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Vol 2 Ch 17 - Laws of Induction

Vol 2 Ch 17 - Laws of Induction - 17 The Laws of Induction...

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Unformatted text preview: 17 The Laws of Induction 17—1 The physics of induction In the last chapter we described many phenomena which show that the effects of induction are quite complicated and interesting. Now we want to discuss the fundamental prinCiples which govern these effects. We have already defined the emf in a conducting Circuit as the total accumulated force on the charges throughout the length of the loop. More specifically, it is the tangential component of the force per unit charge, integrated along the wire once around the Circuit. This quantity is equal, therefore, to the total work done on a single charge that travels once around the circuit. We have also given the “flux rule,” which says that the emf is equal to the rate at which the magnetic flux through such a conducting Circuit is changing. Let’s see if we can understand why that might be. First, we’ll cons1der a case in which the flux changes because a circu1t is moved in a steady field. In Fig. 17—1 we show a Simple loop of wire whose dimensions can be changed. The loop has two parts, a fixed U-shaped part (a) and a movable crossbar (b) that can slide along the two legs of the U. There is always a complete Circuit, but its area is variable. Suppose we now place the loop in a uniform magnetic field with the plane of the U perpendicular to the field. According to the rule, when the cross- bar is moved there should be in the loop an emf that IS proportional to the rate of change of the flux through the loop. This emf will cause a current in the loop. We Will assume that there is enough resrstance in the Wire that the currents are small. Then we can neglect any magnetic field from this current. The flux through the loop is wLB, so the “flux rule” would give for the emf—— which we write as 8— 8 = wBFLL = va, dt where v is the speed of translation of the crossbar. Now we should be able to understand this result from the magnetic v X B forces on the charges in the moving crossbar. These charges will feel a force, tangential to the wire. equal to DB per unit charge. It is constant along the length w of the crossbar and zero elsewhere, so the integral is 8 2 MB, which is the same result we got from the rate of change of the flux. The argument Just given can be extended to any case where there is a fixed magnetic field and the wires are moved. One can prove, in general, that for any Circuit whose parts move in a fixed magnetic field the emf is the time derivative of the flux, regardless of the shape of the Circuit. On the other hand, what happens if the loop is stationary and the magnetic field is changed? We cannot deduce the answer to this question from the same argument. It was Faraday’s discovery—from experiment—that the “flux rule” is still correct no matter why the flux changes. The force on electric charges is given in complete generality by F = q(E + v x B); there are no new speCial “forces due to changing magnetic fields.” Any forces on charges at rest in a stationary wire come from the E term. Faraday’s observations led to the discovery that electric and magnetic fields are related by a new law: in a region where the magnetic field is changing With time, electric fields are generated. It is this electric 17—1 17—1 17—2 17—3 17—4 17—5 17—6 17—7 17—8 The physics of induction Exceptions to the “flux rule” Particle acceleration by an induced electric field; the betatron A paradox Alternating—current generator Mutual inductance Self-inductance Inductance and magnetic energy . to) l . (b w i . ‘ ‘ ‘ V LINES OFB Fig. l7—l. An emf is induced in a loop if the flux is changed by varying the area of the circuit. field which drives the electrons around the wire—and so is responsible for the emf in a stationary circuit when there is a changing magnetic flux. The general law for the electric field associated with a changing magnetic field is 68 VXE———37- (17.1) We will call this Faraday’s law. It was discovered by Faraday but was first written in differential form by Maxwell, as one of his equations. Let’s see how this equation gives the “flux rule” for circuits. Using Stokes’ theorem, this law can be written in integral form as ng'ds=/ (VXE)-nda= —/ 9—B-nda, (17.2) I‘ S Sat where, as usual, I‘ is any closed curve and S is any surface bounded by it. Here, remember, I‘ is a mathematical curve fixed in space, and S is a fixed surface. Then the time derivative can be taken outs1de the integral and we have 6 flE ds —57/;B nda “562 (flux through S). (17.3) ll Applying this relation to a curve P that follows a fixed circuit of conductor, we get the “flux rule” once again. The integral on the left is the emf, and that on the right is the negative rate of change of the flux linked by the circuit. So Eq. (17.1) applied to a fixed circuit is equivalent to the “flux rule.” So the “flux rule”——that the emf in a circuit is equal to the rate of change of the