanical work done during the displacement will be measured by the increase of potential. Although current might in certain cases a displacement in direction or alteration increase the potential, such an alteration would not itself of intensity of the produce work, and due there will be no tendency towards this displacement, for alterations in the current are to electro-motive force, not to electro-magnetic attractions, which can only act on the conductor. Law The VI. electro-motive force on any element of a conductor is measured by the instantaneous rate of change of the electro-tonic intensity on that element, whether in magnitude or dirertio7i. The electro-motive force in a closed conductor is measured by the rate of change of the entire electro-tonic intensity round the circuit referred to unit of time. It is independent of the nature of the conductor, though the current produced varies inversely as the resistance; and it is the same in whatever way the change of electro-tonic intensity has been produced, whether by motion of the conductor or by alterations In tiiese six laws in the external circumstances. have endeavoured to express the idea which I believe to be the mathe- I matical foundation of the modes of thought indicated in the E.rperimental Researches. not think that it contains even the shadow of a true physical theory; in fact, temporary instrument of research There exists so mathematical, is that it however a professedly physical theory of electro-dynamics, wiiich and so entirely different Weber's Electro-dynamic Measurements, from anything in this paper, that I and may be found Leibnitz Society, and of the Royal Society of Sciences of Saxony (1) That two as when particles of electricity at rest, do does not, even in appearance, account for anything. axioms, at the risk of repeating what ought to be well-known. same force its I chief merit as a when but that the force is in It in *. is the The is so elegant, must state its contained in M. W. Transactions of the assumptions are, motion do not repel each other with the altered by a quantity depending on the relative motion of the two particles, so that the expression for the repulsion at distance r • When this was written. I was not aware that part of "SI. Weber's .'Memoir is translated in Taylor's Scientific Memoirs, \'o\. V. Art. XIV. The value of his researches, both experimen- 1 I I tal anil theoretical, every electrician. is renders the study of his theory necessary 10 Mn MAXWELL, ON FARADAY'S LINES OF FORCE. dr ee' / That when (2) electricity is moving of the conductor relatively to the matter is 2 d-r\ , in a conductor, the velocity of the positive fluid equal and opposite to that of the negative Tiie total action of one conducting element on another (3) mutual actions of the masses of electricity of is the fluid. resultant of the both kinds which are in each. Tiie electro-motive force at any point (4) 67 is the difference of the forces acting on the positive and negative fluids. of From these axioms are deducible Ampere's laws of the attraction Neumann and others, for the induction of currents. Here then is satisfying the required conditions better perhaps than of conductors, and those a really physical theory, any yet invented, and put forth by a whose experimental researches form an ample foundation for his mathematical philosopher What investigations. is the use then of imagining an electro-tonic state of which distinctly physical conception, instead of a formula of attraction which stand would answer, that I .'' it is a good thing to admit that there ai-e two ways of looking at to it. we can have two ways of looking Besides, I we have no readily under- at a subject, and do not think that we have any right at present to understand the action of electricity, and I hold that the chief merit of a temporary theory theory when is, that guide experiment, without impeding the progress of the true There are appears. it shall it also objections to making any ultimate on the velocity of the bodies between wiiich they act. depend forces in nature If the forces in nature are to be reduced to forces acting between particles, the principle of the Conservation of Force re- quires that these forces should be in the line joining the particles and functions of the distance The only. experiments of recently repeated M. Weber's With M. Weber on by Professor Tyndall, the reverse polarity of diamagnetics, which have been establish a fact which is equally a consequence of theory of electricity and of the theory of lines of force. respect to the history of the present theory, I may state that the recognition of certain mathematical functions as expressing the " electro-tonic state" of Faraday, and the use of them in determining electro-dynamic potentials and electro-motive forces, aware, original arose in ; my mind from the perusal