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Obtaining Maxwell's equations heuristically Gerhard Diener, Jürgen Weissbarth, Frank Grossmann, and Rüdiger Schmidt Citation: Am. J. Phys. 81, 120 (2013); doi: 10.1119/1.4768196 View online: http://dx.doi.org/10.1119/1.4768196 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v81/i2 Published by the American Association of Physics Teachers Related Articles A low voltage “railgun” Am. J. Phys. 81, 38 (2013) Ampère’s motor: Its history and the controversies surrounding its working mechanism Am. J. Phys. 80, 990 (2012) Examples and comments related to relativity controversies Am. J. Phys. 80, 962 (2012) Inclined Levitron experiments Am. J. Phys. 80, 949 (2012) Quantitative analysis of the damping of magnet oscillations by eddy currents in aluminum foil Am. J. Phys. 80, 804 (2012) Additional information on Am. J. Phys. 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Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission Obtaining Maxwell’s equations heuristically €rgen Weissbarth, Frank Grossmann,b) and Ru €diger Schmidt Gerhard Diener,a) Ju Technische Universit€ at Dresden, Institut f€ ur Theoretische Physik, D-01062 Dresden, Germany (Received 13 February 2012; accepted 2 November 2012) Starting from the experimental fact that a moving charge experiences the Lorentz force and applying the fundamental principles of simplicity (first order derivatives only) and linearity (superposition principle), we show that the structure of the microscopic Maxwell equations for the electromagnetic fields can be deduced heuristically by using the transformation properties of the fields under space inversion and time reversal. Using the experimental facts of charge conservation and that electromagnetic waves propagate with the speed of light, together with Galilean invariance of the Lorentz force, allows us to finalize Maxwell’s equations and to introduce arbitrary electrodynamics units naturally. VC 2013 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4768196] I. INTRODUCTION When teaching electrodynamics one is faced with the question of whether to postulate Maxwell’s equations, as one postulates Newton’s laws in a classical mechanics course, or whether they should be justified from experimental evidence (Coulomb’s law, Faraday’s induction law, Ampère’s law, and the nonexistence of magnetic monopoles). Depending on the choice made, the didactic approach is then either deductive (axiomatic) or inductive. Here, we give a heuristic derivation of the microscopic Maxwell equations. This derivation is based on the principles of simplicity (lowest order in space and time derivatives), linearity (superposition principle), and the transformation properties of the fields under space inversion (~ r ! ~ r ) and time reversal (t ! t). The starting point of the derivation is the experimental fact that a (moving) charge experiences the Lorentz force. In addition, in order to obtain the final form of the Maxwell equations and to introduce electrodynamic units, we use the experimental evidence of charge conservation, the fact that electromagnetic waves propagate at the speed of light c, and the requirement of Galilean invariance of the Lorentz force for low velocities. Several attempts to deduce (or derive) Maxwell’s equations have been published.1–4 The approach presented below is unique because it does not make use of another dynamical equation, such as the time-dependent Schr€odinger equation1 or Newton’s law,2 as a starting point. Our derivation may serve as valuable background information for the lecturer using either an inductive or deductive approach to teaching electrodynamics. It can also be used as an a posteriori justification after the Maxwell equations have been postulated and/or obtained from experimental evidence. The presentation is structured as follows: in Sec. II, we briefly review the classification of vectors and scalars by their behavior under space inversion and time reversal. In Sec. III, we present the deduction of the structure of the Maxwell equations from the principles of simplicity and linearity, with five undetermined multiplicative constants. Three of the five constants are determined in Sec. IV from well-established experimental facts together with the requirement of Galilean invariance of the Lorentz force. Fixing the final two constants leads to the natural introduction of three commonly used systems of units in electrodynamics. 