### Chapter #12 Solutions - Introduction to Electrodynamics - David J Griffiths - 4th Edition

1. Let S be an inertial reference system. Use Galileo’s velocity addition rule. (a) Suppose that ...is also an inertial reference system. [Hint: Use the definition in footnote 1.] (b) Conversely, show that if ...is an inertial system, then it moves with respect to S at constant velocity. Get solution

2. As an illustration of the principle of relativity in classical mechanics, consider the following generic collision: In inertial frame S, particle A (mass mA, velocity uA) hits particle B (mass mB, velocity uB). In the course of the collision some mass rubs off A and onto B, and we are left with particles C (mass mC , velocity uC) and D (mass mD, velocity uD). Assume that momentum (p ≡mu) is conserved in S. (a) Prove that momentum is also conserved in inertial frame ..., which moves with velocity v relative to S. [Use Galileo’s velocity addition rule—this is an entirely classical calculation. What must you assume about mass?] (b) Suppose the collision is elastic in S; show that it is also elastic in... Get solution

3. (a) What’s the percent error introduced when you use Galileo’s rule, instead of Einstein’s, with ... (b) Suppose you could run at half the speed of light down the corridor of a train going three-quarters the speed of light. What would your speed be relative to the ground? (c) Prove, using Eq. 12.3, that if vAB c and vBC c then vAC c. Interpret this result. Equation 12.3 ... Get solution

4. As the outlaws escape in their getaway car, which goes ...the police officer fires a bullet from the pursuit car, which only goes ...(Fig. 12.3). The muzzle velocity of the bullet (relative to the gun) is...Does the bullet reach its target (a) according to Galileo, (b) according to Einstein? ... Get solution

5. Synchronized clocks are stationed at regular intervals, a million km apart, along a straight line. When the clock next to you reads 12 noon: (a) What time do you see on the 90th clock down the line? (b) What time do you observe on that clock? Get solution

6. Every 2 years, more or less, The New York Times publishes an article in which some astronomer claims to have found an object traveling faster than the speed of light. Many of these reports result from a failure to distinguish what is seen from what is observed—that is, from a failure to account for light travel time. Here’s an example: A star is traveling with speed v at an angle θ to the line of sight (Fig. 12.6). What is its apparent speed across the sky? (Suppose the light signal from b reaches the earth at a time Δt after the signal from a, and the star has meanwhile advanced a distance Δs across the celestial sphere; by “apparent speed,” I mean Δs/ Δt.) What angle θ gives the maximum apparent speed? Show that the apparent speed can be much greater than c, even if v itself is less than c. ... Get solution

7. In a laboratory experiment, a muon is observed to travel 800 m before disintegrating. A graduate student looks up the lifetime of a muon (2 × 10−6 s) and concludes that its speed was ... Faster than light! Identify the student’s error, and find the actual speed of this muon. Get solution

8. A rocket ship leaves earth at a speed of ... c. When a clock on the rocket says 1 hour has elapsed, the rocket ship sends a light signal back to earth. (a) According to earth clocks, when was the signal sent? (b) According to earth clocks, how long after the rocket left did the signal arrive back on earth? (c) According to the rocket observer, how long after the rocket left did the signal arrive back on earth? Get solution

9. A Lincoln Continental is twice as long as a VW Beetle, when they are at rest. As the Continental overtakes the VW, going through a speed trap, a (stationary) policeman observes that they both have the same length. The VW is going at half the speed of light. How fast is the Lincoln going? (Leave your answer as a multiple of c.) Get solution

10. A sailboat is manufactured so that the mast leans at an angle with ... respect to the deck. An observer standing on a dock sees the boat go by at speed v (Fig. 12.14). What angle does this observer say the mast makes? ... Get solution

11. A record turntable of radius R rotates at angular velocity ω (Fig. 12.15). The circumference is presumably Lorentz-contracted, but the radius (being perpendicular to the velocity) is not. What’s the ratio of the circumference to the diameter, in terms of ω and R? According to the rules of ordinary geometry, it has to be π. What’s going on here?9 ... Get solution

12. Solve Eqs. 12.18 for x, y, z, t in terms of ... and check that you recover Eqs. 12.19. Reference equation 12.18 ... Reference equation 12.19 ... Get solution

13. Sophie Zabar, clairvoyante, cried out in pain at precisely the instant her twin brother, 500 km away, hit his thumb with a hammer. A skeptical scientist observed both events (brother’s accident, Sophie’s cry) from an airplane traveling at ... to the right (Fig. 12.19). Which event occurred first, according to the scientist? How much earlier was it, in seconds? Reference figure 12.19 ... Get solution