of Prof. of Electric, tation Part I. IS.'i], I I "On a Mechanical Represen- may As an instance of the help which may be derived from other state that after I Diff'raction," Section paper to discuss These are intended merely till papers had investigated the Theorems of 1, this paper Cambridge Transactions, Vol. IX. Part the theory of these functions, considered with reference to electricity, rest of this ; W. Thomson's Mathematical Theory of Magnetism," Philosophical Transac- mathematical ideas to be employed given am Stokes pointed out to me the use which he had made of similar expressions in his "Dynamical Theory of Whether his " Art. 7S, &c. physical investigations, Professor as far as I Magnetic and Galvanic Forces," Cambridge and Dublin Mathematical Journal, January, 1847, and tions, is, but the distinct conception of the possibility of the mathematical expressions in physical research, remains to be seen. may I lead to propose in i. new the a ievi electrical and magnetic problems with reference to spheres. as concrete examples of the methods of which the theory has been I reserve the detailed investigation of cases chosen with special reference to have the means of testing their experiment results. 9—2 Mr maxwell, ON FARADAY'S LINES OF FORCE. 68 ExAMI Theory of Electrical Images. I. The method LES. W. Thomson*, of Electrical Images, due to Prof. spherical conductors simple when we see has been its reduced to by which the theory of great geometrical simplicity, becomes even connexion with the methods of We this paper. more have seen that the pressure at any point in a uniform medium, due to a spherical shell (radius = a) giving out o fluid at the rate of A-n-Pa- units in unit of time, is kP — outside the shell, and kPa inside it, r where r is the distance of the point from the centre of the shell. If there be two shells, one giving out fluid at a rate 47rPa", and the other absorbing at the rate 4nrP'a'-, then the expression for the pressure will be, outside the shells, « , a!' inP — p = iTrP ; , r r where r and r are the distances from the centres of the two to zero we have, Now as the surface of no pressure, that for / P'a'-' r Pa' the surface, for which the distances to of which the centre O is in Equating shells. this expression which two fixed points have a given the line joining the centres of the shells CC ratio, is a sphere produced, so that P'd-'f CC C'O = and its -=r= Pa'^-Pa'"]^ radius If at the centre of this sphere this source must be added to we place another source of the that due to the other depends only on the distance from the centre, where the pressure due to the two other sources We it is will two ; fluid, and then the pressure due to since this additional pressure be constant at the surface of the sphere, zero. have now the means of arranging a system of sources within a given sphere, so that when combined with a given system of sources outside the sphere, they shall produce a given constant pressure at the surface of the sphere. See a series of papers " ning March, 1848. On the Mathematical Theory of Klectricity," in the Cambridge and Dublin Math. Jour., begin- Mr maxwell, ON FARADAY'S LINES OF FORCE. 69 Let a be the radius of the sphere, and p the given pressure, and let the given sources be b^ h.^ &c. from the centre, and let their rates of production be iirPi, ^ttP^ &c. at distances Then centre) — distances if at , - &c. (measured we place negative sources whose the pressure at the =a surface r the same direction as 6,62 in from the Stc. rates are 47rPi will - , - &c. r iirP. Now be reduced to zero. placing a source 47r ~- at K the centre, the pressure at the surface will be uniform and equal to p. The whole amount of fluid emitted by the rates of production of the sources within To apply this result to the case of a The it. conducting sphere, be small electrified bodies, containing e, pose that the whole cliarge of the conducting sphere is iirP-i, ^tt/'o to Then points. all that = a may be found by adding surface r let us suppose the external sources of positive electricity. e.2 = £ Let us also sup- previous to the action of the external required for the complete solution of the problem is the result is is, that the surface of the sphere shall be a surface of equal potential, and that the total charge of the surface shall he E. If by any distribution of imaginary sources within the spherical surface the value of the corresponding potential outside the sphere potential inside the sphere We must therefore point at a" — and At find really be constant the distance b from the , , must , . at that point the centre and equal images of the external centre we must „ . the true and only one. electrified points, that on the same radius we must place a quantity =—e — of imaginary , we must put effect this, The to that at the surface. find a point « . is we can . is, for every at a distance ... electricity. a quantity E' such that E'=E+ei- +e.,~+kc.; Oj then if and R be the distance from the centre, r^r.