120 Am. J. Phys. 81 (2), February 2013 http://aapt.org/ajp II. POLAR AND AXIAL VECTORS AND TIME REVERSAL PROPERTIES In a standard physics curriculum starting with classical mechanics, the necessity of studying electrodynamics can be motivated by the implications of the particle’s charge, be it fixed or moving. Experimental evidence shows that the Lorentz force on a test particle with electric charge q, moving with velocity ~ v, is given by ~ ¼ q ðE ~ þ g~ ~ Þ; F vB (1) where g is a constant. This equation postulates the existence ~ r ; tÞ and a local magnetic field of a local electric field Eð~ ~ Bð~ r ; tÞ at the position of the particle ~ r , generated by all charge-carrying, field-generating particles except the test particle itself. A particle with a magnetic charge has never been observed; thus, the fields are generated only by electric charges with (static) charge density q and moving charges with (electric) current density ~ j. Due to the multiplicative connection between charge and fields in Eq. (1), their units cannot all be chosen arbitrarily. For example, an arbitrarily defined charge unit [q] fixes the dimension of the electric field [E] and that of the product ½gB, with an arbitrary constant g that finally defines [B] and thus fixes the relative dimensions of both fields. Of central importance for what follows are the transformation properties of vectors with respect to a spatial inversion (~ r ! ~ r ):5–7 • Vectors ~ p , that transform according to 0 ~ p ! ~ p p ¼ ~ ð~ r !~ rÞ • (2) under inversion are called polar vectors (or simply vectors). Vectors ~ a , that transform according to 0 ~ a a ¼~ a ! ~ ð~ r !~ rÞ (3) under inversion are called axial vectors (or pseudovectors). They need a (right-) hand rule for their definition. C 2013 American Association of Physics Teachers V Downloaded 25 Jan 2013 to 141.30.17.226. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission 120 long time;9 an early appearance and critical discussion in the literature can be found in Ref. 5. For future reference, we state that the field generating charge density, because of the discrete nature of charge, is defined by q limDV!0 DQ=DV, where DQ is an infinitesimal amount of charge and DV is an infinitesimal volume (such that DV > 0). The charge density is idealized as a continuous scalar field with qð~ r ; tÞ ¼ qð~ r ; tÞ and qð~ r ; tÞ ¼ qð~ r ; tÞ, and thus the field generating current density ~ ~ð~ jð~ r ; tÞ ¼ qð~ r ; tÞ V r ; tÞ; Fig. 1. An example for the transformation properties of polar vectors (position ~ r and momentum ~ p ) and an axial vector (angular momentum ~¼ ~ L r ~ p ) under spatial inversion.8 As an example from classical mechanics, we explicitly depict the (spatial-inversion) transformation properties of ~ in Fig. 1. position ~ r , momentum ~ p , and angular momentum L The vector product of polar and axial vectors can lead to either polar or axial vectors according to the following rules: ~ p1 ~ p2 ¼ ~ a; (4) ~ a1 ~ a2 ¼ ~ a; (5) ~ p1 ¼ ~ p: a1 ~ (6) Furthermore, just as there are vectors and pseudovectors, we can define scalars S as transforming according to 0 S ! S ¼ S; ð~ r !~ rÞ (7) under (spatial) inversion, and pseudoscalars S as transforming according to 0 S ! S ¼ S : ð~ r !