14. (a) In Ex. 12.6 we found how velocities in the x direction transform when you go from S to ... .Derive the analogous formulas for velocities in the y and z directions. (b) A spotlight is mounted on a boat so that its beam makes an angle ...with the deck (Fig. 12.20). If this boat is then set in motion at speed v, what angle θ does an individual photon trajectory make with the deck, according to an observer on the dock? What angle does the beam (illuminated, say, by a light fog) make? Compare Prob. 12.10. ... Reference problem 12.10 A sailboat is manufactured so that the mast leans at an angle with ... respect to the deck. An observer standing on a dock sees the boat go by at speed v (Fig. 12.14). What angle does this observer say the mast makes? ... Get solution

15. You probably did Prob. 12.4 from the point of view of an observer on the ground. Now do it from the point of view of the police car, the outlaws, and the bullet. That is, fill in the gaps in the following table: ... Reference problem 12.4 As the outlaws escape in their getaway car, which goes ...the police officer fires a bullet from the pursuit car, which only goes ...(Fig. 12.3). The muzzle velocity of the bullet (relative to the gun) is...Does the bullet reach its target (a) according to Galileo, (b) according to Einstein? ... Get solution

16. The twin paradox revisited. On their 21st birthday, one twin gets on a moving sidewalk, which carries her out to star X at speed ... her twin brother stays home. When the traveling twin gets to star X, she immediately jumps onto the returning moving sidewalk and comes back to earth, again at speed ... She arrives on her 39th birthday (as determined by her watch). (a) How old is her twin brother? (b) How far away is star X? (Give your answer in light years.) Call the outbound sidewalk system ... at the moment of departure. (c) What are the coordinates (x, t) of the jump (from outbound to inbound sidewalk) in S? (d) What are the coordinates ... (e) What are the coordinates ... of the jump in ... (f) If the traveling twin wants her watch to agree with the clock in ... how must she reset it immediately after the jump? What does her watch then read when she gets home? (This wouldn’t change her age, of course—she’s still 39—it would just make her watch agree with the standard synchronization in ...) (g) If the traveling twin is asked the question, “How old is your brother right now?”, what is the correct reply (i) just before she makes the jump, (ii) just after she makes the jump? (Nothing dramatic happens to her brother during the split second between (i) and (ii), of course; what does change abruptly is his sister’s notion of what “right now, back home” means.) (h) How many earth years does the return trip take? Add this to (ii) from (g) to determine how old she expects him to be at their reunion. Compare your answer to (a). Get solution

17. Check Eq. 12.29, using Eq. 12.27. [This only proves the invariance of the scalar product for transformations along the x direction. But the scalar product is also invariant under rotations, since the first term is not affected at all, and the last three constitute the three-dimensional dot product a · b. By a suitable rotation, the x direction can be aimed any way you please, so the four-dimensional scalar product is actually invariant under arbitrary Lorentz transformations.] Reference equation 12.27 ... Equation 12.29 ... Get solution

18. (a) Write out the matrix that describes a Galilean transformation (Eq. 12.12). (b) Write out the matrix describing a Lorentz transformation along the y axis. (c) Find the matrix describing a Lorentz transformation with velocity v along the x axis followed by a Lorentz transformation with velocity ... along the y axis. Does it matter in what order the transformations are carried out? Equation 12.12 ... Get solution

19. The parallel between rotations and Lorentz transformations is even more striking if we introduce the rapidity: ... (a) Express the Lorentz transformation matrix Λ (Eq. 12.24) in terms of θ, and compare it to the rotation matrix (Eq. 1.29). In some respects, rapidity is a more natural way to describe motion than velocity. 11 For one thing, it ranges from −∞ to +∞, instead of −c to +c. More significantly, rapidities add, whereas velocities do not. (b) Express the Einstein velocity addition law in terms of rapidity. Equation 12.24 ... Equation 12.29 ... Get solution

20. (a) Event A happens at point (xA = 5, yA = 3, z A = 0) and at time tA given by ctA = 15; event B occurs at (10, 8, 0) and ctB = 5, both in system S. (i) What is the invariant interval between A and B? (ii) Is there an inertial system in which they occur simultaneously? If so, find its velocity (magnitude and direction) relative to S. (iii) Is there an inertial system in which they occur at the same point? If so, find its velocity relative to S. (b) Repeat part (a) for A = (2, 0, 0), ct = 1; and B = (5, 0, 0), ct = 3. Get solution

21. The coordinates of event A are (xA, 0, 0), tA, and the coordinates of event B are (xB, 0, 0), tB. Assuming the displacement between them is spacelike, find the velocity of the system in which they are simultaneous. Get solution

