1. Show that the differential equations for V and A (Eqs. 10.4 and
10.5) can be written in the more symmetrical form ... Where ...
Equation 10.4 ... Equation 10.5 ... Get solution
2. For the configuration in Ex. 10.1, consider a rectangular box of length l, width w, and height h, situated a distance d above the yz plane (Fig. 10.2). ... (a) Find the energy in the box at time t1 = d/c, and at t2 = (d + h)/c. (b) Find the Poynting vector, and determine the energy per unit time flowing into the box during the interval t1 t t2. (c) Integrate the result in (b) from t1 to t2, and confirm that the increase in energy (part (a)) equals the net influx. Get solution
3. (a) Find the fields, and the charge and current distributions, corresponding to ... b) Use the gauge function ... to transform the potentials, and comment on the result. Get solution
4. Suppose V = 0 and...where A0, ω, and k are constants. Find E and B, and check that they satisfy Maxwell’s equations in vacuum. What condition must you impose on ω and k? Get solution
5. Which of the potentials in Ex. 10.1, Prob. 10.3, and Prob. 10.4 are in the Coulomb gauge? Which are in the Lorenz gauge? (Notice that these gauges are not mutually exclusive.) Reference prob.10.3(a) Find the fields, and the charge and current distributions, corresponding to ... b) Use the gauge function ... to transform the potentials, and comment on the result. Reference prob.10.4 Suppose V = 0 and...where A0, ω, and k are constants. Find E and B, and check that they satisfy Maxwell’s equations in vacuum. What condition must you impose on ω and k? Get solution
6. In Chapter 5, I showed that it is always possible to pick a vector potential whose divergence is zero (the Coulomb gauge). Show that it is always possible to choose ... as required for the Lorenz gauge, assuming you know how to solve the inhomogeneous wave equation (Eq. 10.16). Is it always possible to pick V = 0? How about A = 0? Reference equation 10.16 ... Get solution
7. A time-dependent point charge q(t) at the origin, ... is fed by a current ... (a) Check that charge is conserved, by confirming that the continuity equation is obeyed. (b) Find the scalar and vector potentials in the Coulomb gauge. If you get stuck, try working on (c) first. (c) Find the fields, and check that they satisfy all of Maxwell’s equations.3 Get solution
8. The vector potential for a uniform magnetostatic field is ... (r×B) (Prob. 5.25). Show that ... in this case, and confirm that Eq. 10.20 yields the correct equation of motion. Reference equation 10.20 ... Reference prob 5.25 If B is uniform, show that ... works. That is, check that ∇ · A = 0 and ∇ × A = B. Is this result unique, or are there other functions with the same divergence and curl? Get solution
9. Derive Eq. 10.23. [Hint: Start by dotting v into Eq. 10.17.] Reference equation 10.23 ... Reference equation 10.17 ... Get solution
10. Confirm that the retarded potentials satisfy the Lorenz gauge condition. [Hint: First show that ... where ∇ denotes derivatives with respect to r, and ∇′ denotes derivatives with respect to r′. Next, noting that ... depends on r′ both explicitly and through ... , whereas it depends on r only through r, confirm that ...confirm that ... Use this to calculate the divergence of A (Eq. 10.26).] Equation 10.26 ... Get solution
11. (a) Suppose the wire in Ex. 10.2 carries a linearly increasing current I ( t ) = kt, for t > 0. Find the electric and magnetic fields generated. (b) Do the same for the case of a sudden burst of current: I ( t ) = q0δ(t). Reference example 10.2 An infinite straight wire carries the current ... That is, a constant current I0 is turned on abruptly at t = 0. Find the resulting electric and magnetic fields. ... Get solution
12. A piece of wire bent into a loop, as shown in Fig. 10.5, carries a current that increases linearly with time: I ( t ) = kt (−∞ t ∞). Calculate the retarded vector potential A at the center. Find the electric field at the center. Why does this (neutral) wire produce an electric field? (Why can’t you determine the magnetic field from this expression for A?) Figure 10.5 ... Get solution
