Electricity and Magnetism

Edward M. Purcell

Chapter 7

Electromagnetic Induction - all with Video Answers

Educators

Chapter Questions

Problem 1

What is the maximum electromotive force induced in a coil of 4000 turns, average radius $12 \mathrm{~cm}$, rotating at 30 revolutions per sec in the earth's magnetic field where the field intensity is $0.5$ gauss?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 2

A long straight wire is parallel to the $y$ axis and passes through the point $z=h$ on the $z$ axis. A current $I$ flows in this wire, returning by a remote conductor whose field we may neglect. Lying in the $x y$ plane is a square loop with two of its sides, of length $b$, parallel to the long wire. This loop slides with constant speed $v$ in the $\hat{\mathbf{x}}$ direction. Find the magnitude of the electromotive force induced in the loop at the moment when the center of the loop crosses the $y$ axis.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 3

In the central region of a solenoid which is connected to a radiofrequency power source, the magnetic field oscillates at $2.5 \times$ $10^{6}$ cycles per sec with an amplitude of 4 gauss. What is the amplitude of the oscillating electric field, in statvolts $/ \mathrm{cm}$, at a point $3 \mathrm{~cm}$ from the axis? (This point lies within the region where the magnetic field is nearly uniform.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 4

Calculate the electromotive force in the moving loop in the figure at the instant when it is in the position there shown. Assume the resistance of the loop is so great that the effect of the current in the loop itself is negligible. Estimate very roughly how large a resistance would be safe, in this respect. Indicate the direction in which current would flow in the loop, at the instant shown.

Dominador Tan

Numerade Educator

Problem 5

Suppose the loop in Fig. $7.6$ has a resistance $R$. Show that whoever is pulling the loop along at constant speed does an amount of work during the interval $d t$ which agrees precisely with the energy dissipated in the resistance during this interval, providing the selfinductance of the loop can be neglected. What is the source of the energy in Fig. $7.14$ where the loop is stationary?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 6

Does the prediction of a simple sinusoidal variation of electromotive force for the rotating loop in Fig. $7.13$ depend on the loop being rectangular, on the magnetic field being uniform, or on both? Explain. Can you suggest an arrangement of rotating loop and stationary coils which will give a definitely nonsinusoidal emf? Sketch the voltagetime curve you would expect to see on the oscilloscope, with that arrangement.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 7

Calculate the self-inductance of a cylindrical solenoid $10 \mathrm{~cm}$ in diameter and 2 meters long. It has a single-layer winding containing a total of 1200 turns. Assume as a first approximation that the magnetic field inside the solenoid is uniform right out to the ends. Estimate roughly the magnitude of the error you will thereby incur. Is the true $L$ larger or smaller than your approximate result?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 8

How could we wind a resistance coil so that its self-inductance would be small?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 9

Derive an approximate formula for the mutual inductance of two circular rings of the same radius $a$, arranged like wheels on the same axle with their centers $b \mathrm{~cm}$ apart. Use an approximation good for $b \gg a$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 10

The coils which first produced a slight but detectable kick in Faraday's galvanometer he describes as made of 203 feet of copper wire each, wound around a large block of wood. The turns of the second spiral (that is, single-layer coil) were interposed between those of the first, but separated from them by twine. The diameter of the copper wire itself was $1 / 20$ inch. He does not give the dimensions of the wooden block or the number of turns in the coils. In the experiment, one of these coils was connected to a "battery of 100 plates." See if you can make a rough estimate of the duration in seconds and magnitude in amperes of the pulse of current that passed through his galvanometer.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 11

