A square loop, side a, resistance R, lies a distance s

Problem 2P A capacitor C has been charged up to potential V0; at time t = 0, it is connected to a resistor R, and begins to discharge (Fig. 7.5a). (a) Determine the charge on the capacitor as a function of time, Q(t). What is the current through the resistor, I (t)? (b) What was the original energy stored in the capacitor (Eq. 2.55)? By integrating Eq. 7.7, confirm that the heat delivered to the resistor is equal to the energy lost by the capacitor. Now imagine charging up the capacitor, by connecting it (and the resistor) to a battery of voltage V0, at time t = 0 (Fig. 7.5b). (c) Again, determine Q(t) and I (t). (d) Find the total energy output of the battery (? V0I dt). Determine the heat delivered to the resistor. What is the final energy stored in the capacitor? What fraction of the work done by the battery shows up as energy in the capacitor? [Notice that the answer is independent of R!] Equation 7.7 Figure 7.5

Problem 3P (a) Two metal objects are embedded in weakly conducting material of conductivity ? (Fig. 7.6). Show that the resistance between them is related to the capacitance of the arrangement by (b) Suppose you connected a battery between 1 and 2, and charged them up to a potential difference V0. If you then disconnect the battery, the charge will gradually leak off. Show that and find the time constant, ?, in terms of ?0 and ?.

Problem 64P (a) Show that Maxwell’s equations with magnetic charge (Eq. 7.44) are invariant under the duality transformation where is an arbitrary rotation angle in “E/B-space.” Charge and current densities transform in the same way as qe and qm. particular, that if you know the fields produced by a configuration of electric charge, you can immediately (using ? = 90?) write down the fields produced by the corresponding arrangement of magnetic charge.] (b) Show that the force law (Prob. 7.38) is also invariant under the duality transformation. Problem 7.38 Assuming that “Coulomb’s law” for magnetic charges (qm) reads work out the force law for a monopole qm moving with velocity v through electric and magnetic fields E and B.26

Problem 1P Two concentric metal spherical shells, of radius a and b, respectively, are separated by weakly conducting material of conductivity ? (Fig. 7.4a). (a) If they are maintained at a potential difference V, what current flows from one to the other? (b) What is the resistance between the shells? (c) Notice that if b? a the outer radius (b) is irrelevant. How do you account for that? Exploit this observation to determine the current flowing between two metal spheres, each of radius a, immersed deep in the sea and held quite far apart (Fig. 7.4b), if the potential difference between them is V. (This arrangement can be used to measure the conductivity of sea water.) Figure 7.4

Problem 4P Suppose the conductivity of the material separating the cylinders in Ex. 7.2 is not uniform; specifically, ?(s) = k/s, for some constant k. Find the resistance between the cylinders. [Hint: Because ? is a function of position, Eq. 7.5 does not hold, the charge density is not zero in the resistive medium, and E does not go like 1/s. But we do know that for steady currents I is the same across each cylindrical surface. Take it from there.]

Problem 5P A battery of emf ? and internal resistance r is hooked up to a variable “load” resistance, R. If you want to deliver the maximum possible power to the load, what resistance R should you choose? (You can’t change E and r , of course.)

Problem 6P A rectangular loop of wire is situated so that one end (height h) is between the plates of a parallel-plate capacitor (Fig. 7.9), oriented parallel to the field E. The other end is way outside, where the field is essentially zero. What is the emf in this loop? If the total resistance is R, what current flows? Explain. [Warning: This is a trick question, so be careful; if you have invented a perpetual motion machine, there’s probably something wrong with it.]

Problem 7P Ametal bar of mass m slides frictionlessly on two parallel conducting rails a distance l apart (Fig. 7.17). A resistor R is connected across the rails, and a uniform magnetic field B, pointing into the page, fills the entire region. (a) If the bar moves to the right at speed v, what is the current in the resistor? In what direction does it flow? (b) What is the magnetic force on the bar? In what direction? (c) If the bar starts out with speed v0 at time t = 0, and is left to slide, what is its speed at a later time t? (d) The initial kinetic energy of the bar was, of course,

Problem 11P A square loop is cut out of a thick sheet of aluminum. It is then placed so that the top portion is in a uniform magnetic field B, and is allowed to fall under gravity (Fig. 7.20). (In the diagram, shading indicates the field region; B points into the page.) If the magnetic field is 1 T (a pretty standard laboratory field), find the terminal velocity of the loop (in m/s). Find the velocity of the loop as a function of time. How long does it take (in seconds) to reach, say, 90% of the terminal velocity? What would happen if you cut a tiny slit in the ring, breaking the circuit? [Note: The dimensions of the loop cancel out; determine the actual numbers, in the units indicated.]