^ " bo &c. the distances from the electrified points, r'l/a the distances from their images at any point outside the sphere, the potential at that point will be E' (1 a \\ n a l\ R Vi o^rj V7-2 62 '•2/ E e,f a = o + r p+ R b^\R b, a\ r, --^ rj e.,1 a L +fb.,\R n\ --+Sjc rj 6» + r^ Mr maxwell, ON FARADAY'S LINES OF FORCE. 70 This is At the surface we have the value of the potential outside the sphere. R=a and 63 — = -n = -a &c. so that at the surface «=— and this must also be the value of p for +— + any point within the sphere. For the application of the principle of Thomson's papers in V-+&C. images the reader electrical is referred to Prof. The the Cnmhridge and Dublin Mathematical Journal. which we shall consider is -^ =/, and that in which 6, is infinitely distant only case along axis of x, and £=0. The value p outside the sphere becomes then p=Lt and inside ^ = 0. force The ill ' the effect of a paramagnetic or diamagnetic sphere in a uniform field of magnetic On IL ,3 *. expression for the potential of a small the direction of the axis of x magnet placed at the origin of co-ordinates is (m\ lm\ X la ix \r) r^ d ' The effect of the sphere in disturbing the lines of force hypothesis to be similar be determined. (We to that of a small magnet the at may be supposed origin, as whose strength a first is to shall find this to be accurately true.) Let the value of the potential undisturbed by the presence of the sphere be p= Let the sphere produce an additional Ix. potential, whicli for external points is —x, p = A ,.3 and let the potential within the sphere be Pi Let k' be the coefficient of resistance = Bx. outside, and k inside the sphere, then the conditions to be fulfilled are, that the interior and exterior potential should coincide at the • See Prof. Thomson, on the Theory of .Magnetic Induction, Pkit. A/ap. March, 1851. Theinductivecapaciti/of thesphere^ according to that paper, the ratio of the quanlity of magnetic induction (not the intensity) within the sphere to that without. It is therefore is equal to ~. B -t- = ^7—r.accordmg to our notation. Mr maxwell, ON FARADAY'S LINES OF FORCE. 71 and that the induction through the surface should be the same whether deduced surface, Putting x = r cos from the external or the internal potential. d, we have for the external potential p= and + Ir { A— \ cos d, for the internal Pi- Br and these must be identical when r = a, cos 9, or I + A = B. The induction through the surface in the external and that through the interior surface medium is is and .-. -,{I-2A) = - B. k k These equations give The moment effect outside the sphere is equal to that of a Suppose in north. this uniform field to — is I and — orI. 2k + k be that due to terrestrial magnetism, then, paramagnetic bodies, the marked end of the equivalent magnet If k is greater than k' as in diamagnetic bodies, the magnet would be turned Magnetic field of variable first k is less than be turned to the unmarked end of the equivalent Intensity. suppose the intensity in the undisturbed magnetic direction from one point to another, and that then, if as a will if to the north. III. Now magnet whose length ml, provided ml = k as little its components field to vary in magnitude and in xyx: are represented by a^fi^yi, approximation we regard the intensity within the sphere as sensibly equal that at the centre, the change of potential outside the sphere arising from to the presence of Mr maxwell, ON FARADAY'S LINES OF FORCE. 72 the sphere, disturbing tlie lines of force, will be the same as that due to three small magnets at the centre, with their axes parallel to x, y, and z, and their k-k' ,a^a. k k-k' 2k + The 2k + distribution to k-k' 2k + k'«V -.a^fi. k' actual distribution of potential within and without the sphere result of a may be conceived as the of imaginary magnetic matter on the surface of the sphere the external effect of this superficial magnetism magnets moments equal is ; but since exactly the same as that of the three small at the centre, the mechanical effect of external attractions will be the same as if the three magnets really existed. Now let three small magnets whose lengths are L l^, and strengths tm, m., m^ exist at the point xyz with their axes parallel to the axes oi xyz; then, resolving the forces on the three ?, magnets in the direction of X, we have da lA Mr maxwell, ON FARADAY'S LINES OF FORCE. Two IV. Spheres in uniform Let two spheres of radius a be connected together and them be suspended tance b, itself would have been let in equilibrium at any part of the produce forces tending to make the balls p — Ir then, althouo-h each sphere cos 9, k-k' a^ , is cos Q. therefore equal to /, , I r + k— _. 