~ rÞ (8) Taking the scalar product of polar and axial vectors leads to both scalars and pseudoscalars according to: ~ p1 ~ p 2 ¼ S; (9) ~ a 2 ¼ S; a1 ~ (10) ~ p1 ~ a 1 ¼ S : (11) We can now identify the vector and scalar character of the quantities appearing in the Lorentz force, together with their behavior under the reversal of time: ~ (i) Because the mass m is a scalar, we see from F 2 2 ¼ md ~ r =dt that force is a polar vector with ~ ~ Therefore, because q in Eq. (1) is a scaFðtÞ ¼ FðtÞ. ~ must be a polar vector with lar, we know that E ~ r ; tÞ ¼ Eð~ ~ r ; tÞ. Eð~ (ii) Because g is a scalar and the velocity ~ v is a polar vector with ~ vðtÞ ¼ ~ vðtÞ, the vector product in ~ is an axial vector with Eq. (1) tells us that B ~ ~ Bð~ r ; tÞ ¼ Bð~ r ; tÞ. This difference in character of the two field vectors of electrodynamics has been known and used in teaching for a 121 Am. J. Phys., Vol. 81, No. 2, February 2013 (12) ~ð~ with V r ; tÞ the velocity field of the charge density, is a polar vector with ~ jð~ r ; tÞ ¼ ~ jð~ r ; tÞ and ~ jð~ r ; tÞ ¼ ~ jð~ r ; tÞ. III. GENERATING THE MAXWELL EQUATIONS: THE BASIC IDEA The classic textbook by Jackson6 contains a discussion showing that the Maxwell equations do indeed equate quantities with the same transformation behavior under time reversal and space inversion. The idea presented here is that the transformation behaviors, together with the Lorentz force and the heuristic demand for simplicity and linearity, allows us to deduce the structure of the Maxwell equations. The principles of simplicity and linearity, together with the symmetry properties of the fields, have already been employed by Migdal10 to derive the homogeneous curl equations. The procedure we follow here is to equate field generating quantities (q and ~ j) with time or space derivatives of the fields up to at most first order and in such a way that the fields and the inhomogeneous terms fulfill the superposition principle.11 First, as already mentioned, there is no experimental evidence for magnetic charges so there cannot exist any fieldgenerating pseudoscalars. The only pseudoscalar [see Eq. (11)] that can be generated and obeys linearity is to take the first-order spatial derivative of an axial field vector. Thus, we quickly conclude that ~ B ~ ¼ 0: r (13) We note in passing that there is a simple physical argu~ if this divergence ment for the vanishing divergence of B: were non-zero, then by analogy with the electric field case, a magnetic field proportional to ~ r would exist and the Lorentz force would contain terms proportional to d~ r =dt ~ r . In other words, there would exist terms proportional to the angular momentum that would lead to an out-of-plane acceleration. However, such trajectories have never been observed for a charged particle in a pure magnetic field. To conclude our discussion of Eq. (13), we mention that mixed terms of the ~ ~ B, ~ would violate correct symmetry, such as ~ j @ B=@t or E the superposition principle.11 It is worth mentioning that, due to an idea proposed by Dirac, magnetic monopoles may explain the discrete nature of the electric charge.12 Therefore, the search for magnetic monopoles remains an active area of research. In addition, magnetic monopoles may also be helpful as a didactic tool.13 ~ E ~ ~ [see Eq. (6)] are the only axial Second, @ B=@t and r vectors without sign change under time reversal so they must appear in the same equation. Because a field generating axial vector does not exist, the heuristic demand for simplicity and linearity leads us to the equation Diener et al. Downloaded 25 Jan 2013 to 141.30.17.226. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission 121 ~ ~ E ~ þ v @ B ¼ 0; (14) r @t where v is an undetermined constant. Third, the only scalar that can be generated from first derivatives of the fields without a sign change under time re~ E. ~ This quantity must be determined by the only versal is r scalar source term q and therefore we can write ~ E ~ ¼ a q; r (15) with a another undetermined constant. Again we note that ~ mixed terms of the correct symmetry, such as ~ j @ E=@t for Eq. (15), would violate the superposition principle,11 whereas a term proportional to @ 2 q=@t2 violates the use of maximally first order derivatives. Finally, the only polar vectors that change sign under time reversal that can be generated from the fields by first order ~ B. ~ ~ These terms must therederivatives are @ E=@t and r fore appear in one equation together with ~ j, giving ~ ~ B ~ þ j @E ¼ b ~ r j; (16) @t where the constants j and b are still to be determined. The homogeneous [Eqs. (13) and (14)] and the inhomogeneous [Eqs. (15) and (16)] “Maxwell equations” constructed in this way contain four undetermined constants: v; a; j, and b. Together with g, appearing in Eq. (1), there are five constants that need to be determined. IV. DETERMINING THE CONSTANTS AND CHOICE OF UNITS Thus, for the purpose of determining v it is sufficient to consider a Galilean transformation: ~ r !