Problem 9P An infinite number of different surfaces can be fit to a given boundary line, and yet, in defining the magnetic flux through a loop, I never specified the particular surface to be used. Justify this apparent oversight.

Problem 10P A square loop (side a) is mounted on a vertical shaft and rotated at angular velocity ? (Fig. 7.19). A uniform magnetic field B points to the right. Find the E(t) for this alternating current generator.

Problem 13P A square loop of wire, with sides of length a, lies in the first quadrant of the xy plane, with one corner at the origin. In this region, there is a nonuniform time-dependent magnetic field (where k is a constant). Find the emf induced in the loop.

Problem 12P A long solenoid, of radius a, is driven by an alternating current, so that the field inside is sinusoidal: A circular loop of wire, of radius a/2 and resistance R, is placed inside the solenoid, and coaxial with it. Find the current induced in the loop, as a function of time.

Problem 14P As a lecture demonstration a short cylindrical bar magnet is dropped down a vertical aluminum pipe of slightly larger diameter, about 2 meters long. It takes several seconds to emerge at the bottom, whereas an otherwise identical piece of unmagnetized iron makes the trip in a fraction of a second. Explain why the magnet falls more slowly.

Problem 17P A long solenoid of radius a, carrying n turns per unit length, is looped by a wire with resistance R, as shown in Fig. 7.28. (a) If the current in the solenoid is increasing at a constant rate (d I/dt = k), what current flows in the loop, and which way (left or right) does it pass through the resistor? (b) If the current I in the solenoid is constant but the solenoid is pulled out of the loop (toward the left, to a place far from the loop), what total charge passes through the resistor?

Problem 15P long solenoid with radius a and n turns per unit length carries a time-dependent current I (t) in the direction. Find the electric field (magnitude and direction) at a distance s from the axis (both inside and outside the solenoid), in the quasistatic approximation.

Problem 20P Imagine a uniform magnetic field, pointing in the z direction and filling all space A positive charge is at rest, at the origin. Now somebody turns off the magnetic field, thereby inducing an electric field. In what direction does the charge move?16

Problem 18P A square loop, side a, resistance R, lies a distance s from an infinite straight wire that carries current I (Fig. 7.29). Now someone cuts the wire, so I drops to zero. In what direction does the induced current in the square loop flow, and what total charge passes a given point in the loop during the time this current flows? If you don’t like the scissors model, turn the current down gradually:

Problem 16P An alternating current I = I0 cos (?t) flows down a long straight wire, and returns along a coaxial conducting tube of radius a. (a) In what direction does the induced electric field point (radial, circumferential, or longitudinal)? (b) Assuming that the field goes to zero as s ??, find E(s, t).15

Problem 19P A toroidal coil has a rectangular cross section, with inner radius a, outer radius a + w, and height h. It carries a total of N tightly wound turns, and the current is increasing at a constant rate (d I/dt = k). If w and h are both much less than a, find the electric field at a point z above the center of the toroid. [Hint: Exploit the analogy between Faraday fields and magnetostatic fields, and refer to Ex. 5.6.]

Problem 22P A small loop of wire (radius a) is held a distance z above the center of a large loop (radius b), as shown in Fig. 7.37. The planes of the two loops are parallel, and perpendicular to the common axis. (a) Suppose current I flows in the big loop. Find the flux through the little loop. (The little loop is so small that you may consider the field of the big loop to be essentially constant.) (b) Suppose current I flows in the little loop. Find the flux through the big loop. (The little loop is so small that you may treat it as a magnetic dipole.) (c) Find the mutual inductances, and confirm that M12 = M21. Reference 7.37

Problem 24P Find the self-inductance per unit length of a long solenoid, of radius R, carrying n turns per unit length.