22. (a) Draw a space-time diagram representing a game of catch (or a conversation) between two people at rest, 10 ft apart. How is it possible for them to communicate, given that their separation is space like? (b) There’s an old limerick that runs as follows: There once was a girl named Ms. Bright, Who could travel much faster than light. She departed one day, The Einsteinian way, And returned on the previous night. What do you think? Even if she could travel faster than light, could she return before she set out? Could she arrive at some intermediate destination before she set out? Draw a space-time diagram representing this trip. Get solution

23. Inertial system...moves in the x direction at speed...relative to system S. (The ... slides long the x axis, and the origins coincide at ... as usual.) (a) On graph paper set up a Cartesian coordinate system with axes ct and x. Carefully draw in lines representing ...and 3. Also draw in the lines corresponding to ...and 3. Label your lines clearly. (b) In ...a free particle is observed to travel from the point ... at time ...to the point...Indicate this displacement on your graph. From the slope of this line, determine the particle’s speed in S. (c) Use the velocity addition rule to determine the velocity in S algebraically, and check that your answer is consistent with the graphical solution in (b). Get solution

24. (a) Equation 12.40 defines proper velocity in terms of ordinary velocity. Invert that equation to get the formula for u in terms of η. (b) What is the relation between proper velocity and rapidity (Eq. 12.34)? Assume the velocity is along the x direction, and find η as a function of θ. Equation 12.34 ... Get solution

25. A car is traveling along the 45˚ line in S (Fig. 12.25), at (ordinary) speed... (a) Find the components ux and uy of the (ordinary) velocity. (b) Find the components ηx and ηy of the proper velocity. (c) Find the zeroth component of the 4-velocity, η0. System ... is moving in the x direction with (ordinary) speed ... relative to S. By using the appropriate transformation laws: (d) Find the (ordinary) velocity components ... ... (e) Find the proper velocity components ... (f) As a consistency check, verify that ... Get solution

26. Find the invariant product of the 4-velocity with itself, ... timelike, spacelike, or lightlike? Get solution

27. A cop pulls you over and asks what speed you were going. “Well, officer, I cannot tell a lie: the speedometer read 4 × 108 m/s.” He gives you a ticket, because the speed limit on this highway is 2.5 × 108 m/s. In court, your lawyer (who, luckily, has studied physics) points out that a car’s speedometer measures proper velocity, whereas the speed limit is ordinary velocity. Guilty, or innocent? Get solution

28. Consider a particle in hyperbolic motion, ... (a) Find the proper time τ as a function of t, assuming the clocks are set so that τ = 0 when t = 0. [Hint: Integrate Eq. 12.37.] (b) Find x and v (ordinary velocity) as functions of τ . (c) Find ημ (proper velocity) as a function of τ . Reference equation 12.37 ... Get solution

29. (a) Repeat Prob. 12.2(a) using the (incorrect) definition p = mu, but with the (correct) Einstein velocity addition rule. Notice that if momentum (so defined) is conserved in S, it is not conserved in .... Assume all motion is along the x axis. (b) Now do the same using the correct definition, p = mη. Notice that if momentum (so defined) is conserved in S, it is automatically also conserved in .... [Hint: Use Eq. 12.43 to transform the proper velocity.] What must you assume about relativistic energy? Reference prob 12.2(a)As an illustration of the principle of relativity in classical mechanics, consider the following generic collision: In inertial frame S, particle A (mass mA, velocity uA) hits particle B (mass mB, velocity uB). In the course of the collision some mass rubs off A and onto B, and we are left with particles C (mass mC , velocity uC) and D (mass mD, velocity uD). Assume that momentum (p ≡mu) is conserved in S. (a) Prove that momentum is also conserved in inertial frame ..., which moves with velocity v relative to S. [Use Galileo’s velocity addition rule—this is an entirely classical calculation. What must you assume about mass?] Equation 12.43 ... Get solution

30. If a particle’s kinetic energy is n times its rest energy, what is its speed? Get solution

31. Suppose you have a collection of particles, all moving in the x direction, with energies E1, E2, E3, . . . and momenta p1, p2, p3, . . . . Find the velocity of the center of momentum frame, in which the total momentum is zero. Get solution

32. Find the velocity of the muon in Ex. 12.8. Reference example 12.8 A pion at rest decays into a muon and a neutrino (Fig. 12.27). Find the energy of the outgoing muon, in terms of the two masses, mπ and mμ (assume mν = 0). ... Get solution

33. A particle of mass m whose total energy is twice its rest energy collides with an identical particle at rest. If they stick together, what is the mass of the resulting composite particle? What is its velocity? Get solution

