1. Calculate the power (energy per unit time) transported down the
cables of Ex. 7.13 and Prob. 7.62, assuming the two conductors are held
at potential difference V, and carry current I (down one and back up the
other). Get solution
2. Consider the charging capacitor in Prob. 7.34. (a) Find the electric and magnetic fields in the gap, as functions of the distance s from the axis and the time t. (Assume the charge is zero at t = 0.) (b) Find the energy density uem and the Poynting vector S in the gap. Note especially the direction of S. Check that Eq. 8.12 is satisfied. (c) Determine the total energy in the gap, as a function of time. Calculate the total power flowing into the gap, by integrating the Poynting vector over the appropriate surface. Check that the power input is equal to the rate of increase of energy in the gap (Eq. 8.9—in this case W = 0, because there is no charge in the gap). [If you’re worried about the fringing fields, do it for a volume of radius b Get solution
3. Calculate the force of magnetic attraction between the northern and southern hemispheres of a uniformly charged spinning spherical shell, with radius R, angular velocity ω, and surface charge density σ. [This is the same as Prob. 5.44, but this time use the Maxwell stress tensor and Eq. 8.21.] Get solution
4. (a) Consider two equal point charges q, separated by a distance 2a. Construct the plane equidistant from the two charges. By integrating Maxwell’s stress tensor over this plane, determine the force of one charge on the other. (b) Do the same for charges that are opposite in sign. Get solution
5. Imagine two parallel infinite sheets, carrying uniform surface charge +σ (on the sheet at z = d) and −σ (at z = 0). They are moving in the y direction at constant speed v (as in Problem 5.17). (a) What is the electromagnetic momentum in a region of area A? (b) Now suppose the top sheet moves slowly down (speed u) until it reaches the bottom sheet, so the fields disappear. By calculating the total force on the charge (q = σ A), show that the impulse delivered to the sheet is equal to the momentum originally stored in the fields. [Hint: As the upper plate passes by, the magnetic field drops to zero, inducing an electric field that delivers an impulse to the lower plate.] reference prob 5.17A large parallel-plate capacitor with uniform surface charge σ on the upper plate and −σ on the lower is moving with a constant speed v, as shown in Fig. 5.43. (a) Find the magnetic field between the plates and also above and below them. (b) Find the magnetic force per unit area on the upper plate, including its direction. (c) At what speed v would the magnetic force balance the electrical force?15 Fig. 5.43. ... Get solution
6. A charged parallel-plate capacitor (with uniform electric field...is placed in a uniform magnetic field...as shown in Fig. 8.6. ... (a) Find the electromagnetic momentum in the space between the plates. (b) Now a resistive wire is connected between the plates, along the z axis, so that the capacitor slowly discharges. The current through the wire will experience a magnetic force; what is the total impulse delivered to the system, during the discharge?7 Get solution
7. Consider an infinite parallel-plate capacitor, with the lower plate (at z = −d/2) carrying surface charge density −σ, and the upper plate (at z = +d/2) carrying charge density +σ. (a) Determine all nine elements of the stress tensor, in the region between the plates. Display your answer as a 3 × 3 matrix: ... (b) Use Eq. 8.21 to determine the electromagnetic force per unit area on the top plate. Compare Eq. 2.51. (c) What is the electromagnetic momentum per unit area, per unit time, crossing the xy plane (or any other plane parallel to that one, between the plates)? (d) Of course, there must be mechanical forces holding the plates apart—perhaps the capacitor is filled with insulating material under pressure. Suppose we suddenly remove the insulator; the momentum flux (c) is now absorbed by the plates, and they begin to move. Find the momentum per unit time delivered to the top plate (which is to say, the force acting on it) and compare your answer to (b). [Note: This is not an additional force, but rather an alternative way of calculating the same force—in (b) we got it from the force law, and in (d) we do it by conservation of momentum.] Reference equation 2.51 ... Reference equation 8.21 ... Get solution