13. Suppose J(r) is constant in time, so (Prob. 7.60) ρ(r, t) = ρ(r, 0) + ρ? (r, 0)t. Show that ... that is, Coulomb’s law holds, with the charge density evaluated at the non-retarded time. Reference Prob. 7.60 Get solution
14. Suppose the current density changes slowly enough that we can (to good approximation) ignore all higher derivatives in the Taylor expansion ... (for clarity, I suppress the r-dependence, which is not at issue). Show that a fortuitous cancellation in Eq. 10.38 yields ... That is: the Biot-Savart law holds, with J evaluated at the non-retarded time. This means that the quasistatic approximation is actually much better than we had any right to expect: the two errors involved (neglecting retardation and dropping the second term in Eq. 10.38) cancel one another, to first order. Equation 10.38 ... Get solution
15. A particle of charge q moves in a circle of radius a at constant angular velocity ω. (Assume that the circle lies in the xy plane, centered at the origin, and at time t = 0 the charge is at (a, 0), on the positive x axis.) Find the Liénard-Wiechert potentials for points on the z axis. Get solution
16. Show that the scalar potential of a point charge moving with constant velocity (Eq. 10.49) can be written more simply as ... where R ≡ r − vt is the vector from the present (!) position of the particle to the field point r, and θ is the angle between R and v (Fig. 10.9). Note that for nonrelativistic velocities (v2 ≫ c2), ... Equation 10.49 ... Get solution
17. I showed that at most one point on the particle trajectory communicates with r at any given time. In some cases there may be no such point (an observer at r would not see the particle—in the colorful language of general relativity, it is “over the horizon”). As an example, consider a particle in hyperbolic motion along the x axis: ... (In special relativity, this is the trajectory of a particle subject to a constant force F = mc2/b.) Sketch the graph of w versus t. At four or five representative points on the curve, draw the trajectory of a light signal emitted by the particle at that point—both in the plus x direction and in the minus x direction. What region on your graph corresponds to points and times (x, t) from which the particle cannot be seen? At what time does someone at point x first see the particle? (Prior to this the potential at x is zero.) Is it possible for a particle, once seen, to disappear from view? Get solution
18. Determine the Liénard-Wiechert potentials for a charge in hyperbolic motion (Eq. 10.52). Assume the point r is on the x axis and to the right of the charge Eqation 10.52 ... Get solution
19. Derive Eq. 10.70. First show that ... Equation 10.70 ... Get solution
20. Suppose a point charge q is constrained to move along the x axis. Show that the fields at points on the axis to the right of the charge are given by ... (Do not assume v is constant!) What are the fields on the axis to the left of the charge? Get solution
21. For a point charge moving at constant velocity, calculate the flux integral ... (using Eq. 10.75), over the surface of a sphere centered at the present location of the charge.21 Reference equation 10.75 ... Get solution
22. (a) Use Eq. 10.75 to calculate the electric field a distance d from an infinite straight wire carrying a uniform line charge λ, moving at a constant speed v down the wire. (b) Use Eq. 10.76 to find the magnetic field of this wire. Equation 10.75 ... Equation 10.76 ... Get solution
23. For the configuration in Prob. 10.15, find the electric and magnetic fields at the center. From your formula for B, determine the magnetic field at the center of a circular loop carrying a steady current I , and compare your answer with the result of Ex. 5.6 Reference prob 10.15 A piece of wire bent into a loop, as shown in Fig. 10.5, carries a current that increases linearly with time: I ( t ) = kt (−∞ t ∞). Calculate the retarded vector potential A at the center. Find the electric field at the center. Why does this (neutral) wire produce an electric field? (Why can’t you determine the magnetic field from this expression for A?) Figure 10.5 ... Get solution
