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1 Natural Sciences I lecture 11: Electricity & Magnetism ELECTRONS as charged particles In the late 1800s, cathode rays were understood to be electric currents. They were known to impart a charge to objects in their path, but because they had not (yet) been observed to be deflected by magnetic fields, their identity as moving charges was not recognized. J.J. Thomson changed that with come classic experiments, which showed unequivocally that cathode rays are particles in motion. Thomson's device looked something like this: evacuated glass tube MAGNET N ANODE wires to high voltage supply ( ) ( ) m ed bea deflect electron beam (+) (-) CATHODE (electron source) S fluorescing screen Thomson observed deflection of cathode rays (i.e., the electron beam) by the electric field, which established that electrons are particles. He was also able to determine the charge-to-mass ratio (e/m) of these particles from the amount of deflection caused by the electric field. Thomson proceeded with attempts to measure the charge separately, but was not completely successful. Knowledge of the charge (and therefore the mass) of the electron would await R.A. Millikan's famous "oil drop" experiments in 1906. voltage supply (variable) (+) (+) (+) (+) oil drop with small electric charge (-) (-) (-) (-) (-) electric field force gravity 2 ELECTRONS as constituents of atoms We saw on page 1 that electrons can be separated from atoms. Indeed, physicists took advantage of this fact in order to measure the properties of electrons. Before we go further in our discussion of electric forces, electricity and magnetism, we should also recognize that electrons are key constituents of atoms. We will examine atomic structure in detail in a couple of weeks, but a quick summary is in order here. proton mass: 1.673E-27 kg charge: (+) 1.602E-19 coulombs neutron mass: 1.6753E-27 kg charge: none electron mass: 9.109E-31 kg charge: (-) 1.602E-19 coulombs The above cartoon of the atom has some major shortcomings. Neutrons and protons are roughly 1840 times as massive as electrons, but the nucleus is nowhere near as big as depicted (in proportion to the overall atom). The nucleus is actually thousands of times smaller than the atom, so there's lots of "empty" space in the volume occupied by the atom. It is also inaccurate to think of electrons orbiting the nucleus in circular paths. [We'll discuss these issues in more detail later in the course.] Under most circumstances, atoms have the same number of protons and electrons, so the overall charge on the atom is neutral. However, electrons can be removed by a variety of processes – some of them "chemical" in nature (as in the case of interactions with atoms of other elements). Electrons can also be "stripped away" from atoms by friction. For ease in discussing atomic charge and charge-transfer processes, we arbitrarily assign a charge of -1 to the electron and +1 to the proton (note the actual units in coulombs above; see also the discussion below about the fundamental unit of charge – the coulomb). 3 Consequently, an atom possessing one fewer electrons than protons has an atomic (ionic) charge of +1. Similarly, an atom that has one excess electron relative to the number of protons has an atomic (ionic) charge of -1: - + + + - 3 protons 3 electrons neutral atom +1 - - -1 + + + - 3 protons 2 electrons atom charge = +1 - + + + - 3 protons 4 electrons atom charge = -1 Your text describes in some detail the processes by which objects acquire static charge: 1. Electrons can be stripped off by friction between two objects, leaving one object with excess electrons (negative charge), the other with a deficit of electrons (positive charge). The objects must be insulators themselves or insulated from their surroundings (see below) – otherwise the charge would be dissipated by flow of electrons through them. 2. Objects can acquire a static charge by contact with other charged objects (electrons transferred from one object to the other). 3. A static charge may be induced on a surface by the close proximity of a charged object: comb with negative static charge (excess electrons) ( )( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) sheet of paper (thickness exaggerated) these electrons have been repelled to the side of the paper away from the comb 4 A brief historical aside... An important discovery about the nature of electrical charges was made by French scientist Charles Dufay in the early 1700s. Dufay rubbed rods made of amber or glass using a silk cloth or a piece of fur. Results: Two charged glass rods repel one another Two charged amber rods repel one another Charged rods of amber and glass attract one another On the basis of these experiments, Dufay concluded that there are two types of electric charge; he called one "resinous electricity" (produced by rubbing amber), the other "vitreous electricity" (produced by rubbing glass). Although fundamentally correct, this conclusion led to confusion: some scientists in Europe who followed Dufay decided that there exist "resinous fluids" and "vitreous fluids" that carry these different charges. What Dufay had in fact discovered is what we now recognize as positive and negative charges. Implicit in his discovery is that like charges repel each other and