Part $(a)$ of the figure shows two coils with self-inductances $L_{1}$ and $L_{2}$. In the relative position shown their mutual inductance is $M$. The positive current direction and the positive electromotive force direction in each coil are defined by the arrows in the figure. The equations relating currents and electromotive forces are
$$\mathscr{E}_{1}=-L_{1} \frac{d I_{1}}{d t} \pm M \frac{d I_{2}}{d t} \quad \text { and } \quad \mathscr{E}_{2}=-L_{2} \frac{d I_{2}}{d t} \pm M \frac{d I_{1}}{d t}$$
Given that $M$ is always to be taken as a positive constant, how must the signs be chosen in these equations? What if we had chosen, as we might have, the other direction for positive current, and for positive electromotive force, in the lower coil? Now connect the two coils together as in part $(b)$ of the figure to form a single circuit. What is the inductance $L^{\prime}$ of this circuit, expressed in terms of $L_{1}, L_{2}$, and $M$ ? What is the inductance $L^{\prime \prime}$ of the circuit formed by connecting the coils as shown in $(c)$ ? Which circuit, $(b)$ or $(c)$, has the greater selfinductance? Considering that the self-inductance of any circuit must be a positive quantity (why couldn't it be negative?), see if you can draw a general conclusion, valid for any conceivable pair of coils, concerning the relative magnitude of $L_{1}, L_{2}$, and $M$.

Vishal Gupta

Numerade Educator

Problem 12

An ocean current flows at a speed of 2 knots (approximately 1 meter $/ \mathrm{sec}$ ) in a region where the vertical component of the earth's magnetic field is $0.35$ gauss. The conductivity of seawater in that region is $0.04(\mathrm{ohm}-\mathrm{cm})^{-1}$. On the assumption that there is no other horizontal component of $\mathbf{E}$ than the motional term $(\mathbf{v} / c) \times \mathbf{B}$, find the density of horizontal electric current in $\mathrm{amps} / \mathrm{m}^{2}$. If you were to carry a bottle of seawater through the earth's field at this speed, would such a current be flowing in it?

Mahnoor Amin

Numerade Educator

Problem 13

A coil with resistance of $0.01$ ohm and self-inductance $0.50$ millihenry is connected across a large 12 -volt battery of negligible internal resistance. How long after the switch is closed will the current reach 90 percent of its final value? At that time, how much energy, in joules, is stored in the magnetic field? How much energy has been withdrawn from the battery up to that time?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 14

A metal crossbar of mass $m$ slides without friction on two long parallel conducting rails a distance $b$ apart. A resistor $R$ is connected across the rails at one end; compared with $R$, the resistance of bar and rails is negligible. There is a uniform field $\mathbf{B}$ perpendicular to the plane of the figure. At time $t=0$ the crossbar is given a velocity $v_{0}$ toward the right. What happens then?
(a) Does the rod ever stop moving? If so when?
(b) How far does it go?
( $c$ ) How about conservation of energy?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 15

A taut wire passes through the gap of a small magnet, where the field strength is 5000 gauss. The length of wire within the gap is $1.8 \mathrm{~cm} .$ Calculate the amplitude of the induced alternating voltage when the wire is vibrating at its fundamental frequency of $2000 \mathrm{~Hz}$ with an amplitude of $0.03 \mathrm{~cm}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 16

The shaded region represents the pole of an electromagnet where there is a strong magnetic field perpendicular to the plane of the paper. The rectangular frame is made of a 5-mm-diameter aluminum rod, bent and with its ends welded together. Suppose that by applying a steady force of 1 newton, starting at the position shown, the frame can be pulled out of the magnet in 1 sec. Then: If the force is doubled, to 2 newtons, the frame will be pulled out in __________ sec. Brass has about twice the resistivity of aluminum. If the frame had been made of a $5-\mathrm{mm}$ brass rod, the force needed to pull it out in 1 sec would be __________ newtons. If the frame had been made newtons. If the frame had been made of a 1-cm-diameter aluminum rod, the force required to pull it out in 1 sec would be __________ newtons. You may neglect in all cases the inertia of the frame.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 17

In the circuit shown in the diagram the 10 -volt battery has negligible internal resistance. The switch $S$ is closed for several seconds, then opened. Make a graph with the abscissa time in milliseconds, showing the potential of point $A$ with respect to ground, just before and then for 10 milliseconds after the opening of switch $S$. Show also the variation of the potential at point $B$ in the same period of time.