34. A neutral pion of (rest) mass m and (relativistic) momentum p = ... decays into two photons. One of the photons is emitted in the same direction as the original pion, and the other in the opposite direction. Find the (relativistic) energy of each photon. Get solution

35. In the past, most experiments in particle physics involved stationary targets: one particle (usually a proton or an electron) was accelerated to a high energy E, and collided with a target particle at rest (Fig. 12.29a). Far higher relative energies are obtainable (with the same accelerator) if you accelerate both particles to energy E, and fire them at each other (Fig. 12.29b). Classically, the energy ... of one particle, relative to the other, is just 4E (why?) . . . not much of a gain (only a factor of 4). But relativistically the gain can be enormous. Assuming the two particles have the same mass, m, show that ... ... Suppose you use protons (mc2 = 1 GeV) with E = 30 GeV. What ... do you get? What multiple of E does this amount to? (1 GeV=109 electron volts.) [Because of this relativistic enhancement, most modern elementary particle experiments involve colliding beams, instead of fixed targets.] Get solution

36. In a pair annihilation experiment, an electron (mass m) with momentum pe hits a positron (same mass, but opposite charge) at rest. They annihilate, producing two photons. (Why couldn’t they produce just one photon?) If one of the photons emerges at 60 ? to the incident electron direction, what is its energy? Get solution

37. In classical mechanics, Newton’s law can be written in the more familiar form F = ma. The relativistic equation, F = dp/dt, cannot be so simply expressed. Show, rather, that ... where a ≡ du/dt is the ordinary acceleration. Get solution

38. Show that it is possible to outrun a light ray, if you’re given a sufficient head start, and your feet generate a constant force. Get solution

39. Define proper acceleration in the obvious way: ... a) Find α0 and α in terms of u and a (the ordinary acceleration). (b) Express α μ α μ in terms of u and a. (c) Show that η μ α μ = 0. (d) Write the Minkowski version of Newton’s second law, Eq. 12.68, in terms of α μ. Evaluate the invariant product K μ η μ. Reference equation 12.68 ... Get solution

40. Show that ... where θ is the angle between u and F. Get solution

41. Show that the (ordinary) acceleration of a particle of mass m and charge q, moving at velocity u under the influence of electromagnetic fields E and B, is given by ... [Hint: Use Eq. 12.74.] ... Get solution

42. Why can’t the electric field in Fig. 12.35b have a z component? After all, the magnetic field does. Figure 12.35(b) ... Get solution

43. A parallel-plate capacitor, at rest in S 0 and tilted at a 45? angle to the x0 axis, carries charge densities }σ0 on the two plates (Fig. 12.41). System S is moving to the right at speed v relative to S 0. Figure 12.41 ... (a) Find E0, the field in S0. (b) Find E, the field in S. (c) What angle do the plates make with the x axis? (d) Is the field perpendicular to the plates in S? Get solution

44. In system S0, a static uniform line charge λ coincides with the z axis. (a) Write the electric field E0 in Cartesian coordinates, for the point (x0, y0, z0). (b) Use Eq. 12.109 to find the electric in S, which moves with speed v in the x direction with respect to S0. The field is still in terms of (x0, y0, z0); express it instead in terms of the coordinates (x, y, z) in S. Finally, write E in terms of the vector S from the present location of the wire and the angle θ between S and ...Does the field point away from the instantaneous location of the wire, like the field of a uniformly moving point charge? Equation 12.109 ... Get solution

45. (a) Charge qA is at rest at the origin in system S; charge qB flies by at speed v on a trajectory parallel to the x axis, but at y = d. What is the electromagnetic force on qB as it crosses the y axis? (b) Now study the same problem from system ... which moves to the right with speed v. What is the force on qB when qA passes the ... axis? [Do it two ways: (i) by using your answer to (a) and transforming the force; (ii) by computing the fields in ... and using the Lorentz force law.] Get solution

46. Two charges, ± q, are on parallel trajectories a distance d apart, moving with equal speeds v in opposite directions. We’re interested in the force on +q due to −q at the instant they cross (Fig. 12.42). Fill in the following table, doing all the consistency checks you can think of as you go along. ... Figure 12.42 ... Get solution

47. (a) Show that (E . B) is relativistically invariant. (b) Show that (E2 − c2B2) is relativistically invariant. (c) Suppose that in one inertial system B = 0 but E ≠ 0 (at some point P). Is it possible to find another system in which the electric field is zero at P? Get solution