24. Suppose you take a plastic ring of radius a and glue charge on it, so that the line charge density is λ0| sin(θ/2)|. Then you spin the loop about its axis at an angular velocity ω. Find the (exact) scalar and vector potentials at the center of the ring. Get solution
25. Figure 2.35 summarizes the laws of electrostatics in a “triangle diagram” relating the source (ρ), the field (E), and the potential (V). Figure 5.48 does the same for magnetostatics, where the source is J, the field is B, and the potential is A. Construct the analogous diagram for electrodynamics, with sources ρ and J (constrained by the continuity equation), fields E and B, and potentials V and A (constrained by the Lorenz gauge condition). Do not include formulas for V and A in terms of E and B. Figure 5.48 ... Figure 2.35 ... Get solution
26. An expanding sphere, radius R(t) = vt (t > 0, constant v) carries a charge Q, uniformly distributed over its volume. Evaluate the integral ... with respect to the center. Show that... Get solution
27. Check that the potentials of a point charge moving at constant velocity (Eqs. 10.49 and 10.50) satisfy the Lorenz gauge condition (Eq. 10.12). Reference equation 10.49 ... Reference equation 10.50 ... Reference equation 10.12 ... Get solution
28. One particle, of charge q1, is held at rest at the origin. Another particle, of charge q2, approaches along the x axis, in hyperbolic motion: ... it reaches the closest point, b, at time t = 0, and then returns out to infinity. (a) What is the force F2 on q2 (due to q1) at time t? (b) What total impulse ... is delivered to q2 by q1? (c) What is the force F1 on q1 (due to q2) at time t? (d) What total impulse ... is delivered to q1 by q2? [Hint: It might help to review Prob. 10.17 before doing this integral. Answer: I2 = −I1 = q1q2/4 ϵ0bc] Reference prob 10.17 Get solution
29. We are now in a position to treat the example in Sect. 8.2.1 quantitatively. Suppose q1 is at x1 = −vt and q2 is at y = −vt (Fig. 8.3, with t 0). Find the electric and magnetic forces on q1 and q2. Is Newton’s third law obeyed? Figure 8.3 ... Get solution
30. A uniformly charged rod (length L, charge density λ) slides out the x axis at constant speed v. At time t = 0 the back end passes the origin (so its position as a function of time is x = vt, while the front end is at x = vt + L). Find the retarded scalar potential at the origin, as a function of time, for t > 0. [First determine the retarded time t1 for the back end, the retarded time t2 for the front end, and the corresponding retarded positions x1 and x2.] Is your answer consistent with the Liénard-Wiechert potential, in the point charge limit (L ≪ vt, with λL = q)? Do not assume v ≪ c. Get solution
31. A particle of charge q is traveling at constant speed v along the x axis. Calculate the total power passing through the plane x = a, at the moment the particle itself is at the origin. Get solution
32. A particle of charge q1 is at rest at the origin. A second particle, of charge q2, moves along the z axis at constant velocity v. (a) Find the force F12(t) of q1 on q2, at time t (when q2 is at z = vt). (b) Find the force F21(t) of q2 on q1, at time t. Does Newton’s third law hold, in this case? (c) Calculate the linear momentum p(t) in the electromagnetic fields, at time t. (Don’t bother with any terms that are constant in time, since you won’t need them in part (d)). (d) Show that the sum of the forces is equal to minus the rate of change of the momentum in the fields, and interpret this result physically. Get solution
33. Develop the potential formulation for electrodynamics with magnetic charge (Eq. 7.44). [Hint: You’ll need two scalar potentials and two vector potentials. Use the Lorenz gauge. Find the retarded potentials (generalizing Eqs. 10.26), and give the formulas for E and B in terms of the potentials (generalizing Eqs. 10.2 and 10.3).] Equation 7.44 ... Equation 10.26 ... Equation 10.2 ... Equation 10.3 ... Get solution
34. Find the (Lorenz gauge) potentials and fields of a time-dependent ideal electric dipole p(t) at the origin.23 (It is stationary, but its magnitude and/or direction are changing with time.) Don’t bother with the contact term. Get solution