8. In Ex. 8.4, suppose that instead of turning off the magnetic field (by reducing I ) we turn off the electric field, by connecting a weakly10 conducting radial spoke between the cylinders. (We’ll have to cut a slot in the solenoid, so the cylinders can still rotate freely.) From the magnetic force on the current in the spoke, determine the total angular momentum delivered to the cylinders, as they discharge (they are now rigidly connected, so they rotate together). Compare the initial angular momentum stored in the fields (Eq. 8.34). (Notice that the mechanism by which angular momentum is transferred from the fields to the cylinders is entirely different in the two cases: in Ex. 8.4 it was Faraday’s law, but here it is the Lorentz force law.) Reference example 8.4 Imagine a very long solenoid with radius R, n turns per unit length, and current I . Coaxial with the solenoid are two long cylindrical (nonconducting) shells of length l—one, inside the solenoid at radius a, carries a charge +Q, uniformly distributed over its surface; the other, outside the solenoid at radius b, carries charge −Q (see Fig. 8.7; l is supposed to be much greater than b). When the current in the solenoid is gradually reduced, the cylinders begin to rotate, as we found in Ex. 7.8. Question: Where does the angular momentum come from?8 ... Get solution
9. Two concentric spherical shells carry uniformly distributed charges +Q (at radius a) and −Q (at radius b > a). They are immersed in a uniform magnetic field ... (a) Find the angular momentum of the fields (with respect to the center). (b) Now the magnetic field is gradually turned off. Find the torque on each sphere, and the resulting angular momentum of the system. Get solution
10. Imagine an iron sphere of radius R that carries a charge Q and a uniform magnetization ... The sphere is initially at rest. (a) Compute the angular momentum stored in the electromagnetic fields. (b) Suppose the sphere is gradually (and uniformly) demagnetized (perhaps by heating it up past the Curie point). Use Faraday’s law to determine the induced electric field, find the torque this field exerts on the sphere, and calculate the total angular momentum imparted to the sphere in the course of the demagnetization. (c) Suppose instead of demagnetizing the sphere we discharge it, by connecting a grounding wire to the north pole. Assume the current flows over the surface in such a way that the charge density remains uniform. Use the Lorentz force law to determine the torque on the sphere, and calculate the total angular momentum imparted to the sphere in the course of the discharge. (The magnetic field is discontinuous at the surface . . . does this matter?) Get solution
11. Derive Eq. 8.39. [Hint: Treat the lower loop as a magnetic dipole.] Reference equation 8.39 ... Get solution
12. Derive Eq. 8.43. [Hint: Use the method of Section 7.2.4, building the two currents up from zero to their final values.] Reference equation 8.43 ... Get solution
13. A very long solenoid of radius a, with n turns per unit length, carries a current Is . Coaxial with the solenoid, at radius b ≫ a, is a circular ring of wire, with resistance R. When the current in the solenoid is (gradually) decreased, a current Ir is induced in the ring. (a) Calculate Ir , in terms of d Is/dt. (b) The power (I 2 r R) delivered to the ring must have come from the solenoid. Confirm this by calculating the Poynting vector just outside the solenoid (the electric field is due to the changing flux in the solenoid; the magnetic field is due to the current in the ring). Integrate over the entire surface of the solenoid, and check that you recover the correct total power. Get solution
14. An infinitely long cylindrical tube, of radius a, moves at constant speed v along its axis. It carries a net charge per unit length λ, uniformly distributed over its surface. Surrounding it, at radius b, is another cylinder, moving with the same velocity but carrying the opposite charge (−λ). Find: (a) The energy per unit length stored in the fields. (b) The momentum per unit length in the fields. (c) The energy per unit time transported by the fields across a plane perpendicular to the cylinders. Get solution
15. A point charge q is located at the center of a toroidal coil of rectangular cross section, inner radius a, outer radius a + w, and height h, which carries a total of N tightly-wound turns and current I . (a) Find the electromagnetic momentum p of this configuration, assuming that w and h are both much less than a (so you can ignore the variation of the fields over the cross section). (b) Now the current in the toroid is turned off, quickly enough that the point charge does not move appreciably as the magnetic field drops to zero. Show that the impulse imparted to q is equal to the momentum originally stored in the electromagnetic fields. [Hint: You might want to refer to Prob. 7.19.] Reference Prob. 7.19 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.] Get solution
