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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 Ushaped 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
Selfinductance 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—Bnda, (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 Eﬁeld 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 Eﬁeld 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. 174. An electron accelerating in an axially
symmetric, timevarying 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 (1712) 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 socalled 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 Alternatingcurrent 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 alternatingcurrent 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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