2. For the configuration in Ex. 10.1, consider a rectangular box of length l, width w, and height h, situated a distance d above the yz plane (Fig. 10.2). ... (a) Find the energy in the box at time t1 = d/c, and at t2 = (d + h)/c. (b) Find the Poynting vector, and determine the energy per unit time flowing into the box during the interval t1 t t2. (c) Integrate the result in (b) from t1 to t2, and confirm that the increase in energy (part (a)) equals the net influx. Get solution
3. (a) Find the fields, and the charge and current distributions, corresponding to ... b) Use the gauge function ... to transform the potentials, and comment on the result. Get solution
4. Suppose V = 0 and...where A0, ω, and k are constants. Find E and B, and check that they satisfy Maxwell’s equations in vacuum. What condition must you impose on ω and k? Get solution
5. Which of the potentials in Ex. 10.1, Prob. 10.3, and Prob. 10.4 are in the Coulomb gauge? Which are in the Lorenz gauge? (Notice that these gauges are not mutually exclusive.) Reference prob.10.3(a) Find the fields, and the charge and current distributions, corresponding to ... b) Use the gauge function ... to transform the potentials, and comment on the result. Reference prob.10.4 Suppose V = 0 and...where A0, ω, and k are constants. Find E and B, and check that they satisfy Maxwell’s equations in vacuum. What condition must you impose on ω and k? Get solution
6. In Chapter 5, I showed that it is always possible to pick a vector potential whose divergence is zero (the Coulomb gauge). Show that it is always possible to choose ... as required for the Lorenz gauge, assuming you know how to solve the inhomogeneous wave equation (Eq. 10.16). Is it always possible to pick V = 0? How about A = 0? Reference equation 10.16 ... Get solution
7. A time-dependent point charge q(t) at the origin, ... is fed by a current ... (a) Check that charge is conserved, by confirming that the continuity equation is obeyed. (b) Find the scalar and vector potentials in the Coulomb gauge. If you get stuck, try working on (c) first. (c) Find the fields, and check that they satisfy all of Maxwell’s equations.3 Get solution
8. The vector potential for a uniform magnetostatic field is ... (r×B) (Prob. 5.25). Show that ... in this case, and confirm that Eq. 10.20 yields the correct equation of motion. Reference equation 10.20 ... Reference prob 5.25 If B is uniform, show that ... works. That is, check that ∇ · A = 0 and ∇ × A = B. Is this result unique, or are there other functions with the same divergence and curl? Get solution
9. Derive Eq. 10.23. [Hint: Start by dotting v into Eq. 10.17.] Reference equation 10.23 ... Reference equation 10.17 ... Get solution
10. Confirm that the retarded potentials satisfy the Lorenz gauge condition. [Hint: First show that ... where ∇ denotes derivatives with respect to r, and ∇′ denotes derivatives with respect to r′. Next, noting that ... depends on r′ both explicitly and through ... , whereas it depends on r only through r, confirm that ...confirm that ... Use this to calculate the divergence of A (Eq. 10.26).] Equation 10.26 ... Get solution
11. (a) Suppose the wire in Ex. 10.2 carries a linearly increasing current I ( t ) = kt, for t > 0. Find the electric and magnetic fields generated. (b) Do the same for the case of a sudden burst of current: I ( t ) = q0δ(t). Reference example 10.2 An infinite straight wire carries the current ... That is, a constant current I0 is turned on abruptly at t = 0. Find the resulting electric and magnetic fields. ... Get solution
12. A piece of wire bent into a loop, as shown in Fig. 10.5, carries a current that increases linearly with time: I ( t ) = kt (−∞ t ∞). Calculate the retarded vector potential A at the center. Find the electric field at the center. Why does this (neutral) wire produce an electric field? (Why can’t you determine the magnetic field from this expression for A?) Figure 10.5 ... Get solution