Khaled Yasein

Numerade Educator

Problem 18

A circular coil of wire, with $N$ turns of radius $a$, is located in the field of an electromagnet. The magnetic field is perpendicular to the coil, and its strength has the constant value $B_{0}$ over that area. The coil is connected by a pair of twisted leads to an external resistance. The totai resistance of this closed circuit, including that of the coil itself, is $R$. Suppose the electromagnet is turned off, its field dropping more or less rapidly to zero. The induced electromotive force causes current to flow around the circuit. Derive a formula for the total charge $Q=\int I d t$ which passes through the resistor, and explain why it does not depend on the rapidity with which the field drops to zero.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 19

Discuss the implications of the theorem $\Phi_{12}=\Phi_{21}$ in the case of the large and small concentric rings in Fig. $7.20$. With fixed current $I_{1}$ in the outer ring, obviously $\Phi_{21}$, the flux through the inner ring, decreases if $R_{1}$ is increased, simply because the field at the center gets weaker. But with fixed current in the inner ring, why should $\Phi_{12}$, the flux through the outer ring, decrease as $R_{1}$ increases, holding $R_{2}$ constant? It must do so to satisfy our theorem.

Mahnoor Amin

Numerade Educator

Problem 20

Can you devise a way to use the theorem $\Phi_{21}=\Phi_{12}$ to find the magnetic field strength due to a ring current at points in the plane of the ring at a distance from the ring much greater than the ring radius? (Hint: Consider the effect of a small change $\Delta R_{1}$ in the radius of the outer ring in Fig. $7.20$; it must have the same effect on $\Phi_{12}$ as on $\Phi_{21} .$ )

Keshav Singh

Numerade Educator

Problem 21

The figure shows a solenoid of radius $a_{1}$ and length $b_{1}$ located inside a longer solenoid of radius $a_{2}$ and length $b_{2}$. The total number of turns is $N_{1}$ on the inner coil, $N_{2}$ on the outer. Work out a formula for the mutual inductance $M$.

Vishal Gupta

Numerade Educator

Problem 22

A thin ring of radius $a$ carries a static charge $q$. This ring is in a magnetic field of strength $B_{0}$, parallel to the ring's axis, and is supported so that it is free to rotate about that axis. If the field is switched off, how much angular momentum will be added to the ring? Supposing the mass of the ring to be $m$, show that the ring, if initially at rest, will acquire an angular velocity $\omega=q B_{0} / 2 m c$. Notice that, as in Problem $7.18$, the result depends only on the initial and final values of the field strength, and not the rapidity of change.

Mahnoor Amin

Numerade Educator

Problem 23

A magnetic field exists in most of the interstellar space in our galaxy. There is evidence that its strength in most regions is between $10^{-6}$ and $10^{-5}$ gauss. Adopting $3 \times 10^{-6}$ gauss as a typical value, find, in order of magnitude, the total energy stored in the magnetic field of the galaxy. For this purpose you may assume the galaxy is a disk roughly $10^{23} \mathrm{~cm}$ in diameter and $10^{21} \mathrm{~cm}$ thick. To see whether the magnetic energy amounts to much, on that scale, you might consider the fact that all the stars in the galaxy are radiating about $10^{44}$ ergs $/$ sec. How many years of starlight is the magnetic energy worth?

Mahnoor Amin

Numerade Educator

Problem 24

A superconducting solenoid designed for whole-body imaging by nuclear magnetic resonance is $0.9$ meters in diameter and $2.2$ meters long. The field at its center is $0.4$ tesla. Estimate roughly the energy stored in the field of this coil, in joules.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 25

It has been estimated that the magnetic field strength at the surface of a neutron star, or pulsar, may be as high as $10^{12}$ gauss. What is the energy density in such a field? Express it, using the massenergy equivalence, in grams per $\mathrm{cm}^{3}$.