magnetic flux through the circuit—applies whether the flux changes because the field changes or because the circuit moves (or both). The two pOSSibilities— “Circuit moves” or “field changes”~—are not distinguished in the statement of the rule. Yet in our explanation of the rule we have used two completely distinct laws for the two cases—v X B for “Circuit moves” and V X E = —63/61 for “field changes.” We know of no other place in physics where such a simple and accurate general prinCiple requires for its real understanding an analysis in terms of two diflerent phenomena. Usually such a beautiful generalization is found to stem from a Single deep underlying principle. Nevertheless, in this case there does not appear to be any such profound implication. We have to understand the “rule” as the combined effects of two quite separate phenomena. We must look at the “flux rule” in the following way. In general, the force per unit charge is F/q = E + v x B. In moving wires there is the force from the second term. Also, there is an E-field if there is somewhere a changing magnetic field. They are independent effects, but the emf around the loop of wire is always equal to the rate of change of magnetic flux through it. 17—2 Exceptions to the “flux rule” We will now give some examples, due in part to Faraday, which show the importance of keeping clearly in mind the distinction between the two effects re- sponsible for induced emf‘s. Our examples involve Situations to which the “flux rule” cannot be applied—either because there is no Wire at all or because the path taken by induced currents moves about Within an extended volume of a conductor. We begin by making an important point: The part of the emf that comes from the E-field does not depend on the ex1stence of a phySical wire (as does the v X B part). The E—field can exist in free space, and its line integral around any imaginary line fixed in space is the rate of change of the flux of B through that line. (Note that this is quite unlike the E—field produced by static charges, for in that case the line integral of E around a closed loop is always zero.) 1 7—2 BAR MAGNET Now we will describe a Situation in which the flux through a circuit does not change, but there is nevertheless an emf. Figure 17—2 shows a conducting disc which can be rotated on a fixed axis in the presence of a magnetic field. One contact is made to the shaft and another rubs on the outer periphery of the disc. A circu1t is completed through a galvanometer. As the disc rotates, the “circuit,” in the sense of the place in space where the currents are, is always the same. But the part of the “circuit” in the disc is in material which is mOVing. Although the flux through the “circuit” is constant, there is still an emf, as can be observed by the deflection ofthe galvanometer. Clearly, here is a case where the v x B force in the movmg disc gives rise to an emf which cannot be equated to a change of flux. Now we consider. as an oppos1te example, a somewhat unusual situation in which the flux through a “Circuit” (again in the sense ofthe place where the current 15) changes but where there is no emf. Imagine two metal plates with slightly curved edges, as shown in Fig. 17—3, placed in a uniform magnetic field perpendicular to their surfaces. Each plate is connected to one of the terminals of a galvanometer, as shown. The plates make contact at one point P. so there is a complete CirCUit If the plates are now rocked through a small angle, the point of contact will move to P’. If we imagine the “Circuit” to be completed through the plates on the dotted line shown in the figure, the magnetic flux through this Circu1t changes by a large amount as the plates are rocked back and forth. Yet the rocking can be done With small motions, so that v X B is very small and there is practically no emf. The “flux rule” does not work in this case. It must be applied to Circuits in which the material of the Circuit remains the same. When the material of the Circuit is chang— ing, we must return to the basic laws. The correct physics is always given by the two basic laws F=q(E+v><B), 6 v><E_——5- 17—3 Particle acceleration by an induced electric field; the betatron We have said that the electromotive force generated by a changing magnetic field can ex1st even without conductors; that is, there can be magnetic induction without Wires. We may still imagine an electromotive force around an arbitrary mathematical curve in space. It is defined as the tangential component of E integrated around the curve. Faraday’s law says that this line integral is equal to the rate of change of the magnetic flux through the Closed curve, Eq. (17.3). As an example of the effect of such an induced electric field, we want now to conSider the motion of an electron in a changing magnetic field. We imagine a magnetic field which, everywhere on a plane, points in a vertical direction, as shown in Fig. 17—4. The magnetic field is produced by an electromagnet, but we Will not worry about the details For our example we Will imagine that the magnetic