16. A sphere of radius R carries a uniform polarization P and a uniform magnetization M (not necessarily in the same direction). Find the electromagnetic momentum of this configuration. Get solution
17. Picture the electron as a uniformly charged spherical shell, with charge e and radius R, spinning at angular velocity ω. (a) Calculate the total energy contained in the electromagnetic fields. (b) Calculate the total angular momentum contained in the fields. (c) According to the Einstein formula (E = mc2), the energy in the fields should contribute to the mass of the electron. Lorentz and others speculated that the entire mass of the electron might be accounted for in this way: Uem = mec2. Suppose, moreover, that the electron’s spin angular momentum is entirely attributable to the electromagnetic fields: Lem = ¯h/2. On these two assumptions, determine the radius and angular velocity of the electron. What is their product, ωR? Does this classical model make sense? Get solution
18. Work out the formulas for u, S, g, and ... in the presence of magnetic charge. [Hint: Start with the generalized Maxwell equations (7.44) and Lorentz force law (Eq. 8.44), and follow the derivations in Sections 8.1.2, 8.2.2, and 8.2.3.] Equation 7.44 ... Equation 8.44 ... Get solution
19. Suppose you had an electric charge qe and a magnetic monopole qm. The field of the electric charge is ... (of course), and the field of the magnetic monopole is ... Find the total angular momentum stored in the fields, if the two charges are separated by a distance d. Get solution
20. Consider an ideal stationary magnetic dipole m in a static electric field E. Show that the fields carry momentum ... [Hint: There are several ways to do this. The simplest method is to start with ... and use integration by parts to show that ... So far, this is valid for any localized static configuration. For a current confined to an infinitesimal neighborhood of the origin we can approximate V(r) ≈ V(0) − E(0) . r. Treat the dipole as a current loop, and use Eqs. 5.82 and 1.108.]21 Equation 5.82 ... Get solution
21. Because the cylinders in Ex. 8.4 are left rotating (at angular velocities ωa and ωb, say), there is actually a residual magnetic field, and hence angular momentum in the fields, even after the current in the solenoid has been extinguished. If the cylinders are heavy, this correction will be negligible, but it is interesting to do the problem without making that assumption.22 (a) Calculate (in terms of ωa and ωb) the final angular momentum in the fields. [Define ... so ωa and ωb could be positive or negative.] (b) As the cylinders begin to rotate, their changing magnetic field induces an extra azimuthal electric field, which, in turn, will make an additional contribution to the torques. Find the resulting extra angular momentum, and compare it with your result in (a). Equation 8.24 ... Get solution
22. A point charge q is a distance a > R from the axis of an infinite solenoid (radius R, n turns per unit length, current I ). Find the linear momentum and the angular momentum (with respect to the origin) in the fields. (Put q on the x axis, with the solenoid along z; treat the solenoid as a nonconductor, so you don’t need to worry about induced charges on its surface.) Get solution
23. (a) Carry through the argument in Sect. 8.1.2, starting with Eq. 8.6, but using J f in place of J. Show that the Poynting vector becomes S = E × H, and the rate of change of the energy density in the fields is ... For linear media, show that24 ... (b) In the same spirit, reproduce the argument in Sect. 8.2.2, starting with Eq. 8.15, with ρ f and J f in place of ρ and J. Don’t bother to construct the Maxwell stress tensor, but do show that the momentum density is25 g = D × B. Reference equation 8.6 ... Reference equation 8.15 ... Get solution
24. A circular disk of radius R and mass M carries n point charges (q), attached at regular intervals around its rim. At time t = 0 the disk lies in the xy plane, with its center at the origin, and is rotating about the z axis with angular velocity ω0, when it is released. The disk is immersed in a (time-independent) external magnetic field ... where k is a constant. (a) Find the position of the center if the ring, z(t), and its angular velocity, ω(t), as functions of time. (Ignore gravity.) (b) Describe the motion, and check that the total (kinetic) energy—translational plus rotational—is constant, confirming that the magnetic force does no work.26 Get solution