13. Suppose J(r) is constant in time, so (Prob. 7.60) ρ(r, t) = ρ(r, 0) + ρ? (r, 0)t. Show that ... that is, Coulomb’s law holds, with the charge density evaluated at the non-retarded time. Reference Prob. 7.60 Get solution
14. Suppose the current density changes slowly enough that we can (to good approximation) ignore all higher derivatives in the Taylor expansion ... (for clarity, I suppress the r-dependence, which is not at issue). Show that a fortuitous cancellation in Eq. 10.38 yields ... That is: the Biot-Savart law holds, with J evaluated at the non-retarded time. This means that the quasistatic approximation is actually much better than we had any right to expect: the two errors involved (neglecting retardation and dropping the second term in Eq. 10.38) cancel one another, to first order. Equation 10.38 ... Get solution
15. A particle of charge q moves in a circle of radius a at constant angular velocity ω. (Assume that the circle lies in the xy plane, centered at the origin, and at time t = 0 the charge is at (a, 0), on the positive x axis.) Find the Liénard-Wiechert potentials for points on the z axis. Get solution
16. Show that the scalar potential of a point charge moving with constant velocity (Eq. 10.49) can be written more simply as ... where R ≡ r − vt is the vector from the present (!) position of the particle to the field point r, and θ is the angle between R and v (Fig. 10.9). Note that for nonrelativistic velocities (v2 ≫ c2), ... Equation 10.49 ... Get solution
17. I showed that at most one point on the particle trajectory communicates with r at any given time. In some cases there may be no such point (an observer at r would not see the particle—in the colorful language of general relativity, it is “over the horizon”). As an example, consider a particle in hyperbolic motion along the x axis: ... (In special relativity, this is the trajectory of a particle subject to a constant force F = mc2/b.) Sketch the graph of w versus t. At four or five representative points on the curve, draw the trajectory of a light signal emitted by the particle at that point—both in the plus x direction and in the minus x direction. What region on your graph corresponds to points and times (x, t) from which the particle cannot be seen? At what time does someone at point x first see the particle? (Prior to this the potential at x is zero.) Is it possible for a particle, once seen, to disappear from view? Get solution
18. Determine the Liénard-Wiechert potentials for a charge in hyperbolic motion (Eq. 10.52). Assume the point r is on the x axis and to the right of the charge Eqation 10.52 ... Get solution
19. Derive Eq. 10.70. First show that ... Equation 10.70 ... Get solution
20. Suppose a point charge q is constrained to move along the x axis. Show that the fields at points on the axis to the right of the charge are given by ... (Do not assume v is constant!) What are the fields on the axis to the left of the charge? Get solution
21. For a point charge moving at constant velocity, calculate the flux integral ... (using Eq. 10.75), over the surface of a sphere centered at the present location of the charge.21 Reference equation 10.75 ... Get solution
22. (a) Use Eq. 10.75 to calculate the electric field a distance d from an infinite straight wire carrying a uniform line charge λ, moving at a constant speed v down the wire. (b) Use Eq. 10.76 to find the magnetic field of this wire. Equation 10.75 ... Equation 10.76 ... Get solution
23. For the configuration in Prob. 10.15, find the electric and magnetic fields at the center. From your formula for B, determine the magnetic field at the center of a circular loop carrying a steady current I , and compare your answer with the result of Ex. 5.6 Reference prob 10.15 A piece of wire bent into a loop, as shown in Fig. 10.5, carries a current that increases linearly with time: I ( t ) = kt (−∞ t ∞). Calculate the retarded vector potential A at the center. Find the electric field at the center. Why does this (neutral) wire produce an electric field? (Why can’t you determine the magnetic field from this expression for A?) Figure 10.5 ... Get solution