opposite charges attract. CONDUCTORS and INSULATORS These are probably familiar to you in a general way. Conductors allow charged particles (usually electrons) to move through them readily. Two factors are important in determining whether a given material is a good electrical conductor: (next page) 5 conductors (cont'd) 1. availability of mobile electrons (metals have lots) 2. "regularity" of the atomic structure In insulators, the availability of mobile electrons is limited – the electrons are tightly bound to the atoms. Glasses are particularly good insulators: not only are the electrons tightly bound in typical oxide glasses, the atomic structure is also "chaotic" (in contrast to the structure shown above). Electrons can still be added or removed from the surfaces of insulators, so they can readily acquire a static charge. Semiconductors, as their name suggests, fall in between conductors and insulators. Their properties can be changed by "doping" with extra electrons that may be very mobile, or by creating electron "holes" that promote mobility. Silicon and germanium are the semiconductors commonly used in technological applications. COULOMB'S LAW: the force exerted by charges More than a century before J.J. Thomson showed that electrons are particles, French physicist Charles Coulomb had worked out the relationship between the magnitude of two charges and the forces between them. He used a torsion balance... q = magnitude of charge long fiber F ( ) fixed rod q2 ( ) ( ) d q1 ) ( ) ( ) ( insulating rod F 6 Coulomb knew the force required to twist the fiber by a given amount, so he could use the simple device shown on page 5 to measure the force pulling opposite charges together and pushing like charges apart. He learned two things about electric force, Fe: when the charges on the two spheres are maintained at fixed values, Fe 1/d 2 (Another inverse square law!) and when the distance, R, between the centers of the two spheres is maintained at a fixed value, Fe q1 q2 Putting these two results together, Coulomb obtained the law that bears his name... Fe = kcq1 q2 d 2 Coulomb's Law has the same form as the law of universal gravitation The constant of proportionality, kc, is analogous to G in the equation for universal gravitation. 9 2 kc = 9 10 N m / C 2 18 One coulomb (C) of charge is the charge on 6.24 x 10 electrons The coulomb is the SI unit of charge; it is a fundamental property. We can turn this around and observe that the charge on the electron, e = 1.6022 x 10 -19 coulombs q=ne number of electrons charge on the electron 7 FORCE FIELDS On the previous page, the similarity between the equations governing gravitational forces and electrical forces was noted. This is satisfying in a way, because both gravity and electric charges are capable of exerting forces at a distance – it's tempting to consider them as having something in common. To pre-19th century physicists trying to come to terms with the behavior of natural systems, the notion of forces acting at a distance was difficult to rationalize (as it still is for today for students of science). The equations had been worked out by Newton (in the case of gravity) and by Coulomb (in the case of electric charge), but there existed no intuitively satisfying model to explain the phenomena of "non-contact" forces. In the early 19th century, Michael Faraday and James Maxwell developed a rather different way of looking at forces associated with charged objects (actually, it was Faraday who provided most of the fundamental observations; Maxwell formalized the mathematics). The conceptual breakthrough was that an electric charge came to be viewed as affecting the space around it in a particular way. Just as a large mass carries with it a gravitational field, an electric charge is surrounded by an electric field (and a magnet is surrounded by a magnetic field). All three are characterized as force fields. The concept of a force field may seem like a subtle change in thinking, but it freed scientists from the idea that gravitational and electric forces are like invisible ropes connecting objects together. Of the three types listed above, the magnetic force field is the easiest to "see"... iron filings bar magnet N S 8 Evaluating Electric Fields Electric fields are more complex to represent than are gravitational fields, in part because there exist both positive and negative charges. Electric fields are represented as circles or spheres with lines of force radiating toward or away from them: Remember that forces are vector quantities. By convention, the lines of force are directed away from the charge if it is positive, and toward the charge if it is negative -q +q The manner in which a charge is evaluated is by bringing a test charge near it (the test charge is positive by convention). Consider the electric force Fe acting on a test charge q that is brought near another charge. The electric field at the location of the test charge is defined as E Fe / q Since Fe is a vector, E is also a vector – it has direction as well as magnitude. We can rearrange this equation to answer the question What is the electric force on a charged body (of charge q) interacting with the electric field due to other charged bodies?: Fe qE This equation can be combined with Coulomb's law to allow calculation of the electric