2k + k' a^\ ,, - cos ^ = p. k' r^ — k' a^\ k-k' k a^ ^ ( = /^( 1 - 2 — -, cos dr \ 2k + k ry dp 1 dp r d9 „ dp •'''''= dr ^( = -'{ by the disturbance of the field will to the sphere is , potential field, as origin, then the undisturbed potential P = I 2k v-'—, + k r^ The whole field, set in a particular direction. Let the centre of one of the spheres be taken and the potential due Jield. so that their centres are kept at a dis- uniform magnetic in a 73 k - 9, k' a^\ dp 2k + k r^J dtf) = 0, is Mb maxwell, ON FARADAY'S LINES OF FORCE. 74 The the middle of the line joining expression for the potential, / we this y/c^+r'+2 cos 6cr) 'iM- I r- r^ \ c \ c' c- J 6 is increased to c* turn a pair of spheres (radius a, distance 26) in the direction in which is k-Jc APa'V +k 2k This force, i -18—-r- sin 20, I-T^= dd and the moment : P, find as the value of .-. poles being the 1_ 1 \\/cr+r'^-2 cos 6cr From the which tends to turn the ^ . c^ line of centres equatoreally for diamagnetic and axially for magnetic spheres, varies directly as the square of the strength of the magnet, the cube of the radius of the spheres and the square of the distance of their centres, and inversely as the sixth power of the distance of the poles of the magnet, considered these poles are near each other this action of the poles will be as points. much As long as stronger than the mutual action of the spheres, so that as a general rule we may say. that elongated bodies set axially or equatoreally between the or diamagnetic. If, they had been placed in a uniform by a magnet according poles of a as they are magnetic between two poles very near to each other, instead of being placed such as that of terrestrial magnetism or that produced field Ex. VIII.), an elongated body would spherical electro-magnet (see set axially whether magnetic or diamagnetic. In these cases the all phenomena depend on t — magnetically or diamagnetically according as diamagnetic than the medium in which VI. On the it is for the axes of a\ y, z. and more or less is different let k' Let k^, k-,, throughout the sphere, and be I, field into which the sphere field them is Let / introduced, and m, n. Let us now take the case of a homogeneous sphere whose tlie let be the coefficients of resistance in these three be that of the external medium, and a the radius of the sphere. let its direction-cosines uniform magnetic coefficient in different directions. parallel k^, be the undisturbed magnetic intensity of the outside magnetic, or less or more placed. Let the axes of magnetic resistance be directions, so that the sphere conducts itself Magnetic Phenomena of a Sphere cut from a substance whose of resistance be taken is it k', whose intensity is // in the direction of x. sphere would be / coefficient is ki- k' a'\ The A;, placed in a resultant potential Mr maxwell, ON FARADAY'S LINES OF FORCE. 75 and for internal points — ^^1 7 7 11 Pi= 2ki So that It is in tiie interior of w. the sphere the magnetization is entirely in the direction of therefore quite independent of the coefficients of resistance in the directions of which may be changed from We -, +k may ^•, into k^ and therefore treat the sphere as but we must use a different p= I {Lv + ks without disturbing this distribution of homogeneous We coefficient for each. — my + nz + + Ix K'ih+k' L for each of tlie three w and .v. y, magnetism. components of /, find for external points Zk^+k' my + -^ ^ Qk,+ , nss\-). k' Jr'j' and for internal points - / 3k^ 3h, \2k^+k' The external effect is the same 2k, + '' k' 3ks \ 2k^+k' I that which would have been produced if the small as magnet whose moments are 2ki + k had been placed effect 2k2+ 2k^+ at the origin with their directions coinciding with of the original force / in turning the axes of x, y, z. sphere about the axis of w tlie The may be found by magnets. The taking the moments of the components of that force on these equivalent moment of the force in the direction of y acting on the third magnet 2*3+ k and that of the force in , is mnl'a « on the second magnet is k.,- k' ; 2k,+ V The whole couple about the axis of x is therefore 3k' (kj {2k3 + mnPw'. - k,) k')(2t, + mnPa?, k') tending to turn the sphere round from the axis of y towards that of to be suspended so that the axis of x is vertical, and let angle which the axis of y makes with the direction of expression for the / be /, m= z. Suppose the sphere horizontal, cos 0, then n = — if d be the sin 9, and the moment becomes / w,; 1 T-i 2 {2k^+]c)(2k2+ k) , tending to increase 9. The axis of least resistance V sin 29 therefore sets axially, but with either end indifferently towards the north. Since in all bodies, except iron, the values of k are nearly the same as in a vacuum, 10—2 Mr maxwell, ON FARADAY'S LINES OF FORCE. 