Problem 25P Try to compute the self-inductance of the “hairpin” loop shown in Fig. 7.38. (Neglect the contribution from the ends; most of the flux comes from the long straight section.) You’ll run into a snag that is characteristic of many self inductance calculations. To get a definite answer, assume the wire has a tiny radius , and ignore any flux through the wire itself.

Problem 26P An alternating current I (t) = I0 cos(?t) (amplitude 0.5 A, frequency 60 Hz) flows down a straight wire, which runs along the axis of a toroidal coil with rectangular cross section (inner radius 1 cm, outer radius 2 cm, height 1 cm, 1000 turns). The coil is connected to a 500 ?resistor. (a) In the quasistatic approximation, what emf is induced in the toroid? Find the current, IR(t), in the resistor. (b) Calculate the back emf in the coil, due to the current IR(t). What is the ratio of the amplitudes of this back emf and the “direct” emf in (a)?

Problem 23P A square loop of wire, of side a, lies midway between two long wires, 3a apart, and in the same plane. (Actually, the long wires are sides of a large rectangular loop, but the short ends are so far away that they can be neglected.) A clockwise current I in the square loop is gradually increasing: d I/dt = k (a constant). Find the emf induced in the big loop. Which way will the induced current flow?

Problem 28P Find the energy stored in a section of length l of a long solenoid (radius R, current I , n turns per unit length), (a) using Eq. 7.30 (you found L in Prob. 7.24); (b) using Eq. 7.31 (we worked out A in Ex. 5.12); (c) using Eq. 7.35; (d) using Eq. 7.34 (take as your volume the cylindrical tube from radius a < R out to radius b > R). Equation 7.30 Equation 7.31 Equation 7.35 Equation 7.34

Problem 29P Calculate the energy stored in the toroidal coil of Ex. 7.11, by applying Eq. 7.35. Use the answer to check Eq. 7.28. Reference: Eq. 7.35. Reference: Eq.7.28. Reference: Ex. 7.11 Find the self-inductance of a toroidal coil with rectangular cross section (inner radius a, outer radius b, height h), that carries a total of N turns.

Problem 27P A capacitor C is charged up to a voltage V and connected to an inductor L, as shown schematically in Fig. 7.39. At time t = 0, the switch S is closed. Find the current in the circuit as a function of time. How does your answer change if a resistor R is included in series with C and L?

Problem 30P A long cable carries current in one direction uniformly distributed over its (circular) cross section. The current returns along the surface (there is a very thin insulating sheath separating the currents). Find the self-inductance per unit length.

Problem 34P A fat wire, radius a, carries a constant current I , uniformly distributed over its cross section. A narrow gap in the wire, of width w ?a, forms a parallel-plate capacitor, as shown in Fig. 7.45. Find the magnetic field in the gap, at a distance s < a from the axis.

Problem 31P Suppose the circuit in Fig. 7.41 has been connected for a long time when suddenly, at time t = 0, switch S is thrown from A to B, bypassing the battery. (a) What is the current at any subsequent time t? (b) What is the total energy delivered to the resistor? (c) Show that this is equal to the energy originally stored in the inductor.

Problem 32P Two tiny wire loops, with areas a1 and a2, are situated a displacement apart (Fig. 7.42). (a) Find their mutual inductance. [Hint: Treat them as magnetic dipoles, and use Eq. 5.88.] Is your formula consistent with Eq. 7.24? (b) Suppose a current I1 is flowing in loop 1, and we propose to turn on a current I2 in loop 2. How much work must be done, against the mutually induced emf, to keep the current I1 flowing in loop 1? In light of this result, comment on Eq. 6.35. Equation 5.88

Problem 37P Suppose (The theta function is defined in Prob. 1.46b). Show that these fields satisfy all of Maxwell’s equations, and determine ? and J. Describe the physical situation that gives rise to these fields. Reference: Prob. 1.46b. (b) Let ?(x) be the step function: Show that d?/dx = ?(x).