2. Consider the charging capacitor in Prob. 7.34. (a) Find the electric and magnetic fields in the gap, as functions of the distance s from the axis and the time t. (Assume the charge is zero at t = 0.) (b) Find the energy density uem and the Poynting vector S in the gap. Note especially the direction of S. Check that Eq. 8.12 is satisfied. (c) Determine the total energy in the gap, as a function of time. Calculate the total power flowing into the gap, by integrating the Poynting vector over the appropriate surface. Check that the power input is equal to the rate of increase of energy in the gap (Eq. 8.9—in this case W = 0, because there is no charge in the gap). [If you’re worried about the fringing fields, do it for a volume of radius b Get solution
3. Calculate the force of magnetic attraction between the northern and southern hemispheres of a uniformly charged spinning spherical shell, with radius R, angular velocity ω, and surface charge density σ. [This is the same as Prob. 5.44, but this time use the Maxwell stress tensor and Eq. 8.21.] Get solution
4. (a) Consider two equal point charges q, separated by a distance 2a. Construct the plane equidistant from the two charges. By integrating Maxwell’s stress tensor over this plane, determine the force of one charge on the other. (b) Do the same for charges that are opposite in sign. Get solution
5. Imagine two parallel infinite sheets, carrying uniform surface charge +σ (on the sheet at z = d) and −σ (at z = 0). They are moving in the y direction at constant speed v (as in Problem 5.17). (a) What is the electromagnetic momentum in a region of area A? (b) Now suppose the top sheet moves slowly down (speed u) until it reaches the bottom sheet, so the fields disappear. By calculating the total force on the charge (q = σ A), show that the impulse delivered to the sheet is equal to the momentum originally stored in the fields. [Hint: As the upper plate passes by, the magnetic field drops to zero, inducing an electric field that delivers an impulse to the lower plate.] reference prob 5.17A large parallel-plate capacitor with uniform surface charge σ on the upper plate and −σ on the lower is moving with a constant speed v, as shown in Fig. 5.43. (a) Find the magnetic field between the plates and also above and below them. (b) Find the magnetic force per unit area on the upper plate, including its direction. (c) At what speed v would the magnetic force balance the electrical force?15 Fig. 5.43. ... Get solution
6. A charged parallel-plate capacitor (with uniform electric field...is placed in a uniform magnetic field...as shown in Fig. 8.6. ... (a) Find the electromagnetic momentum in the space between the plates. (b) Now a resistive wire is connected between the plates, along the z axis, so that the capacitor slowly discharges. The current through the wire will experience a magnetic force; what is the total impulse delivered to the system, during the discharge?7 Get solution
7. Consider an infinite parallel-plate capacitor, with the lower plate (at z = −d/2) carrying surface charge density −σ, and the upper plate (at z = +d/2) carrying charge density +σ. (a) Determine all nine elements of the stress tensor, in the region between the plates. Display your answer as a 3 × 3 matrix: ... (b) Use Eq. 8.21 to determine the electromagnetic force per unit area on the top plate. Compare Eq. 2.51. (c) What is the electromagnetic momentum per unit area, per unit time, crossing the xy plane (or any other plane parallel to that one, between the plates)? (d) Of course, there must be mechanical forces holding the plates apart—perhaps the capacitor is filled with insulating material under pressure. Suppose we suddenly remove the insulator; the momentum flux (c) is now absorbed by the plates, and they begin to move. Find the momentum per unit time delivered to the top plate (which is to say, the force acting on it) and compare your answer to (b). [Note: This is not an additional force, but rather an alternative way of calculating the same force—in (b) we got it from the force law, and in (d) we do it by conservation of momentum.] Reference equation 2.51 ... Reference equation 8.21 ... Get solution