76 the coefficient of this quantity can be but The value in space. medium independent of the external The k homogeneous intensity within the resistance of the shell is infinite, shell is by supposing it would have been effect the shell if When be diamagnetic or paramagnetic. vanishes, the intensity within the shell is the zero. of the shell on any point, internal or external, a superficial stratum of magnetic matter spread over the by the equation outer surface, the density being given = 3/ jo Suppose the it than what less shell and when In the case of no resistance the entire represented shell. diamagnetic or paramagnetic substance presents no shell of a were away, whether the substance of the may be k' iok, the *. Permanent magnetism in a spherical VII. difficulty. by clianging the value of t^=/Vsin20, *^ case of a altered expression then becomes J The little cos B. now to be converted into a permanent magnet, so that the distribution of shell imaginary magnetic matter invariable, then the external potential is p = - I- due to the shell will be cos 0, r~ p^= - Ir cos and the internal potential Now let us investigate the effect of the resistance magnetized is k^ shell the resistance in the external may be 9. up the filling with some substance of which shell medium being The k'. thickness of the Let the magnetic moment of the permanent magnetism neglected. be la^, and that of the imaginary superficial distribution due to the medium k = Aa^. Then the potentials are external The distribution of medium k, so that + A) — magnetism is p = real {I v or cos 9, internal ;j, = (I + A) r cos 9. the same before and after the introduction of the A=— -, I. 2k + k The magnetized effect of the greater or less than //. It external matter, and diminished by • is shell is increased or therefore increased filling it by filling diminished according as k up the with paramagnetic matter, such Taking the more general case of magnetic induction re(2tl), we find, in the expression for the moment ferred to in Art. in nature, we must admit in the case of electric those terms which depend on sines and cosines of through which heat or is, The result that in every complete revolution in the negative direction round the axis of T, a certain positive amount of work is gained; but, since no inexhaustible source of work can exist diamagnetic as iron. that 7' = in all substances, with This argument does not hold conduction, or in the case of a body respect to magnetic induction. of the magnetic forces, a constant term depending on T, besides ti. shell with is electricity is passing, for such states are maintained by the continual expenditure of work. See Prof. Thomson, P/ii/. Mag. Blarch, 1851, p. 186. Mr maxwell, ON FARADAY'S LINES OF FORCE. VIII. Electro-magnetic spherical Let us take as an example of the magnetic in the form of a thin spherical Let shell. external effect be that of a magnet whose may be the magnetic effect Let p, radius be a, and moment is magnetic be the external and internal potentials, 2>i a' • p^ no permanent magnetism, is A= we draw any whicli 3la cos and d, 6 as the the quantity of the current at any point which flows through the element fdd shell referred to unit of area of section = a, the equator, at and at this curve sin 0d9, some other will be current which flows through it, may be found by - Sla is r magnetic intensity round total a measure of the total electric is tliis when , -21. curve cutting closed known, then the is cos 9, dr dr point for = Ar .dp' — = —dpi- • 1 and since there If thickness cannot be represented by a potential. effects p = I~cos6, , an electro-mao-net t and let its Both within and without the shell /a'. its represented by a potential, but within the substance of the shell, electric currents, the where there are shell. effects of electric currents its 77 The differentiation. so that quantity the quantity of the current is - 31- sin e. t If the shell be composed of a wire round the sphere so that the number of coiled to the inch varies as the sine of 9, then the external made of the shell had been effect we have the if just given. wound round If a wire conducting a current of strength /„ be that coils be nearly the same as uniform conducting substance, and the currents had been a distributed according to the law so will a sphere of radius a measured along the axis of x distance between successive coils So is — then n there will be n coils altogether, and the value of /j for the resulting electro-magnet will be ba The The potentials, external interior of the shell and internal, will be p= /, is let The coil. Pi =- 2/, - - cos 9. "6a field. Effect of the core of the electro-magnet. P = external effect is, i-esult n , The 0, us suppose a sphere of diamagnetic or paramagnetic matter introduced into the electro-magnetic than k, that — cos therefore a uniform masnetic IX. Now - 'or- Ji is - — may be 3k' greater a" 7 — or obtained as in the ^ cos 9, less Pi than according as the interior of the =— last case, n sphere and the potentials become r ^ cos 2Z, before, 3k according as is 9. k' magnetic or is greater or less diamagnetic with Mr maxwell, ON FARADAY'S LINES OF FORCE. 