Problem 38P Assuming that “Coulomb’s law” for magnetic charges (qm) reads work out the force law for a monopole qm moving with velocity v through electric and magnetic fields E and B.26

Problem 35P The preceding problem was an artificial model for the charging capacitor, designed to avoid complications associated with the current spreading out over the surface of the plates. For a more realistic model, imagine thin wires that connect to the centers of the plates (Fig. 7.46a). Again, the current I is constant, the radius of the capacitor is a, and the separation of the plates is w ? a. Assume that the current flows out over the plates in such a way that the surface charge is uniform, at any given time, and is zero at t = 0. (a) Find the electric field between the plates, as a function of t. (b) Find the displacement current through a circle of radius s in the plane midway between the plates. Using this circle as your “Amperian loop,” and the flat surface that spans it, find the magnetic field at a distance s from the axis. (c) Repeat part (b), but this time use the cylindrical surface in Fig. 7.46(b), which is open at the right end and extends to the left through the plate and terminates outside the capacitor. Notice that the displacement current through this surface is zero, and there are two contributions to Ienc

Problem 40P Sea water at frequency

Problem 39P Suppose a magnetic monopole qm passes through a resistanceless loop of wire with self-inductance L. What current is induced in the loop?

Problem 42P A rare case in which the electrostatic field E for a circuit can actually be calculated is the following:28 Imagine an infinitely long cylindrical sheet, of uniform resistivity and radius a. A slot (corresponding to the battery) is maintained at and a steady current flows over the surface, as indicated in Fig. 7.51. According to Ohm’s law, then, a) Use separation of variables in cylindrical coordinates to determine V (s, ?) inside and outside the cylinder. b) Find the surface charge density on the cylinder.

Problem 43P The magnetic field outside a long straight wire carrying a steady current I is The electric field inside the wire is uniform: where ? is the resistivity and a is the radius (see Exs. 7.1 and 7.3). Question: What is the electric field outside the wire?29 The answer depends on how you complete the circuit. Suppose the current returns along a perfectly conducting grounded coaxial cylinder of radius b (Fig. 7.52). In the region a < s < b, the potential V (s, z) satisfies Laplace’s equation, with the boundary conditions This does not suffice to determine the answer—we still need to specify boundary conditions at the two ends (though for a long wire it shouldn’t matter much). In the literature, it is customary to sweep this ambiguity under the rug by simply stipulating that V(s, z) is proportional to z: V(s, z) = z f (s). On this assumption: (a) Determine f (s). (b) Find E(s, z). (c) Calculate the surface charge density ?(z) on the wire Reference example 7.1 A cylindrical resistor of cross-sectional area A and length L is made from material with conductivity ?. (See Fig. 7.1; as indicated, the cross section need not be circular, but I do assume it is the same all the way down.) If we stipulate that the potential is constant over each end, and the potential difference between the ends is V, what current flows?

Problem 41P Two long, straight copper pipes, each of radius a, are held a distance 2d apart (see Fig. 7.50). One is at potential V0, the other at ?V0. The space surrounding the pipes is filled with weakly conducting material of conductivity ?. Find the current per unit length that flows from one pipe to the other. [Hint: Refer to Prob. 3.12.] Figure 7.50 Reference prob 3.12 Two long, straight copper pipes, each of radius R, are held a distance 2d apart. One is at potential V0, the other at ?V0 (Fig. 3.16). Find the potential everywhere. [Hint: Exploit the result of Prob. 2.52.]

Problem 45P A familiar demonstration of superconductivity (Prob. 7.44) is the levitation of a magnet over a piece of superconducting material. This phenomenon can be analyzed using the method of images.31 Treat the magnet as a perfect dipole m, a height z above the origin (and constrained to point in the z direction), and pretend that the superconductor occupies the entire half-space below the xy plane. Because of the Meissner effect, B = 0 for z ? 0, and since B is divergenceless, the normal (z) component is continuous, so Bz = 0 just above the surface. This boundary condition is met by the image configuration in which an identical dipole is placed at ?z, as a stand-in for the superconductor; the two arrangements therefore produce the same magnetic field in the region z > 0. (a) Which way should the image dipole point (+z or ?z)? (b) Find the force on the magnet due to the induced currents in the superconductor (which is to say, the force due to the image dipole). Set it equal to Mg (where M is the mass of the magnet) to determine the height h at which the magnet will “float.” (c) The induced current on the surface of the superconductor (the xy plane) can be determined from the boundary condition on the tangential component of B (Eq. 5.76): Using the field you get from the image configuration, show that where r is the distance from the origin. Refer Prob. 6.3.] Find the force of attraction between two magnetic dipoles, m1 and m2, oriented as shown in Fig. 6.7, a distance r apart, (a) using Eq. 6.2, and (b) using Eq. 6.3. Eq. 6.2. Eq.6.3

Problem 46P If a magnetic dipole levitating above an infinite superconducting plane (Prob. 7.45) is free to rotate, what orientation will it adopt, and how high above the surface will it float?