Mahnoor Amin

Numerade Educator

Problem 26

Faraday describes in the following words an unsuccessful attempt to detect a current induced when part of a circuit consists of water moving through the earth's magnetic field:
I made experiments therefore (by favour) at Waterloo Bridge, extending a copper wire nine hundred and sixty feet in length upon the parapet of the bridge, and dropping from its extremities other wires with extensive plates of metal attached to them to complete contact with the water. Thus the wire and the water made one conducting circuit; and as the water ebbed or flowed with the tide, I hoped to obtain currents analogous to those of the brass ball. I constantly obtained deflections at the galvanometer, but they were irregular, and were, in succession, referred to other causes than that sought for. The different condition of the water as to purity on the two sides of the river; the difference in temperature; slight differences in the plates, in the solder used, in the more or less perfect contact made by twisting or otherwise; all produced effects in turn: and though I experimented on the water passing through the middle arches only; used platina plates instead of copper; and took every other precaution, I could not after three days obtain any satisfactory results. ("Experimental Researches in Electricity," vol. I, London, $1839, \mathrm{p} .55 .$ )
Assume the vertical component of the field was $0.5$ gauss, make a reasonable guess about the velocity of tidal currents in the Thames and estimate the magnitude of the induced voltage Faraday was trying to detect.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 27

We can think of a voltmeter as a device which registers the line integral $\int \mathbf{E} \cdot d \mathbf{s}$ along a path $C$ from the clip at the end of its $(+)$ lead to the clip at the end of its (-) lead. Part of $C$ lies inside the voltmeter itself. Path $C$ may also be part of a loop which is completed by some external path from the $(-)$ clip to the $(+)$ clip. With that in mind, consider the arrangement in the figure. The solenoid is so long that its external magnetic field is negligible. Its cross section is $20 \mathrm{~cm}^{2}$ in area, and the field inside is toward the right and increasing at the rate of 100 gauss/sec. Two identical voltmeters are connected as shown to points on the loop which encloses the solenoid and contains the two 50 -ohm resistors. The voltmeters are capable of reading microvolts and have high internal resistance. What will each voltmeter read? Make sure your answer is consistent, from every point of view, with Eq. 25 .

Dominador Tan

Numerade Educator

Problem 28

Magnetic fields inside good conductors cannot change quickly. We found that current in a simple inductive circuit decays exponentially with characteristic time $L / R$ (Eq. 66). In a large conducting body such as the metallic core of the earth the "circuit" is not easy to identify. Nevertheless, we can find the order of magnitude of the decay time, and what it depends on, by making some reasonable approximations. Consider the solid doughnut of square cross section made of material with conductivity $\sigma$, in $\sec ^{-1}$. A current $I$ flows around it. Of course $I$ is spread out in some manner over the cross section, but we shall assume the resistance is that of a wire of area $a^{2}$ and length $\pi a$, that is, $R \approx \pi / a \sigma$. For the field $B$ we adopt the field at the center of a ring with current $I$ and radius $a / 2$. For the stored energy $U$ a reasonable estimate would be $B^{2} / 8 \pi$ times the volume of the doughnut. Since $d U / d t=-I^{2} R$, the decay time of the energy $U$ will be $\tau \approx U / I^{2} R$. Show that, except for some numerical factor depending on our various approximations, $\tau \approx \sigma a^{2} / c^{2}$. The radius of the earth's core is $3000 \mathrm{~km}$, and its conductivity is believed to be $10^{16}$ $\sec ^{-1}$, roughly one-tenth that of iron at room temperature. Evaluate $\tau$ in centuries.

Dominador Tan

Numerade Educator

Problem 29

The constant $c$ which turns up in Maxwell's equations can be determined by electrical experiments involving low-frequency fields only. Consider the arrangement shown in the figure. The force between capacitor plates is balanced against the force between parallel wires carrying current in the same direction. A voltage alternating sinusoidally at a frequency $f$ cycles per sec is applied to the parallel-plate capacitor $C_{1}$ and also to the capacitor $C_{2}$. The charge flowing into and out of $C_{2}$ constitutes the current in the rings. Suppose that $C_{2}$ and the various distances involved have been adjusted so that the time-average downward force on the upper plate of $C_{1}$ exactly balances the time-average downward force on the upper ring. (Of course, the weights of the two sides should be adjusted to balance with the voltage turned off.) Show that under these conditions the constant $c$ can be computed from measured quantities as follows:
$$c=(2 \pi)^{3 / 2} a\left(\frac{b}{h}\right)^{1 / 2}\left(\frac{C_{2}}{C_{1}}\right) f \quad(\mathrm{~cm} / \mathrm{sec})$$
Note that only measurements of distance and time (or frequency) are required, apart from a measurement of the ratio of the two capacitances $C_{1}$ and $C_{2}$. Electrical units, as such, are not involved in the result. (The experiment is actually feasible at a frequency as low as 60 cycles $/ \mathrm{sec}$ if $C_{2}$ is made, say, $10^{6}$ times $C_{1}$ and the current rings are made with several turns to multiply the effect of a small current.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 30