78 medium, and the external respect to the beinir sreatest for a A; the effect of the core k' and the 0, The the core effect is may be of the core each = 0, it effect of the electro-magnet As k has always has witliout the core. a finite value, this. increased is by surrounding the with a shell of iron. coil on internal points would be thick, the effect the If tripled. greater in the case of a cylindric magnet, and greatest of all when Electro-tonic functions in spherical electro-magnet. to this electro-magnet. form oo = some function of r. and due find the electro-tonic functions will be of the is of introducing an iron core into an electro-magnet. effect a ring of soft iron. Let us now where w the opposite direction, electro-magnet we have a uniform field of magnetic force, shell infinitely X. They than less is the interior of the intensity of which = altered in were to vanish altogether, the for the core would be three times that which In is diamagnetic medium. This investigation explains the If the value of internal effect Oj = fio Where 7o = - lo^, <^yj there are no electric currents, we must have Oj, b^, c, this implies d the solution of which dw\ [ is r^ Within w the shell a must vanish at an cannot become infinite therefore ; w = Q is the solution, and outside infinite distance, so that C2 is The magnetic the solution outside. _ 2 ~^' 6a 2k quantity within the shell + k'~ '~ dy ~ dr is found by last article to be ' ' therefore within the sphere 0)0= — 3k 2ar. Outside the sphere we must determine value. The external value is w k' so as to coincide at the surface with the internal therefore I^n "^ where the + shell containing the currents is a' 1 ~ ~ 2a 3^ + k' r' made up of n coils of wire, conducting a current of total quantity I^- Let another wire be coiled round the number of coils be 71 ; shell according to the same law, and then the total electro-tonic intensity found by integrating EI^ = / wa smdds. EL let the total round the second coil is Mr maxwell, ON FARADAY'S LINES OF FORCE. The along the whole length of the wire. equation of the wire = -fn TT cos where w is a large number ; = a may be called £jln= the find The 2,r 3 3 — oia^n = ,1 ann 1-- 3^- -, + ^ Spherical electro-magnetic Coil-Machine. action of each coil on itself is found by putting n^ or n"' for nri . effects on both wires, supposing their / and — N stand for 3 (oft to R be Let us and R', and the then the electro-motive force of the , wire on the second first is ft ) - Nnn Thai of resistances /'. -7- + total on coil Let the be connected with an apparatus producing a variable electro-motive force F. quantity of the currents Let 47r have now obtained the electro-tonic function which defines the action of the one the other. first coil TT ?,\x\^6d$, the electro-tonic coefficient for the particular wire. XI. We , sin Qd(p, = - an E is and therefore ds .•. 79 the second on itself — dt is dt The equation of the current in the second wire -Nnn'~equation of the current in the - Nn^ Eliminating the differential therefore Nn''^-^ = R'I' dt The is (1)' dt first wire - Nnn' -~ dt ^ is — + F = RI (2)' ^ dt coefficients, we get nnn A and from which of to find /and T. AT iV — + ^— \ (^ ; m] \R For this dl ^ F — +7= — + JV— R R' n'^ ; dt dF — dt . . (s\ ^ ' purpose we require to know the value of F in terms t. Let us case first take the case in which F is constant and of an electro-magnetic coil-machine at the galvanic trough. I and moment when /' initially = the connexion 0. is This is the made with the Mr maxwell, ON FARADAY'S LINES OF FORCE. 80 Putting i T for iV (-^ + we -^.1 find F it The primary current increases very rapidly from F n — and speedily — . . — , , owing vanishes, The whole work done by is n O to p — t being to the value of , R and the secondary commences at generally very small. either current in heating the wire or in any other kind of action found from the expression f" Rdt. The total quantity of current is /to Idt. For the secondary current we R n- Jf, The work done and of quantity I' = find 4 Rn v/j 2 the quantity of the current are therefore the same as if a current Fn' ^, liad passed through the wire for a time t, where This method of considering a variable current of short duration is due to Weber, whose experimental methods render the determination of the equivalent current a matter of great precision. Now is /„ and let in the secondary = 0. equivalent currents are When of R. suddenly cease while the current in the primary wire Then we I-Ioe The F the electro-motive force ^ shall have for the subsequent time ^ , Ig and ^ -^o "57 Rf ~ n > R' ^^^ will is produce a change in the value of t, so that strength