48. An electromagnetic plane wave of (angular) frequency ω is traveling in the x direction through the vacuum. It is polarized in the y direction, and the amplitude of the electric field is E0. (a) Write down the electric and magnetic fields, E(x, y, z, t) and B(x, y, z, t). [Be sure to define any auxiliary quantities you introduce, in terms of ω, E0, and the constants of nature.] (b) This same wave is observed from an inertial system...moving in the x direction with speed v relative to the original system S. Find the electric and magnetic fields in ...and express them in terms of the...coordinates: ...and....[Again, be sure to define any auxiliary quantities you introduce.] (c) What is the frequency...? Interpret this result. What is the wavelength ...of the wave in ... determine the speed of the waves in ...Is it what you expected? (d) What is the ratio of the intensity in...to the intensity in S? As a youth, Einstein wondered what an electromagnetic wave would look like if you could run along beside it at the speed of light. What can you tell him about the amplitude, frequency, and intensity of the wave, as v approaches c? Get solution

49. Work out the remaining five parts to Eq. 12.118. ... Get solution

50. Prove that the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation (that is: if tμν is symmetric, show that ... is also symmetric, and likewise for antisymmetric). Get solution

51. Recall that a covariant 4-vector is obtained from a contravariant one by changing the sign of the zeroth component. The same goes for tensors: When you “lower an index” to make it covariant, you change the sign if that index is zero. Compute the tensor invariants ... in terms of E and B. Compare Prob. 12.47. Reference prob 12.47 (a) Show that (E . B) is relativistically invariant. (b) Show that (E2 − c2B2) is relativistically invariant. (c) Suppose that in one inertial system B = 0 but E ≠ 0 (at some point P). Is it possible to find another system in which the electric field is zero at P? Get solution

52. A straight wire along the z axis carries a charge density λ traveling in the +z direction at speed v. Construct the field tensor and the dual tensor at the point (x, 0, 0). Get solution

53. Obtain the continuity equation (Eq. 12.126) directly from Maxwell’s equations (Eq. 12.127). Equation 12.126 ... Equation 12.127 ... Get solution

54. Show that the second equation in Eq. 12.127 can be expressed in terms of the field tensor Fμν as follows: ... Equation 12.127 ... Get solution

55. Work out, and interpret physically, the μ = 0 component of the electromagnetic force law, Eq. 12.128. Equation 12.128 ... Get solution

56. You may have noticed that the four-dimensional gradient operator ∂/∂xμ functions like a covariant 4-vector—in fact, it is often written ∂μ, for short. For instance, the continuity equation, ∂μ J μ = 0, has the form of an invariant product of two vectors. The corresponding contravariant gradient would be ∂μ ≡ ∂/∂xμ. Prove that ∂μφ is a (contravariant) 4-vector, if φ is a scalar function, by working out its transformation law, using the chain rule. Get solution

57. Show that the potential representation (Eq. 12.133) automatically satisfies ∂Gμν/∂xν = 0. [Suggestion: Use Prob. 12.54.] Equation 12.133 ... Reference prob 12.54 Show that the second equation in Eq. 12.127 can be expressed in terms of the field tensor Fμν as follows: ... Get solution

58. Show that the Liénard-Wiechert potentials (Eqs. 10.46 and 10.47) can be expressed in relativistic notation as ... where ... equation 10.46 ... Equation 10.47 ... Get solution

59. Inertial system ... moves at constant velocity ... ... as usual. Find the Lorentz transformation matrix Λ (Eq. 12.25). Get solution

60. Calculate the threshold (minimum) momentum the pion must have in order for the process π + p → K + ∑ to occur. The proton p is initially at rest. Use mπ c2 = 150, mK c2 = 500, mpc2 = 900, m∑ c2= 1200 (all in MeV). [Hint: To formulate the threshold condition, examine the collision in the center-ofmomentum frame (Prob. 12.31). Reference equation 12.31 Suppose you have a collection of particles, all moving in the x direction, with energies E1, E2, E3, . . . and momenta p1, p2, p3, . . . . Find the velocity of the center of momentum frame, in which the total momentum is zero. Get solution

61. A particle of mass m collides elastically with an identical particle at rest. Classically, the outgoing trajectories always make an angle of 90 ˚. Calculate this angle relativistically, in terms of φ, the scattering angle, and v, the speed, in the center-of-momentum frame. Get solution

62. Find x as a function of t for motion starting from rest at the origin under the influence of a constant Minkowski force in the x direction. Leave your answer in implicit form (t as a function of x). Get solution