~ r0 ¼~ r ~ v 0 t; ~ v 0 ¼~ v ~ v 0 . When applied to Eq. (1) we find ~ ¼ qðE ~ þ g~ ~ Þ ¼ qðE ~ þ g~ ~ þ g~ ~Þ F vB v0 B v0B ~ 0 Þ: ~0 þ g~ v0B ¼ qðE For the fields, the postulated invariance of the Lorentz force then gives ~0 ¼ E ~ þ g~ ~; E v0 B For the five unknown constants we therefore have three independent equations, meaning there is a degree of ambiguity in this system. This ambiguity will be resolved with the discussion of three frequently used systems of units, the metric system (SI units), the Gaussian system (cgs units), and the Heaviside-Lorentz system. A. Charge conservation Taking the divergence of Eq. (16), together with Eq. (15), leads to ~ ~ ~ ~ @ E ¼ ja @q : br j ¼ jr (17) @t @t ~ ~ Therefore, the continuity equation @q=@t þ r j ¼ 0 will be automatically fulfilled if we require ja ¼ b: (18) B. Invariance of the Lorentz force For small velocities (v c), the Lorentz invariance of the Lorentz force, to leading order, gives Galilean invariance. 122 Am. J. Phys., Vol. 81, No. 2, February 2013 ~0 ¼ B: ~ B (20) Using a calculation similar to that in Ref. 14, we find, on the other hand, that ~ Eð~ ~ 0 Eð~ ~ r ; tÞ ¼ r ~ r 0 þ~ r v 0 t; tÞ ~ r ; tÞ ~ r 0 þ~ @ Bð~ v 0 t; tÞ ¼ @ Bð~ ~ 0 ÞB ~ v0 r 0 ð~ @t ~r @t ~ r ~ r 0 þ~ @ Bð~ v 0 t;tÞ ~ 0 v0 B ~Þ; ¼ 0 þ r ð~ @t ~ r (21) (22) ~ 0 means to differentiate with respect to where the notation r the primed coordinates, and j~r reminds us to hold the vector ~ r fixed. Using these results in Eq. (14) leads to ~ ~ @B @B 0 ~ ~ ~ ~ ~ ¼ r ðE þ v~ v 0 B Þ þ v rEþv @t @t ~r 0 The undetermined constants must be chosen so that Eqs. (13)–(16) satisfy additional constraints. We will use the following three experimental facts: (A) Charge conservation—the field generating quantities (q and ~ j) must satisfy the continuity equation, (B) Invariance of the Lorentz force—the Lorentz force must be invariant under a Galilean transformation for v c, where c is the speed of light, (C) Wave equation—the universal propagation speed c must be contained in the wave equation derivable from the equations obtained in Sec. III. (19) ~0 ~0 E ~ 0 þ v @ B ¼ 0; ¼r @t (23) from which we see that ~ þ v~ ~ ~0 ¼ E v 0 B; E ~0 ¼ B: ~ B (24) Comparison with Eq. (20) then shows that v ¼ g: (25) C. Wave equation The wave equation for vanishing inhomogeneities q and ~ j follows by taking the curl of Eq. (14) and then using ~ ðr ~ EÞ ~ Eqs. (15) and (16). Using the identity r 2 ~ r ~ E ~ Þ r E, ~ this leads to ¼ rð 2~ ~ ~ @ B ¼ r2 E ~ r ~ E ~þ v r ~ vj @ E ¼ 0: ~ Þ r2 E rð @t @t2 (26) Finally, making use of Eq. (25) we find that gj ¼ 1 ; c2 (27) where the velocity c ¼ 2.9986 108 m=s, is determined by measuring the velocity of electromagnetic waves in vacuum. It is worthwhile to note that a velocity with the same Diener et al. Downloaded 25 Jan 2013 to 141.30.17.226. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission 122 numerical value can be determined by pure electrostatic and magnetostatic measurements (“c equivalence principle”).15,16 In this way, the units are introduced naturally by fixing two remaining constants appearing in the Maxwell equations and the Lorentz force. D. Final choice of the system of units The constant v is now fixed by Eq. (25) and for j; a; b, and g we have Eqs. (18) and (27) that lead to a=ðgbÞ ¼ c2 (see also Ref. 3). Two of these constants can be chosen arbitrarily; we discuss three common choices below. (i) SI units: Choose g ¼ 1 to get j ¼ 1=c2 , and b ¼ 4p 107 N=A2 l0 to get a ¼ l0 c2 1=e0 , and thus a Lorentz force ~ ¼ q½Eð~ ~ r ; tÞ þ~ ~ r ; tÞ; F v Bð~ (28) and the rationalized Maxwell equations (without explicit appearance of p) (ii) ~ r ; tÞ ~ Eð~ ~ r ; tÞ ¼ @ Bð~ r @t (29) ~ r ; tÞ @ Eð~ ~ Bð~ ~ r ; tÞ ¼ l0~ jð~ r ; tÞ þ l0 e0 r @t (30) r ; tÞ ~ Eð~ ~ r ; tÞ ¼ qð~ r e0 (31) ~ Bð~ ~ r ; tÞ ¼ 0 r (32) cgs units: Choose g ¼ 1=c to get j ¼ 1=c, and b ¼ 4p=c to get a ¼ 4p and a Lorentz force ~ v ~ ~ ~ r ; tÞ ; (33) F ¼ q Eð~ r ; tÞ þ Bð~ c where the electric and magnetic fields have the same units. The Maxwell equations then read (iii) 123 ~ r ; tÞ ~ Eð~ ~ r ; tÞ ¼ 1 @ Bð~ r ; c @t (34) ~ r ; tÞ 1 @ Eð~ ~ Bð~ ~ r ; tÞ ¼ 4p ~ jð~ r ; tÞ þ ; r c c @t (35) ~ Eð~ ~ r ; tÞ ¼ 4pqð~ r r ; tÞ; (36) ~ Bð~ ~ r ; tÞ ¼ 0: r (37) Heaviside-Lorentz units: Choose g ¼ 1=c to get j ¼ 1=c, and b ¼ 1=c to get a ¼ 1. This choice again gives the Lorentz force in Eq. (33) but with different field units (although the electric and magnetic fields again have the same units), and leads to rationalized Maxwell equations ~ r ; tÞ ~ Eð~ ~ r ; tÞ ¼ 1 @ Bð~ r ; c @t (38) ~ r ; tÞ 1 @ Eð~ ~ Bð~ ~ r ; tÞ ¼ 1 ~ jð~ r ; tÞ þ ; r c c @t (39) ~ Eð~ ~ r ; tÞ ¼ qð~ r r ; tÞ; (40) ~ Bð~ ~ r ; tÞ ¼ 0: r (41) Am. J. Phys., Vol. 81, No. 2, February 2013 V. SUMMARY Starting from the Lorentz force as an experimental fact and using the principles of simplicity and linearity, we showed that the structure of the Maxwell equations for the electromagnetic fields in vacuum can be deduced. These are two (pseudo)scalar equations for the divergence of the respective fields and two (pseudo)vector equations for the curl of the fields, in accord with the fundamental theorem of vector calculus. The vector character (polar or axial) of the fields and their transformation properties under time reversal are at the heart of our approach. Charge conservation, the Galilean invariance of the Lorentz force, and the fact that electromagnetic waves propagate at the speed of light enabled us to fix three of the five undetermined constants in the Lorentz force and the four deduced Maxwell equations. As an additional useful result of the calculation, arbitrary systems of units can be introduced, which is didactically more pleasing than to postulate the units a priori. ACKNOWLEDGMENTS Helpful comments on the manuscript by Larry Schulman and valuable discussions with Klaus Becker are gratefully acknowledged. a) Deceased on August 17, 2009 Electronic mail: [email protected] 1 Donald H. Kobe, “Derivation of Maxwell’s equations from the local gauge invariance of quantum mechanics,” Am J. Phys. 46, 342–348 (1978). 2 Freeman J. Dyson, “Feynman’s proof of the Maxwell equations,” Am. J. Phys. 58, 209–211 (1990). 3 Jose A. Heras, “Can Maxwell’s equations be obtained from the continuity equation?” Am. J. Phys. 75, 652–657 (2007). 4 Jose A. Heras, “How to obtain the covariant form of Maxwell’s equations from the continuity equation,” Eur. J. Phys. 30, 845–854 (2009). 5 Fritz Emde, “Polare und axiale Vektoren in der Physik,” Z. Phys. A 12, 258–264 (1923). 6 John David Jackson, Classical Electrodynamics, 2nd. ed. (Wiley, New York, 1975). 7 J. W. Norbury, “The invariance of classical electromagnetism under charge conjugation, parity and time reversal (CPT) transformations,” Eur. J. Phys. 11, 99–102 (1990). 8 Adapted from <http://commons.wikimedia.org/wiki/File:Angular_ momentum_circle.png>. 9 ~ and B ~ is apparent also in the construction The different nature of E of the electromagnetic field tensor in the covariant formulation of electrodynamics. 10 Arkadi B. Migdal, Qualitative Methods in Quantum Theory (Benjamin, Reading MA, 1977). 11 The superposition principle for an inhomogeneous, partial differential equation with linear differential operator L states that with two solutions obeying Lu1 ¼ g1 and Lu2 ¼ g2 , the sum u1 þ u2 is also a solution for the inhomogeneity g1 þ g2 . 12 Julian Schwinger, “A Magnetic Model of Matter,” Science 165, 757–761 (1969). 13 Frank S. Crawford, “Magnetic monopoles, Galilean invariance, and Maxwell’s equations,” Am. J. Phys. 60, 109–114 (1992). 14 Max Jammer and John Stachel, “If Maxwell has worked between Ampère and Faraday: An historical fable with a pedagogical moral,” Am. J. Phys. 48, 5–7 (1980). 15 Jose A. Heras, “The Galilean limits of Maxwell’s equations,” Am. J. Phys. 78, 1048–1055 (2010). 16 Jose A. Heras, “The c equivalence principle and the correct form of writing Maxwell’s equations,” Eur. J. Phys. 31, 1177–1185 (2010). b) Diener et al. Downloaded 25 Jan 2013 to 141.30.17.226. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission 123