field is symmetric about some axis, ie., that the strength of the magnetic field Will depend only on the distance from the axis. The magnetic field is also varying With time We now imagine an electron that IS movmg in this field on a path that is a circle of constant radius with its center at the axis of the field. (We Will see later 17—3 Fig. l7-—2. When the disc rotates there is an emf from v X B, but with GALVANOMETER no change in the linked flux. GALVANOMETER Fig. 17—3. When the plates are rocked in a uniform magnetic field, there can be a large change in the flux linkage without the generation of an emf. SIDE VIEW LINES OF 8 TOP VIEW Fig. 17-4. An electron accelerating in an axially symmetric, time-varying magnetic field. how this motion can be arranged.) Because of the changing magnetic field, there will be an electric field E tangential to the electron’s orbit which will drive it around the circle. Because of the symmetry, this electric field will have the same value everywhere on the circle. If the electron’s orbit has the radius r, the line integral of E around the orbit is equal to the rate of change of the magnetic flux through the circle. The line integral of E is Just its magnitude times the circumference of the circle, 27rr. The magnetic flux must, in general, be obtained from an integral. For the moment, we let BM, represent the average magnetic field in the interior of the circle; then the flux is this average magnetic field times the area of the circle. We Will have _ a o 2 27rrE — at (Bflv 7rr ). Since we are assuming r is constant, E 18 proportional to the time derivative of the average field: rdBav E — 5 dt (17.4) The electron will feel the electric force qE and Wlll be accelerated by it. Remember- ing that the relativistically correct equation of motion is that the rate of change of the momentum is proportional to the force, we have 45 : _. (17.5) For the c1rcular orbit we have assumed, the electric force on the electron is always in the direction of its motion, so its total momentum will be increasmg at the rate given by Eq. (17.5). Combining Eqs. (17.5) and (17.4), we may relate the rate of change of momentum to the change of the average magnetic field: dp qr dBa ~ = w V. 7. dt 2 dt (1 6) Integrating with respect to t, we find for the electron’s momentum where p0 is the momentum With which the electrons start out, and ABM, is the sub- sequent change in Ba... The operation of a betatron—a machine for accelerating electrons to high energies—is based on this idea. To see how the betatron operates in detail, we must now examine how the electron can be constrained to move on a Circle. We have discussed in Chapter 11 of Vol. I the principle involved. If we arrange that there is a magnetic field B at the orbit of the electron, there will be a transverse force qv X B which, for a suit- 17—4 ably chosen B, can cause the electron to keep movmg on its assumed orbit. In the betatron this transverse force causes the electron to move in a Circular orbit of constant radius. We can find out what the magnetic field at the orbit must be by using again the relatiVistic equation of motion, but this time, for the transverse component of the force. In the betatron (see Fig 17—4), B is at right angles to u, so the transverse force is qu. Thus the force is equal to the rate of change of the trans- verse component pt of the momentum: _ div» qu —— d1 (17.8) When a particle is movmg in a circle, the rate of change of its transverse momentum is equal to the magnitude of the total momentum times to, the angular velocity of rotation (followmg the arguments of Chapter 11, Vol. I): dpt _ dt — cop, (17.9) where, Since the motion is circular, or = g (17.10) Setting the magnetic force equal to the transverse acceleration, we have u qUBorbit = P ;’ (17.11) where Borblt is the field at the radius r. As the betatron operates, the momentum of the electron grows in proportion to B“, according to Eq. (17.7), and if the electron is to continue to move in its proper Circle, Eq. (17.11) must continue to hold as the momentum of the electron increases. The value of Borb1t must increase in proportion to the momentum p. Comparing Eq. (17.11) with Eq. (17.7), which determines p, we see that the follow- ing relation must hold between B.,,,, the average magnetic field Inside the orbit at the radius r, and the magnetic field Borblt at the orbit: ABav : 2ABorbit- (17-12) The correct operation of a betatron requires that the average magnetic field lnSldC the orbit increase at tw1ce the rate of the magnetic field at the orbit itself. In these Circumstances, as the energy of the particle 15 increased by the induced electric field the magnetic field at the orbit increases at just the rate required to keep the particle mOVing in a circle. The betatron is used to accelerate electrons to energies of tens of millions of volts, or even to hundreds of millions of volts. However, it becomes impractical for the acceleration of electrons to energies much higher than a few hundred million volts for several reasons. One of them is the practical difiiculty of attaining the required high average value for the magnetic field inside the orbit. Another is that Eq. (17.6) 18 no longer correct at very high energies because it does not include the loss of energy from the particle due to its radiation of electromagnetic energy (the so-called synchrotron radiation discussed in Chapter 36, Vol. I). For these reasons, the acceleration of electrons to the highest energies—to many billions of electron voltsfiis accomplished by means of a different kind of machine, called a synchrotron. 