field E at a point distance d from charge q: E = kc q d 2 = 9 10 9 q d 2 9 Electric fields (cont'd) Let's look at the interaction between two charges, first considering opposite charges of equal magnitude... At any point in the "neighborhood" of the two charges, the total force field is represented by the vector sum of contributions from the two charges (this diagram has bilateral symmetry because the two charges are equal, but this does not have to be the case) -q2 +q1 vector sum vector sum q2 q1 contri bution contribution q1 contribution q2 contribution Now consider two charges that are equal in magnitude and charge... Because the force-field vectors are pointing outward from both charges, their sums in the region around (and especially between) the two charges have a very different orientation. vector sum q1 contribution q2 contribution +q1 +q2 10 Electric fields In the examples on page 9, the electric field E at any point in the vicinity of the two charges q1 and q2 can be calculated from the relation: E = n1 kc q1 d1 2 + n2 kc q2 d2 2 You don't need to remember this equation if the diagrams on page 9 make sense to you where d1 and d2 are the distances to charges q1 and q2, respectively, and n1 and n2 are vectors indicating the direction to the two charges. This is the vector sum shown schematically for each case on page 9. Electric Potential Two or three weeks ago, we talked about the creation of gravitational potential energy by raising an object in a gravitational field (PE = mgh). Given the mathematical similarities between gravitational and electric forces, it probably won't come as a surprise to you that we also speak in terms of electric potential energy. A charge (or charged particle) is surrounded by an electric field that is conceptually similar to the Earth's gravitational field. Just as moving an object in the Earth's gravity field changes the potential energy of that object, so too does moving a charge in the electric field of another charge. Bringing charged particle into the electric field of another particle of the same charge involves the performance of work: likecharged particles repel each other, so work must be done against the repulsive force to bring the particles together. Squeezing two tennis balls together is not a bad analogy. Electric potential energy is usually measured in terms of a potential difference (PD), which is defined as the work required to create the potential divided by the charge moved: PD = W/q 11 Consider two spheres, one anchored by an insulating rod, the other moveable. Both are positively charged; the moveable one at 1 coulomb (C). At the point where 1 joule of work has been done in bringing the 1-C sphere closer to the stationary one, we have created a potential difference of 1 volt. 1 volt = 1 joule / 1 C (+) 1.0 C (+) 1 joule of work done to bring the moveable sphere into position In the set-up at the bottom of page 10, we created a potential difference of 1 volt by moving the charged spheres together. The electric potential energy we created is now capable of performing 1 joule of work. In other words, if we let go of the moveable sphere (allowing the repulsive force to push it away) that force could be used to do 1 joule of work – just as the energy stored in the compressed tennis balls could also do work. ELECTRIC CURRENT Up to this point we have been considering charges that are static in nature. As we saw last week, these static charges can be produced by physical processes that involve motion, but apart from the experiments in which we deliberately moved charges together to assess their character, the charges themselves did not move. The sustained flow or movement of charge is referred to as electric current, and it is given the symbol I. 12 Electric circuits If you acquire a static charge on your clothing by rubbing on a carpet or chair seat, you may become aware of the fact when your hand approaches a conductor such as a metal doorknob. Suddenly a pathway is available for electrons to move, so the static charge is quickly dissipated by flow of electrons – often with an uncomfortable spark. The resulting shock you feel is very brief, however, because the electric potential is eliminated by flow of electrons. In order for current to flow continuously, you need two things: a circuit and a sustained voltage source, such as a battery or generator. In many respects, electric current is like the flow of water: PUMP Water will keep flowing in this system as long as the pump is on. The pump maintains the potential energy that sustains flow. Most car batteries have a potential of 12 volts. This means that each coulomb of charge that moves through the circuit can do 12 joules of work. Voltage describes the potential difference between any two points in the circuit. It has aspects in common conducting wire with water pressure or water voltage source: voltage drop "force", and so is sometimes (maintains potential; (work is done here) called electromotive force does work) (emf). A typical electric circuit 13 The flow rate, I, of the electric current is the amount of charge that passes through a given point in the circuit in a given amount of time: electric current = I = quantity of charge time q t The units of charge are therefore coulombs per second; a coulomb per second is called an ampere (amp for short). 