Problem 47P A perfectly conducting spherical shell of radius a rotates about the z axis with angular velocity ?, in a uniform magnetic field Calculate the emf developed between the “north pole” and the equator.

In a perfect conductor, the conductivity is infinite, so \(E=0\) (Eq. 7.3), and any net charge resides on the surface (just as it does for an imperfect conductor, in electrostatics). (a) Show that the magnetic field is constant (\(\partial \mathrm{B} / \partial \mathrm{t}=0)\)), inside a perfect conductor. (b) Show that the magnetic flux through a perfectly conducting loop is constant. A superconductor is a perfect conductor with the additional property that the (constant) B inside is in fact zero. (This “flux exclusion” is known as the Meissner effect.) (c) Show that the current in a superconductor is confined to the surface. (d) Superconductivity is lost above a certain critical temperature (\(T_c\)), which varies from one material to another. Suppose you had a sphere (radius \(a\)) above its critical temperature, and you held it in a uniform magnetic field \(B_{0} \hat{\mathbf{z}}\) while cooling it below \(T_c\). Find the induced surface current density K, as a function of the polar angle \(\theta\).

Problem 50P Electrons undergoing cyclotron motion can be sped up by increasing the magnetic field; the accompanying electric field will impart tangential acceleration. This is the principle of the betatron. One would like to keep the radius of the orbit constant during the process. Show that this can be achieved by designing a magnet such that the average field over the area of the orbit is twice the field at the circumference (Fig. 7.53). Assume the electrons start from rest in zero field, and that the apparatus is symmetric about the center of the orbit. (Assume also that the electron velocity remains well below the speed of light, so that nonrelativistic mechanics applies.) [Hint: Differentiate Eq. 5.3 with respect to time, and use F = ma = qE.]

Problem 48P Refer to Prob. 7.11 (and use the result of Prob. 5.42): How long does is take a falling circular ring (radius a, mass m, resistance R) to cross the bottom of the magnetic field B, at its (changing) terminal velocity? Refer Prob. 7.11 A square loop is cut out of a thick sheet of aluminum. It is then placed so that the top portion is in a uniform magnetic field B, and is allowed to fall under gravity (Fig. 7.20). (In the diagram, shading indicates the field region; B points into the page.) If the magnetic field is 1 T (a pretty standard laboratory field), find the terminal velocity of the loop (in m/s). Find the velocity of the loop as a function of time. How long does it take (in seconds) to reach, say, 90% of the terminal velocity? What would happen if you cut a tiny slit in the ring, breaking the circuit? [Note: The dimensions of the loop cancel out; determine the actual numbers, in the units indicated.]

Problem 51P An infinite wire carrying a constant current I in the direction is moving in the y direction at a constant speed v. Find the electric field, in the quasistatic approximation, at the instant the wire coincides with the z axis (Fig. 7.54).

Problem 53P The current in a long solenoid is increasing linearly with time, so the flux is proportional to Two voltmeters are connected to diametrically opposite points (A and B), together with resistors (R1 and R2), as shown in Fig. 7.55. What is the reading on each voltmeter? Assume that these are ideal voltmeters that draw negligible current (they have huge internal resistance), and that a voltmeter registers E l between the terminals and through the meter. [Answer: ; . Notice that , even though they are connected to the same points.]