63. An electric dipole consists of two point charges (±q), each of mass m, fixed to the ends of a (massless) rod of length d. (Do not assume d is small.) (a) Find the net self-force on the dipole when it undergoes hyperbolic motion (Eq. 12.61) along a line perpendicular to its axis. [Hint: Start by appropriately modifying Eq. 11.90.] (b) Notice that this self-force is constant (t drops out), and points in the direction of motion—just right to produce hyperbolic motion. Thus it is possible for the dipole to undergo self-sustaining accelerated motion with no external force at all!28 [Where do you suppose the energy comes from?] Determine the self-sustaining force, F, in terms of m, q, and d. Equation 12.61 ... Equation 11.90 ... Get solution

64. An ideal magnetic dipole moment m is located at the origin of an inertial system ... that moves with speed v in the x direction with respect to inertial system S. In ... the vector potential is ... (Eq. 5.85), and the scalar potential ...is zero. (a) Find the scalar potential V in S. (b) In the nonrelativistic limit, show that the scalar potential in S is that of an ideal electric dipole of magnitude ... located at ... equation 5.85 ... Get solution

65. A stationary magnetic dipole, ... is situated above an infinite uniform surface current, ... (Fig. 12.44). (a) Find the torque on the dipole, using Eq. 6.1. (b) Suppose that the surface current consists of a uniform surface charge σ, moving at velocity ... so that K = σv, and the magnetic dipole consists of a uniform line charge λ, circulating at speed v (same v) around a square loop of side l, as shown, so that m = λvl2. Examine the same configuration from the point of view of system ... moving in the x direction at speed v. In ...the surface charge is at rest, so it generates no magnetic field. Show that in this frame the current loop carries an electric dipole moment, and calculate the resulting torque, using Eq. 4.4. Equation 12.44 ... Reference Equation 6.1 ... Reference Equation 4.4 ... Get solution

66. In a certain inertial frame S, the electric field E and the magnetic field B are neither parallel nor perpendicular, at a particular space-time point. Show that in a different inertial system ... moving relative to S with velocity v given by ... the fields ... are parallel at that point. Is there a frame in which the two are perpendicular? Get solution

67. Two charges ±q approach the origin at constant velocity from opposite directions along the x axis. They collide and stick together, forming a neutral particle at rest. Sketch the electric field before and shortly after the collision (remember that electromagnetic “news” travels at the speed of light). How would you interpret the field after the collision, physically?29 Get solution

68. “Derive” the Lorentz force law, as follows: Let charge q be at rest in ... move with velocity ... with respect to S. Use the transformation rules (Eqs. 12.67 and 12.109) to rewrite ... in terms of F, and ... in terms of E and B. From these, deduce the formula for F in terms of E and B. equation 12.67 ... equation 12.109 ... Get solution

69. A charge q is released from rest at the origin, in the presence of a uniform electric field ...and a uniform magnetic field...Determine the trajectory of the particle by transforming to a system in which E = 0, finding the path in that system and then transforming back to the original system. Assume E0 cB0. Compare your result with Ex. 5.2. Get solution

70. (a) Construct a tensor Dμν (analogous to Fμν) out of D and H. Use it to express Maxwell’s equations inside matter in terms of the free current density J μ f . (b) Construct the dual tensor Hμν (analogous to Gμν). (c) Minkowski proposed the relativistic constitutive relations for linear media: ... Where is the proper30 permittivity, μ is the proper permeability, and ημ is the 4-velocity of the material. Show that Minkowski’s formulas reproduce Eqs. 4.32 and 6.31, when the material is at rest. (d) Work out the formulas relating D and H to E and B for a medium moving with (ordinary) velocity u. Reference equation 4.32 ... Reference equation 6.31 B = μH, Get solution

71. Use the Larmor formula (Eq. 11.70) and special relativity to derivethe Liénard formula (Eq. 11.73). Equation 11.70 ... Equation 11.73 ... Get solution

72. The natural relativistic generalization of the Abraham-Lorentz formula (Eq. 11.80) would seem to be ... This is certainly a 4-vector, and it reduces to the Abraham-Lorentz formula in the nonrelativistic limit v ≪ c. (a) Show, nevertheless, that this is not a possible Minkowski force. [Hint: See Prob. 12.39d.] (b) Find a correction term that, when added to the right side, removes the objection you raised in (a), without affecting the 4-vector character of the formula or its nonrelativistic limit. Reference prob 12.39d Define proper acceleration in the obvious way: ...(d) Write the Minkowski version of Newton’s second law, Eq. 12.68, in terms of Get solution

73. Generalize the laws of relativistic electrodynamics (Eqs. 12.127 and 12.128) to include magnetic charge. [Refer to Sect. 7.3.4.] Equation 12.127 ... Equation 12.128 ... Get solution

### Chapter #11 Solutions - Introduction to Electrodynamics - David J Griffiths - 4th Edition

1. Check that the retarded potentials of an oscillating dipole (Eqs. 11.12 and 11.17) satisfy the Lorenz gauge condition. Do not use approximation 3. Reference equation 11.12 ... Reference equation 11.17 ... Get solution