8. In Ex. 8.4, suppose that instead of turning off the magnetic field (by reducing I ) we turn off the electric field, by connecting a weakly10 conducting radial spoke between the cylinders. (We’ll have to cut a slot in the solenoid, so the cylinders can still rotate freely.) From the magnetic force on the current in the spoke, determine the total angular momentum delivered to the cylinders, as they discharge (they are now rigidly connected, so they rotate together). Compare the initial angular momentum stored in the fields (Eq. 8.34). (Notice that the mechanism by which angular momentum is transferred from the fields to the cylinders is entirely different in the two cases: in Ex. 8.4 it was Faraday’s law, but here it is the Lorentz force law.) Reference example 8.4 Imagine a very long solenoid with radius R, n turns per unit length, and current I . Coaxial with the solenoid are two long cylindrical (nonconducting) shells of length l—one, inside the solenoid at radius a, carries a charge +Q, uniformly distributed over its surface; the other, outside the solenoid at radius b, carries charge −Q (see Fig. 8.7; l is supposed to be much greater than b). When the current in the solenoid is gradually reduced, the cylinders begin to rotate, as we found in Ex. 7.8. Question: Where does the angular momentum come from?8 ... Get solution
9. Two concentric spherical shells carry uniformly distributed charges +Q (at radius a) and −Q (at radius b > a). They are immersed in a uniform magnetic field ... (a) Find the angular momentum of the fields (with respect to the center). (b) Now the magnetic field is gradually turned off. Find the torque on each sphere, and the resulting angular momentum of the system. Get solution
10. Imagine an iron sphere of radius R that carries a charge Q and a uniform magnetization ... The sphere is initially at rest. (a) Compute the angular momentum stored in the electromagnetic fields. (b) Suppose the sphere is gradually (and uniformly) demagnetized (perhaps by heating it up past the Curie point). Use Faraday’s law to determine the induced electric field, find the torque this field exerts on the sphere, and calculate the total angular momentum imparted to the sphere in the course of the demagnetization. (c) Suppose instead of demagnetizing the sphere we discharge it, by connecting a grounding wire to the north pole. Assume the current flows over the surface in such a way that the charge density remains uniform. Use the Lorentz force law to determine the torque on the sphere, and calculate the total angular momentum imparted to the sphere in the course of the discharge. (The magnetic field is discontinuous at the surface . . . does this matter?) Get solution
11. Derive Eq. 8.39. [Hint: Treat the lower loop as a magnetic dipole.] Reference equation 8.39 ... Get solution
12. Derive Eq. 8.43. [Hint: Use the method of Section 7.2.4, building the two currents up from zero to their final values.] Reference equation 8.43 ... Get solution
13. A very long solenoid of radius a, with n turns per unit length, carries a current Is . Coaxial with the solenoid, at radius b ≫ a, is a circular ring of wire, with resistance R. When the current in the solenoid is (gradually) decreased, a current Ir is induced in the ring. (a) Calculate Ir , in terms of d Is/dt. (b) The power (I 2 r R) delivered to the ring must have come from the solenoid. Confirm this by calculating the Poynting vector just outside the solenoid (the electric field is due to the changing flux in the solenoid; the magnetic field is due to the current in the ring). Integrate over the entire surface of the solenoid, and check that you recover the correct total power. Get solution
14. An infinitely long cylindrical tube, of radius a, moves at constant speed v along its axis. It carries a net charge per unit length λ, uniformly distributed over its surface. Surrounding it, at radius b, is another cylinder, moving with the same velocity but carrying the opposite charge (−λ). Find: (a) The energy per unit length stored in the fields. (b) The momentum per unit length in the fields. (c) The energy per unit time transported by the fields across a plane perpendicular to the cylinders. Get solution
15. A point charge q is located at the center of a toroidal coil of rectangular cross section, inner radius a, outer radius a + w, and height h, which carries a total of N tightly-wound turns and current I . (a) Find the electromagnetic momentum p of this configuration, assuming that w and h are both much less than a (so you can ignore the variation of the fields over the cross section). (b) Now the current in the toroid is turned off, quickly enough that the point charge does not move appreciably as the magnetic field drops to zero. Show that the impulse imparted to q is equal to the momentum originally stored in the electromagnetic fields. [Hint: You might want to refer to Prob. 7.19.] Reference Prob. 7.19 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.] Get solution