Problem 49P (a) Referring to Prob. 5.52(a) and Eq. 7.18, show that for Faraday-induced electric fields. Check this result by taking the divergence and curl of both sides. (b) A spherical shell of radius R carries a uniform surface charge ?. It spins about a fixed axis at an angular velocity ?(t) that changes slowly with time. Find the electric field inside and outside the sphere. [Hint: There are two contributions here: the Coulomb field due to the charge, and the Faraday field due to the changing B. Refer to Ex. 5.11.] Reference prob 5.52 iv class="question"> (a) One way to fill in the “missing link” in Fig. 5.48 is to exploit the analogy between the defining equations for A (viz. ? · A = 0, ? × A = B) and Maxwell’s equations for B (viz. ? · B = 0, ? × B = ?0J). Evidently A depends on B in exactly the same way that B depends on ?0J (to wit: the Biot-Savart law). Use this observation to write down the formula for A in terms of B. (b) The electrical analog to your result in (a) is Derive it, by exploiting the appropriate analogy. Figure 5.48 Refer Ex. 5.11 A spherical shell of radius R, carrying a uniform surface charge ?, is set spinning at angular velocity ?. Find the vector potential it produces at point r (Fig. 5.45). Equation 7.18

Problem 52P An atomic electron (charge q) circles about the nucleus (charge Q) in an orbit of radius r ; the centripetal acceleration is provided, of course, by the Coulomb attraction of opposite charges. Now a small magnetic field dB is slowly turned on, perpendicular to the plane of the orbit. Show that the increase in kinetic energy, dT, imparted by the induced electric field, is just right to sustain circular motion at the same radius r. (That’s why, in my discussion of diamagnetism, I assumed the radius is fixed. See Sect. 6.1.3 and the references cited there.)

Problem 55P In the discussion of motional emf (Sect. 7.1.3) I assumed that the wire loop (Fig. 7.10) has a resistance R; the current generated is then I = vBh/R. But what if the wire is made out of perfectly conducting material, so that R is zero? In that case, the current is limited only by the back emf associated with the self inductance L of the loop (which would ordinarily be negligible in comparison with I R). Show that in this régime the loop (mass m) executes simple harmonic motion, and find its frequency

Problem 54P A circular wire loop (radius r , resistance R) encloses a region of uniform magnetic field, B, perpendicular to its plane. The field (occupying the shaded region in Fig. 7.56) increases linearly with time (B = ?t). An ideal voltmeter (infinite internal resistance) is connected between points P and Q. (a) What is the current in the loop? (b) What does the voltmeter read?

Problem 57P Two coils are wrapped around a cylindrical form in such a way that the same flux passes through every turn of both coils. (In practice this is achieved by inserting an iron core through the cylinder; this has the effect of concentrating the flux.) The primary coil has N1 turns and the secondary has N2 (Fig. 7.57). If the current I in the primary is changing, show that the emf in the secondary is given by where E1 is the (back) emf of the primary. [This is a primitive transformer—a device for raising or lowering the emf of an alternating current source. By choosing the appropriate number of turns, any desired secondary emf can be obtained. If you think this violates the conservation of energy, study Prob. 7.58.] figure 7.57

Problem 58P A transformer (Prob. 7.57) takes an input AC voltage of amplitude V1, and delivers an output voltage of amplitude V2, which is determined by the turns ratio (V2/V1 = N2/N1). If N2 > N1, the output voltage is greater than the input voltage. Why doesn’t this violate conservation of energy? Answer: Power is the product of voltage and current; if the voltage goes up, the current must come down. The purpose of this problem is to see exactly how this works out, in a simplified model. (a) In an ideal transformer, the same flux passes through all turns of the primary and of the secondary. Show that in this case M² = L1L2, where M is the mutual inductance of the coils, and L1, L2 are their individual self-inductances. (b) Suppose the primary is driven with AC voltage Vin = V1 cos (?t), and the secondary is connected to a resistor, R. Show that the two currents satisfy the relations (c) Using the result in (a), solve these equations for I1(t) and I2(t). (Assume I1 has no DC component.) (d) Show that the output voltage (Vout = I2R) divided by the input voltage (Vin) is equal to the turns ratio: Vout/Vin = N2/N1. (e) Calculate the input power (Pin = Vin I1) and the output power (Pout = Vout I2), and show that their averages over a full cycle are equal.