2. Equation 11.14 can be expressed in “coordinate-free” form by writing ... Do so, and likewise for Eqs. 11.17, 11.18. 11.19, and 11.21. Reference equation 11.14 ... Reference equation 11.17 ... Reference equation 11.18 ... Reference equation 11.19 ... Reference equation 11.21 ... Get solution

3. Find the radiation resistance of the wire joining the two ends of the dipole. (This is the resistance that would give the same average power loss—to heat—as the oscillating dipole in fact puts out in the form of radiation.) Show that R = 790 (d/λ)2 Ω, where λ is the wavelength of the radiation. For the wires in an ordinary radio (say, d = 5 cm), should you worry about the radiative contribution to the total resistance? Get solution

4. A rotating electric dipole can be thought of as the superposition of two oscillating dipoles, one along the x axis and the other along the y axis (Fig. 11.7), with the latter out of phase by 90˚: ... Using the principle of superposition and Eqs. 11.18 and 11.19 (perhaps in the form suggested by Prob. 11.2), find the fields of a rotating dipole. Also find the Poynting vector and the intensity of the radiation. Sketch the intensity profile as a function of the polar angle θ, and calculate the total power radiated. Does the answer seem reasonable? (Note that power, being quadratic in the fields, does not satisfy the superposition principle. In this instance, however, it seems to. How do you account for this?) Reference equation 11.18 ... Reference equation 11.19 ... Reference prob 11.2 Equation 11.14 can be expressed in “coordinate-free” form by writing ... Do so, and likewise for Eqs. 11.17, 11.18. 11.19, and 11.21. Get solution

5. Calculate the electric and magnetic fields of an oscillating magnetic dipole without using approximation 3. [Do they look familiar? Compare Prob. 9.35.] Find the Poynting vector, and show that the intensity of the radiation is exactly the same as we got using approximation 3. Reference prob.9.35 Suppose ... (This is, incidentally, the simplest possible spherical wave. For notational convenience, let (kr − ωt) ≡ u in your calculations.) (a) Show that E obeys all four of Maxwell’s equations, in vacuum, and find the associated magnetic field. (b) Calculate the Poynting vector. Average S over a full cycle to get the intensity vector I. (Does it point in the expected direction? Does it fall off like r−2, as it should?) (c) Integrate I · da over a spherical surface to determine the total power radiated. Get solution

6. Find the radiation resistance (Prob. 11.3) for the oscillating magnetic dipole in Fig. 11.8. Express your answer in terms of λ and b, and compare the radiation resistance of the electric dipole. figure 11.8 ... Reference Prob. 11.3 Find the radiation resistance of the wire joining the two ends of the dipole. (This is the resistance that would give the same average power loss—to heat—as the oscillating dipole in fact puts out in the form of radiation.) Show that R = 790 (d/λ)2 Ω, where λ is the wavelength of the radiation. For the wires in an ordinary radio (say, d = 5 cm), should you worry about the radiative contribution to the total resistance? Get solution

7. Use the “duality” transformation of Prob. 7.64, together with the fields of an oscillating electric dipole (Eqs. 11.18 and 11.19), to determine the fields that would be produced by an oscillating “Gilbert” magnetic dipole (composed of equal and opposite magnetic charges, instead of an electric current loop). Compare Eqs. 11.36 and 11.37, and comment on the result. Reference equation 11.36 and 11.37 ... Reference prob 7.64. (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 α = 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. Get solution

8. A parallel-plate capacitor C, with plate separation d, is given an initial charge (±)Q0. It is then connected to a resistor R, and discharges, Q(t) = ... (a) What fraction of its initial energy (Q20 /2C) does it radiate away? (b) If C = 1 pF, R = 1000 Ω, and d = 0.1 mm, what is the actual number? In electronics we don’t ordinarily worry about radiative losses; does that seem reasonable, in this case? Get solution

9. Apply Eqs. 11.59 and 11.60 to the rotating dipole of Prob. 11.4. Explain any apparent discrepancies with your previous answer. Reference equation 11.59 ... Reference equation 11.60 ... Reference Prob. 11.4. A rotating electric dipole can be thought of as the superposition of two oscillating dipoles, one along the x axis and the other along the y axis (Fig. 11.7), with the latter out of phase by 90˚: ... Using the principle of superposition and Eqs. 11.18 and 11.19 (perhaps in the form suggested by Prob. 11.2), find the fields of a rotating dipole. Also find the Poynting vector and the intensity of the radiation. Sketch the intensity profile as a function of the polar angle θ, and calculate the total power radiated. Does the answer seem reasonable? (Note that power, being quadratic in the fields, does not satisfy the superposition principle. In this instance, however, it seems to. How do you account for this?) Get solution