24. Suppose you take a plastic ring of radius a and glue charge on it, so that the line charge density is λ0| sin(θ/2)|. Then you spin the loop about its axis at an angular velocity ω. Find the (exact) scalar and vector potentials at the center of the ring. Get solution
25. Figure 2.35 summarizes the laws of electrostatics in a “triangle diagram” relating the source (ρ), the field (E), and the potential (V). Figure 5.48 does the same for magnetostatics, where the source is J, the field is B, and the potential is A. Construct the analogous diagram for electrodynamics, with sources ρ and J (constrained by the continuity equation), fields E and B, and potentials V and A (constrained by the Lorenz gauge condition). Do not include formulas for V and A in terms of E and B. Figure 5.48 ... Figure 2.35 ... Get solution
26. An expanding sphere, radius R(t) = vt (t > 0, constant v) carries a charge Q, uniformly distributed over its volume. Evaluate the integral ... with respect to the center. Show that... Get solution
27. Check that the potentials of a point charge moving at constant velocity (Eqs. 10.49 and 10.50) satisfy the Lorenz gauge condition (Eq. 10.12). Reference equation 10.49 ... Reference equation 10.50 ... Reference equation 10.12 ... Get solution
28. One particle, of charge q1, is held at rest at the origin. Another particle, of charge q2, approaches along the x axis, in hyperbolic motion: ... it reaches the closest point, b, at time t = 0, and then returns out to infinity. (a) What is the force F2 on q2 (due to q1) at time t? (b) What total impulse ... is delivered to q2 by q1? (c) What is the force F1 on q1 (due to q2) at time t? (d) What total impulse ... is delivered to q1 by q2? [Hint: It might help to review Prob. 10.17 before doing this integral. Answer: I2 = −I1 = q1q2/4 ϵ0bc] Reference prob 10.17 Get solution
29. We are now in a position to treat the example in Sect. 8.2.1 quantitatively. Suppose q1 is at x1 = −vt and q2 is at y = −vt (Fig. 8.3, with t 0). Find the electric and magnetic forces on q1 and q2. Is Newton’s third law obeyed? Figure 8.3 ... Get solution
30. A uniformly charged rod (length L, charge density λ) slides out the x axis at constant speed v. At time t = 0 the back end passes the origin (so its position as a function of time is x = vt, while the front end is at x = vt + L). Find the retarded scalar potential at the origin, as a function of time, for t > 0. [First determine the retarded time t1 for the back end, the retarded time t2 for the front end, and the corresponding retarded positions x1 and x2.] Is your answer consistent with the Liénard-Wiechert potential, in the point charge limit (L ≪ vt, with λL = q)? Do not assume v ≪ c. Get solution
31. A particle of charge q is traveling at constant speed v along the x axis. Calculate the total power passing through the plane x = a, at the moment the particle itself is at the origin. Get solution
32. A particle of charge q1 is at rest at the origin. A second particle, of charge q2, moves along the z axis at constant velocity v. (a) Find the force F12(t) of q1 on q2, at time t (when q2 is at z = vt). (b) Find the force F21(t) of q2 on q1, at time t. Does Newton’s third law hold, in this case? (c) Calculate the linear momentum p(t) in the electromagnetic fields, at time t. (Don’t bother with any terms that are constant in time, since you won’t need them in part (d)). (d) Show that the sum of the forces is equal to minus the rate of change of the momentum in the fields, and interpret this result physically. Get solution
33. Develop the potential formulation for electrodynamics with magnetic charge (Eq. 7.44). [Hint: You’ll need two scalar potentials and two vector potentials. Use the Lorenz gauge. Find the retarded potentials (generalizing Eqs. 10.26), and give the formulas for E and B in terms of the potentials (generalizing Eqs. 10.2 and 10.3).] Equation 7.44 ... Equation 10.26 ... Equation 10.2 ... Equation 10.3 ... Get solution
34. Find the (Lorenz gauge) potentials and fields of a time-dependent ideal electric dipole p(t) at the origin.23 (It is stationary, but its magnitude and/or direction are changing with time.) Don’t bother with the contact term. Get solution