Problem 61P The magnetic field of an infinite straight wire carrying a steady current I can be obtained from the displacement current term in the Ampère/Maxwell law, as follows: Picture the current as consisting of a uniform line charge ? moving along the z axis at speed v (so that I = ?v), with a tiny gap of length ? , which reaches the origin at time t = 0. In the next instant (up to t = ?/v) there is no real current passing through a circular Amperian loop in the xy plane, but there is a displacement current, due to the “missing” charge in the gap. (a) Use Coulomb’s law to calculate the z component of the electric field, for points in the xy plane a distance s from the origin, due to a segment of wire with uniform density ?? extending from z1 = vt ? ? to z2 = vt. (b) Determine the flux of this electric field through a circle of radius a in the xy plane. (c) Find the displacement current through this circle. Show that Id is equal to Id, in the limit as the gap width (?) goes to zero.

Problem 60P Suppose J(r) is constant in time but ?(r, t) is not—conditions that might prevail, for instance, during the charging of a capacitor. (a) Show that the charge density at any particular point is a linear function of time: where is the time derivative of ? at t = 0. [Hint: Use the continuity equation.] This is not an electrostatic or magnetostatic configuration;34 nevertheless, rather surprisingly, both Coulomb’s law (Eq. 2.8) and the Biot-Savart law (Eq. 5.42) hold, as you can confirm by showing that they satisfy Maxwell’s equations. In particular: (b) Show that obeys Ampère’s law with Maxwell’s displacement current term.

Problem 56P (a) Use the Neumann formula (Eq. 7.23) to calculate the mutual inductance of the configuration in Fig. 7.37, assuming a is very small (a ? b, a ? z). Compare your answer to Prob. 7.22. (b) For the general case (not assuming a is small), show that Equation 7.23 Reference prob 7.22 p>A small loop of wire (radius a) is held a distance z above the center of a large loop (radius b), as shown in Fig. 7.37. The planes of the two loops are parallel, and perpendicular to the common axis. (a) Suppose current I flows in the big loop. Find the flux through the little loop. (The little loop is so small that you may consider the field of the big loop to be essentially constant.) (b) Suppose current I flows in the little loop. Find the flux through the big loop. (The little loop is so small that you may treat it as a magnetic dipole.) (c) Find the mutual inductances, and confirm that M12 = M21.

Problem 62P A certain transmission line is constructed from two thin metal “ribbons,” of width w, a very small distance apart. The current travels down one strip and back along the other. In each case, it spreads out uniformly over the surface of the ribbon. (a) Find the capacitance per unit length, C. (b) Find the inductance per unit length, L. (c) What is the product LC, numerically? [L and C will, of course, vary from one kind of transmission line to another, but their product is a universal constant— check, for example, the cable in Ex. 7.13—provided the space between the conductors is a vacuum. In the theory of transmission lines, this product is related to the speed with which a pulse propagates down the line: (d) If the strips are insulated from one another by a nonconducting material of permittivity ? and permeability ?, what then is the product LC?What is the propagation speed? [Hint: see Ex. 4.6; by what factor does L change when an inductor is immersed in linear material of permeability ??]

Problem 63P Prove Alfven’s theorem: In a perfectly conducting fluid (say, a gas of free electrons), the magnetic flux through any closed loop moving with the fluid is constant in time. (The magnetic field lines are, as it were, “frozen” into the fluid.) (a) Use Ohm’s law, in the form of Eq. 7.2, together with Faraday’s law, to prove that if ? =?and J is finite, then (b) Let S be the surface bounded by the loop (?P) at time t, and S ? a surface bounded by the loop in its new position (P?) at time t + dt (see Fig. 7.58). The change in flux is Figure 7.58

Problem 36P Refer to Prob. 7.16, to which the correct answer was (a) Find the displacement current density Jd . (b) Integrate it to get the total displacement current, (c) Compare Id and I . (What’s their ratio?) If the outer cylinder were, say, 2 mm in diameter, how high would the frequency have to be, for Id to be 1% of I? [This problem is designed to indicate why Faraday never discovered displacement currents, and why it is ordinarily safe to ignore them unless the frequency is extremely high.] Reference problem 7.16 An alternating current I = I0 cos (?t) flows down a long straight wire, and returns along a coaxial conducting tube of radius a. (a) In what direction does the induced electric field point (radial, circumferential, or longitudinal)? (b) Assuming that the field goes to zero as s ??, find E(s, t).15