Consider the loop of wire shown in the figure. Suppose we want to calculate the flux of $\mathbf{B}$ through this loop. Two surfaces bounded by the loop are shown, in parts $(a)$ and $(b)$, respectively.
What is the essential difference between them? Which, if either, is the correct surface to use in performing the surface integral $\int \mathbf{B} \cdot$ da to find the flux? Describe the corresponding surface for a three-turn coil. Show that this is all consistent with our previous assertion that for a compact coil of $N$ turns the electromotive force is just $N$ times what it would be for a single loop of the same size and shape.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 31

In this question the term dynamo will be used for a generator which works in the following way. By some external agency-the shaft of a steam turbine, for instance- a conductor is driven through a magnetic field, inducing an electromotive force in a circuit of which that conductor is part. The source of the magnetic field is the current which is caused to flow in that circuit by that electromotive force. An electrical engineer would call it a self-excited dc generator. One of the simplest dynamos conceivable is sketched below. It has only two essential parts. One part is a solid metal disk and axle which can be driven in rotation. The other is a two-turn "coil" which is stationary but is connected by sliding contacts, or "brushes," to the axle and to the rim of the revolving disk. One of the two devices pictured is, at least potentially, a dynamo. The other is not. Which is the dynamo? Note that the answer to this question cannot depend on any convention about handedness or current directions. An intelligent extraterrestrial being inspecting the sketches could give the answer, provided only that it knows about arrows! What do you think determines the direction of the current in such a dynamo? What will determine the magnitude of the current?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 32

A dynamo like the one in the preceding problem has a certain critical speed $\omega_{0}$. If the disk revolves with an angular velocity less than $\omega_{0}$, nothing happens. Only when that speed is attained is the induced $E$ large enough to make the current large enough to make the magnetic field large enough to induce an $\mathscr{E}$ of that magnitude. The critical speed can only depend on the size and shape of the conductors and the conductivity $\sigma$. Remember that $\sigma$ has the dimensions $\mathrm{sec}^{-1}$. Let $d$ be some characteristic dimension expressing the size of the dynamo, such as the radius of the disk in our example. Show by a dimensional argument that $\omega_{0}$ must be given by a relation of this form: $\omega_{0}=$ $K c^{2} / d^{2} \sigma$, where $K$ is some dimensionless numerical factor that depends only on the arrangement and relative size of the various parts of the dynamo. For a dynamo of modest size made wholly of copper, the critical speed $\omega_{0}$ would be practically unattainable. It is ferromagnetism that makes possible the ordinary dc generator by providing a magnetic field much stronger than the current in the coils, unaided, could produce. For an earth-sized dynamo, however, with $d$ measured in hundreds of kilometers rather than meters, the critical speed is very much smaller. The earth's magnetic field is almost certainly produced by a nonferromagnetic dynamo involving motions in the fluid metallic core. That fluid happens to be molten iron, but it is not even slightly ferromagnetic because it is too hot. (That will be explained in Chapter
11.) We don't know how the conducting fluid moves, or what configuration of electric currents and magnetic fields its motion generates in the core. The magnetic field we observe at the earth's surface is the external field of the dynamo in the core. The direction of the earth's field a million years ago is preserved in the magnetization of rocks that solidified at that time. That magnetic record shows that the field has reversed its direction nearly 200 times in the last 100 million years. Although a reversal cannot have been instantaneous (see Problem 7.28), it was a relatively sudden event on the geological time scale. The immense value of paleomagnetism as an indelible record of our planet's history is well explained in Chapter 18 of Earth, by Frank Press and Raymond Siever, second edition, 1978 (W. H. Freeman).

Lainey Roebuck

Numerade Educator