1 amp (A) = 1 coulomb (C) 1 second (s) The Nature of Electric Current Because of the manner in which our understanding of electricity progressed, we are left today with two different conventions for describing the flow of current in a circuit: 1. Conventional Current refers to the flow of positive charges from the positive to the negative terminal (pre-dates discovery of the electron – but still used in circuit diagrams!) 2. Electron Current refers to flow of negative charges. conventional current )+( ( ) (+) )+( electron current )+( (+) ( ) (+) ( ) ( ) (+) ( ) ( ) ( ) (+) ( ) The electron current is actually the "correct" description of flow in the circuit, because it refers to the drift of electrons from the negative to the positive terminal in a battery. However, the two "models" are mathematically equivalent, and there's no difference between them when it comes to calculating current in circuits. 14 Physical details Our everyday experience is that electric current appears to travel very fast in wires and other conductors. After an electrical outage, for example, when the power is turned back on, the lights come on over an entire city instantaneously. What we actually observe in such a case is not incredibly fast flow of electrons, but the near-instantaneoous establishment of the electric field. SECTION OF METAL CONDUCTOR no electric field electron ion electric field present Unattached electrons move about randomly as in a gas. Motion is chaotic and collisions are frequent – no net movement of electrons in any direction Motion of electrons retains its random, chaotic aspect, but there is also some net motion (flow) due to the presence of the field – the electrons make some progress... It is the net motion of electrons that constitutes the electric current; the drift velocity is ~ 0.01 cm/s. The electric field that causes the net motion propagates much, much faster – close to the speed of light. Properties of electric current – Summary a potential difference establishes an electric field throughout a circuit; this occurs very rapidly (near light-speed) net motion of electrons results from the presence of the field – this is the electric current the net electron motion is relatively slow (~0.01 cm/s on average) (collisions occur among electrons and between electrons and positivelycharged atom nuclei – this is the cause of resistance to electron flow) 15 RESISTANCE Now that you are familiar with the physical nature of electric current, we are almost ready to discuss the properties of circuits and do some example calculations. Let's first look at resistance a little more closely... When a current flows in a conductor, random collisions of mobile electrons with positive ions result in loss of energy from the electrons. Kinetic energy of the electrons is partially transferred to the ions, whose increased vibrational amplitude causes the temperature of the conductor to rise. This can be a sort of "runaway" effect, because the resistance of a conductor increases with increasing temperature: the "thermal agitation" of the positive ions increases the likelihood of collisions with electrons. As noted last week, electrical resistance varies greatly from one material to another, due to differences in the electronic structure of the atoms making up the material and their physical arrangement. For a given potential difference, the resistance of a circuit affects the current passing through it, as we'll see momentarily... Let's return briefly to the analogy between current in an electric circuit and water flow in a plumbing "circuit". If the potential difference between two points in an electric circuit is analogous to a water pressure difference, it follows that I PD I 1/R It is reasonable, also, that where R is the resistance of the circuit. The "water-flow" analogy is valid here, too (see next page)... 16 A constriction in a pipe restricts the flow of water A high-resistance circuit reduces the current (at a given PD) I I (-) (+) (-) low R A large diameter pipe can carry more water than a small one (at a fixed pressure difference) (+) high R A large diameter wire can carry more current than a small one of the same material (for a given PD) I (-) (+) I (-) (+) Combining the two proportionalities from p. 15, we obtain I = PD / R R = PD / I or which many of you will recognize as Ohm's law. It is usually written V = IR The unit of resistance is the ohm: 1 ohm (W) = 1 volt (V) 1 amp (A) This relationship indicates that the resistance of a conductor is 1 ohm if a potential difference of 1 volt will maintain a current of 1 amp through that conductor. Summarizing (and extending) conclusions from the previous page... the resistance depends on four factors: material identity; length; crosssectional area; and temperature length area 17 The resistances of materials vary over many orders of magnitude. Most metals are reasonably good conductors; most oxides and organic materials are reasonably good insulators (i.e., they have high resistances). The best conductors (rank order) incandescent light filaments conductors insulators silver rubber copper glass gold ceramics aluminum diamond graphite plastics tungsten wood iron most rocks Nichrome dry air nickel/chromium alloy used in toaster windings and the like the length factor is a direct multiplier – i.e., twice the length means twice the resistance (all else being equal) The resistance of all conductors decreases at low temperatures; in the case of superconductors, it goes to zero. DIRECT CURRENT (DC) vs. ALTERNATING CURRENT (AC) Up to this point we have spoken only about direct current (DC), which involves the actual flow of charges (electrons) through conductors, always in the same direction. In the case of alternating current (AC), the potential difference between the two terminals of the voltage source switches back and forth between positive (+) and negative (-) –-- hence the term "alternating" current. In the U.S., the frequency of the alternation is 60 changes from (+) to (-) and 60 changes from (-) to (+) per second, for 60 complete cycles (60 Hz) each second (in the U.K., the frequency is 50 Hz; which is why some electrical devices made for the American market require an adaptor). 18 1/60 sec +120 potential difference (volts) -120 time Alternating current was adopted for long-distance power transmission, regional grids, and household current because it is more efficient than direct current (resistance of the wires does not take as big a toll). The switching of the potential in AC means that there is no net electron drift. The electric field moves back and forth through the circuit at near-light speed. An interesting historical note... Thomas Edison is one of the huge names in electricity. We probably owe more to him than to any other single individual for the range and breadth of his electrical inventions. The one big battle he lost, however, was that he was a proponent of DC power. He set up a small DC grid in northern New Jersey that was never a financial success. George Westinghouse carried the day in his advocacy of AC power, mainly because the voltage can easily be stepped up and down for different applications. We'll talk more about AC circuits next lecture, in connection with transformers, induction and related topics... 19 ELECTRICAL POWER and ELECTRICAL WORK (equations and example applications) As summarized on page 3, electrical circuits consist of three types of components: a voltage source conducting wires an electrical device (can be more than one) In a DC circuit, the electric field moves from the voltage source through one length of wire to the electrical device (e.g., a light bulb) and back along another length of wire to the other terminal of the voltage source, maintaining the potential difference across the device: conducting wire voltage source electrical device In an AC circuit, one wire (the black or "hot" one) supplies the alternating electric field to the device. A second wire from the device is "grounded" – that is, connected to a metal rod hammered into the ground. As noted on page 3, it is the voltage source that does work in an electrical circuit. This voltage source could be a battery (DC) or an electric power generator (AC). The total work done by the voltage source is... work done in the device by the electric field + energy lost to resistance in the wires (this is kept small) Resistance is analogous to friction in a mechanical device – we'll ignore it as we often do in the case of friction. 20 We already know (page 10) that potential difference = work / charge PD = W/q so we can write W = (PD) q Note the units: joules = joules coulomb coulomb As in mechanical processes, a joule is a unit of work, However, as we learned yesterday, an "electrical" joule is related to moving a charge to a higher potential rather than to a force acting through a distance (these two "kinds" of joules are really the same thing, of course, as you could discover with the appropriate electromechanical device. Remember from a few weeks ago that power is work per unit time P = W/t so electrical power must be P = power = J/s = watts (PD) q q = PD t t current (C/s) = I voltage (V = J/C) We end up with... power = current x potential difference = amps x volts P = IV If W = P x t, then W = IVt units: C s J C s = J 21 Power is what the utility company bills you for: Power is work per unit time, so what the utility company wants to know is the amount of power you used times the amount of time you used it: power x time = kilowatt x hour = kWhr But remember that P = IV so kWhr can be calculated by multiplying amps x volts x time = watts x hr If you were to use 1000 watts (1 kilowatt) of power for 1 hour, you would be billed for 1 kilowatt-hour (kWhr). As a point of calibration, a typical hair dryer is 1000-1500 watts (let's say 1200 on average). Note that if the hair dryer is operating at 120 volts, this means it draws a current of 10 amps (P = I V). EXAMPLE CALCULATIONS Following on the hair dryer example above, let's do another hair dryer calculation in which we keep track of all the units. If you use a 1,100-watt hair dryer designed to operate at 120 V, how much current does this hair dryer draw? given P = 1,100 W V = 120 V I = ?A solution P=IV joule P = second joule V = coulomb I = P/V joule 1,100 second = joule 120 coulomb = 9.2 C/s = 9.2 A 22 Another example... What is the cost of operating a 75-watt light bulb for 24 hours if the cost of electricity is $0.12 per kWhr? given P = 75 watts t = 24 hr rate = $0.12 / kWhr cost = ?$ solution kWhr = P t = (watts) (time) = (75 W) (24 hr) = 1,800 Whr = 1.8 kWhr 1.8 kWhr x $0.12 kWhr = $0.22