of the secondary current will be increased, and case in the ordinary coil-machines. The quantity N ' ' their duration is t. the communication with the source of the current This n its cut if off, R there will be a change be suddenly increased, the duration diminished. This is the depends on the form of the machine, and may be determined by experiment for a machine of any shape. Mu MAXWELL, ON FARADAY'S LINES OF FORCE. XII. Spherical shell revolving in magnetic Let us next take the case of a revolving Experimental Researches, Series II., explained by Faraday in are and references are there given Let the axis of z be the axis of revolution, and let be represented in quantity by the magnetism of the field field. of conducting matter under the influence shell The phenomena of a uniform field of magnetic force. 81 the angular velocity be w. /, his to previous experiments. inclined at an angle Let to the direction of z, in the plane of zx. Let R Oo' /3o, 7oi be the radius of the spherical and shell, T the thickness. be the electro-tonic functions at any point of space; of magnetic quantity and intensity; Let P2 ^^ '^^ a^, b^, Cj, Oj, /Sj, a^, 6„ e„ Let the quantities a,, /3i, 7, symbols 72 of electric quantity and intensity. any point, electric tension at dp^ = 02 ka^ V J aw •(0 72=— +^2 The expressions for oo, da,, db,. dc„ dx dy dz da, dRo dyn dx dy dz aa= ^e magnetism of*the <lue to the 70 /3o, (2); -^0 +~ = B^ + y cos - {z s,\n y«= Co-~y sin Aq, Bq, Co being constants; and the dx ~= - G, 9 - X cos d). 0, velocities of the particles of the revolving sphere are dy wy, dt We = dz — = dt wx. dt have therefore for the electro-motive forces 02 /33 = - = - 72 = - field are 1 doo 47r dt 1 / cos 9 wx. 0. Ma MAXWELL, ON FARADAY'S LINES OF FORCE. Returning to equations . we (l), \dz dyl idc, dao\ -^ — i i- /dwj ^ _ db^\ dy dy, da^ — 17 = sin Ow, 47r 2 dar ^_^ ^ dx ^ dxj V dy dz = -f^ dx dz Kdx From which get Q dy with equation (2) we find ^ * = 62 0, 11/.. = Cj sin ytoa?, 47r 4 A; —— = P2 4 47r 7(0 + (,r^ \ lOTT cos B 3/^) - xz sm6\. These expressions would determine completely the motion of sphere if we neglect the action of of circular currents about the axis proportional to the distance from of a small tliese of that magnet whose moment electricity in a revolving currents on themselves. the quantity of y, The axis. external They current magnetic express a system any point at effect will being be that to7 sin 0, with its direction along the axis of y, is 487rA; magnetism of the so that the The existence functions, and field would tend to turn it back to the axis of a;*. of these currents will of course alter the distribution of the electro-tonic so they will react ^n themselves. Let the final result of this action be a system of currents about an axis in the plane of xy inclined to the axis of x at an angle •ind producing an external The effect equal to that of a magnet whose magnetic inductive components within the 7i sin 9 — 27' cos - zT 7, cos Each of these would produce and these would give rise to new its moment is I'R^. shell are ^ in x, sin (p in y, 6 in z. own system of distributions currents when the sphere of magnetism, which, is in when the motion, velocity is uniform, must be the same as the original distribution, (/i sin 6 - 2I' cos (p) in (- 27' 7i cos * it 9 The in sin (p) in x produces 2 y produces T 2 9 - 27* cos 0) w (7, sin w (27' sin (p) in x; in y, 487r/c z produces no currents. expression for p^ indicates a variable electric tension in the shell, so that currents might be collected by wires touching at the equator and poles. Mk maxwell, on FARADAY'S LINES OF FORCE. We must therefore have the following equations, since the state of the shell 83 is the same at every instant, /, sin 9 - 2/' cos - = T ^ = 2/' sin T + /, sin sin tt)(Zj wS/ sin Q - 9.1' rf» cos ^), whence T r=x ..,. I V1+ To understand the meaning of these expressions Let the axis of the revolving west. Let I be the /cos0 is The small angle of the rotation T = let vertical, tan"^ -w to is to tlie us take a particular case. and total intensity of the terrestrial the horizontal component in result be shell -w 247^^• let the revolution be from north to magnetism, and let the dip be 9, then direction of magnetic north. produce currents in the the south shell of magnetic west, and about an axis inclined at a the external effect of these 247r«; currents is the same as that of a magnet whose 2 loss of TV of the couple due to terrestrial magnetism tending to stop the rotation work due is Tu> 247r^ and the is ^mi cos 9. 2 V'247rA;]-'+ The moment moment Tw 1 247rA;|- + TV R'Pcos^9, to this in unit of time is Taf" 247r>i R'rcos^9. 2 This loss of work is 24,rA:] made up by an proved by a recent experiment of + T'w^ evolution of heat in the substance of the shell, as M. Foucault, (see Comptes Rendus, xli. p. 450). 11—2 is