17—4 A paradox We would now like to describe for you an apparent paradox. A paradox is a Situation which gives one answer when analyzed one way, and a different answer when analyzed another way, so that we are left in somewhat of a quandary as to actually what should happen. Of course, in phySics there are never any real para- doxes because there is only one correct answer; at least we believe that nature Will 1 7—5 CHARGED METAL SPHERES COIL OF WIRE PLASTIC DISC Fig. l7—5. Will the disc rotate if the current I is stopped? Fig. 17—6. A coil of wire rotating in a uniform magnetic field—the basic idea of the ac generator. act in only one way (and that is the right way, naturally). So in phySics a paradox is only a confusion in our own understanding. Here is our paradox. Imagine that we construct a deVice like that shown in Fig. 17—5. There IS a thin, Circular plastic disc supported on a concentric shaft With excellent bearings, so that it is quite free to rotate. On the disc is a 0011 of wire in the form of a short solenoid concentric With the aXis of rotation. This solenoid carries a steady current I provided by a small battery, also mounted on the disc. Near the edge of the disc and spaced uniformly around its circumference are a number of small metal spheres insulated from each other and from the solenOid by the plastic material of the disc. Each of these small conducting spheres is charged with the same electrostatic charge Q. Everything is quite stationary, and the disc is at rest. Suppose now that by some accident—or by prearrangement—the current in the solenOid is inter- rupted, Without, however, any intervention from the outside. So long as the current continued, there was a magnetic flux through the solenOid more or less parallel to the ax1s of the disc. When the current is interrupted. this flux must go to zero. There will, therefore, be an electric field induced which Will Circulate around in circles centered at the axis. The charged spheres on the perimeter of the disc Will all experience an electric field tangential to the perimeter of the disc. This electric force is in the same sense for all the charges and so Will result in a net torque on the disc. From these arguments we would expect that as the current in the solenOid disappears, the disc would begin to rotate. If we knew the moment of inertia of the disc, the current in the solenOId, and the charges on the small spheres, we could compute the resulting angular velocity. But We could also make a different argument. Usmg the principle of the con- servation of angular momentum, we could say that the angular momentum of the disc With all its equipment is initially zero, and so the angular momentum of the assembly should remain zero. There should be no rotation when the current is stopped. Which argument is correct" Will the disc rotate or Will it not” We Will leave this question for you to think about. We should warn you that the correct answer does not depend on any non- essential feature, such as the asymmetric pos1tion of a battery, for example. In fact, you can imagine an ideal situation such as the followmg‘ The solenOid is made of superconducting Wire through which there is a current. After the disc has been carefully placed at rest. the temperature ofthe solenOid is allowed to rise slowly When the temperature of the Wire reaches the tranSition temperature between superconductiVity and normal conductiVity, the current in the solenOid Will be brought to zero by the re51stance of the Wire. The flux will, as before, fall to zero, and there Will be an electric field around the fins. We should also warn you that the solution 18 not easy, nor is it a trick. When you figure it out, you Will have dis- covered an important principle of electromagnetism. 17—5 Alternating-current generator In the remainder of this chapter we apply the prinCiples of Section I7—l to analyze a number ofthe phenomena discussed in Chapter 16. We first look in more detail at the alternating-current generator. Such a generator consists basically of at cm] of Wire rotating in a uniform magnetic field. The same result can also be achieved by a fixed (2011 in a magnetic field whose direction rotates in the manner described in the last chapter. We Will consider only the former case. Suppose we have a Circular 0011 of Wire which can be turned on an ax1s along one of its diam- eters. Let this 0011 be located in a uniform magnetic field perpendicular to the sum of ‘rotation, as in Fig. 17—6 We a...
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