16. A sphere of radius R carries a uniform polarization P and a uniform magnetization M (not necessarily in the same direction). Find the electromagnetic momentum of this configuration. Get solution
17. Picture the electron as a uniformly charged spherical shell, with charge e and radius R, spinning at angular velocity ω. (a) Calculate the total energy contained in the electromagnetic fields. (b) Calculate the total angular momentum contained in the fields. (c) According to the Einstein formula (E = mc2), the energy in the fields should contribute to the mass of the electron. Lorentz and others speculated that the entire mass of the electron might be accounted for in this way: Uem = mec2. Suppose, moreover, that the electron’s spin angular momentum is entirely attributable to the electromagnetic fields: Lem = ¯h/2. On these two assumptions, determine the radius and angular velocity of the electron. What is their product, ωR? Does this classical model make sense? Get solution
18. Work out the formulas for u, S, g, and ... in the presence of magnetic charge. [Hint: Start with the generalized Maxwell equations (7.44) and Lorentz force law (Eq. 8.44), and follow the derivations in Sections 8.1.2, 8.2.2, and 8.2.3.] Equation 7.44 ... Equation 8.44 ... Get solution
19. Suppose you had an electric charge qe and a magnetic monopole qm. The field of the electric charge is ... (of course), and the field of the magnetic monopole is ... Find the total angular momentum stored in the fields, if the two charges are separated by a distance d. Get solution
20. Consider an ideal stationary magnetic dipole m in a static electric field E. Show that the fields carry momentum ... [Hint: There are several ways to do this. The simplest method is to start with ... and use integration by parts to show that ... So far, this is valid for any localized static configuration. For a current confined to an infinitesimal neighborhood of the origin we can approximate V(r) ≈ V(0) − E(0) . r. Treat the dipole as a current loop, and use Eqs. 5.82 and 1.108.]21 Equation 5.82 ... Get solution
21. Because the cylinders in Ex. 8.4 are left rotating (at angular velocities ωa and ωb, say), there is actually a residual magnetic field, and hence angular momentum in the fields, even after the current in the solenoid has been extinguished. If the cylinders are heavy, this correction will be negligible, but it is interesting to do the problem without making that assumption.22 (a) Calculate (in terms of ωa and ωb) the final angular momentum in the fields. [Define ... so ωa and ωb could be positive or negative.] (b) As the cylinders begin to rotate, their changing magnetic field induces an extra azimuthal electric field, which, in turn, will make an additional contribution to the torques. Find the resulting extra angular momentum, and compare it with your result in (a). Equation 8.24 ... Get solution
22. A point charge q is a distance a > R from the axis of an infinite solenoid (radius R, n turns per unit length, current I ). Find the linear momentum and the angular momentum (with respect to the origin) in the fields. (Put q on the x axis, with the solenoid along z; treat the solenoid as a nonconductor, so you don’t need to worry about induced charges on its surface.) Get solution
23. (a) Carry through the argument in Sect. 8.1.2, starting with Eq. 8.6, but using J f in place of J. Show that the Poynting vector becomes S = E × H, and the rate of change of the energy density in the fields is ... For linear media, show that24 ... (b) In the same spirit, reproduce the argument in Sect. 8.2.2, starting with Eq. 8.15, with ρ f and J f in place of ρ and J. Don’t bother to construct the Maxwell stress tensor, but do show that the momentum density is25 g = D × B. Reference equation 8.6 ... Reference equation 8.15 ... Get solution
24. A circular disk of radius R and mass M carries n point charges (q), attached at regular intervals around its rim. At time t = 0 the disk lies in the xy plane, with its center at the origin, and is rotating about the z axis with angular velocity ω0, when it is released. The disk is immersed in a (time-independent) external magnetic field ... where k is a constant. (a) Find the position of the center if the ring, z(t), and its angular velocity, ω(t), as functions of time. (Ignore gravity.) (b) Describe the motion, and check that the total (kinetic) energy—translational plus rotational—is constant, confirming that the magnetic force does no work.26 Get solution