10. An insulating circular ring (radius b) lies in the xy plane, centered at the origin. It carries a linear charge density λ = λ0 sin φ, where λ0 is constant and φ is the usual azimuthal angle. The ring is now set spinning at a constant angular velocity ω about the z axis. Calculate the power radiated. Get solution

11. A current I (t) flows around the circular ring in Fig. 11.8. Derive the general formula for the power radiated (analogous to Eq. 11.60), expressing your answer in terms of the magnetic dipole moment, m(t), of the loop. Figure 11.8 ... Equation 11.60 ... Get solution

12. An electron is released from rest and falls under the influence of gravity. In the first centimeter, what fraction of the potential energy lost is radiated away? Get solution

13. A positive charge q is fired head-on at a distant positive charge Q (which is held stationary), with an initial velocity v0. It comes in, decelerates to v = 0, and returns out to infinity. What fraction of its initial energy ...is radiated away? Assume v0 ≪ c, and that you can safely ignore the effect of radiative losses on the motion of the particle. Get solution

14. In Bohr’s theory of hydrogen, the electron in its ground state was supposed to travel in a circle of radius 5 × 10−11m, held in orbit by the Coulomb attraction of the proton. According to classical electrodynamics, this electron should radiate, and hence spiral in to the nucleus. Show that v ≪ c for most of the trip (so you can use the Larmor formula), and calculate the lifespan of Bohr’s atom. (Assume each revolution is essentially circular.) Get solution

15. Find the angle θmax at which the maximum radiation is emitted, in Ex. 11.3 (Fig. 11.13). Show that for ultrarelativistic speeds (v close to c), ... ... What is the intensity of the radiation in this maximal direction (in the ultrarelativistic case), in proportion to the same quantity for a particle instantaneously at rest? Give your answer in terms of γ . Figure 11.13 ... Get solution

16. In Ex. 11.3 we assumed the velocity and acceleration were (instantaneously, at least) collinear. Carry out the same analysis for the case where they are perpendicular. Choose your axes so that v lies along the z axis and a along the x axis (Fig. 11.14), so that ... Check that P is consistent with the Liénard formula ... Figure 11.14 and 11.5 ... For relativistic velocities (β ≈ 1) the radiation is again sharply peaked in the forward direction (Fig. 11.15). The most important application of these formulas is to circular motion—in this case the radiation is called synchrotron radiation. For a relativistic electron, the radiation sweeps around like a locomotive’s headlight as the particle moves.] Get solution

17. (a) A particle of charge q moves in a circle of radius R at a constant speed v. To sustain the motion, you must, of course, provide a centripetal force mv2/R; what additional force (Fe) must you exert, in order to counteract the radiation reaction? [It’s easiest to express the answer in terms of the instantaneous velocity v.] What power (Pe) does this extra force deliver? Compare Pe with the power radiated (use the Larmor formula). (b) Repeat part (a) for a particle in simple harmonic motion with amplitude A and angular frequency ... Explain the discrepancy. (c) Consider the case of a particle in free fall (constant acceleration g). What is the radiation reaction force? What is the power radiated? Comment on these results. Get solution

18. A point charge q, of mass m, is attached to a spring of constant k. At time t = 0 it is given a kick, so its initial energy is...Now it oscillates, gradually radiating away this energy. (a) Confirm that the total energy radiated is equal to U0. Assume the radiation damping is small, so you can write the equation of motion as ... and the solution as ... with...and γ ≪ω0 (drop γ2 in comparison to ω2 0, and when you average over a complete cycle, ignore the change in e−γ t ). (b) Suppose now we have two such oscillators, and we start them off with identical kicks. Regardless of their relative positions and orientations, the total energy radiated must be 2U0. But what if they are right on top of each other, so it’s equivalent to a single oscillator with twice the charge; the Larmor formula says that the power radiated is four times as great, suggesting that the total will be 4U0. Find the error in this reasoning, and show that the total is actually 2U0, as it should be.13 Get solution

19. With the inclusion of the radiation reaction force (Eq. 11.80), Newton’s second law for a charged particle becomes ... where F is the external force acting on the particle. (a) In contrast to the case of an uncharged particle (a = F/m), acceleration (like position and velocity) must now be a continuous function of time, even if the force changes abruptly. (Physically, the radiation reaction damps out any rapid change in a.) Prove that a is continuous at any time t, by integrating the equation of motion above from ... and taking the limit ... (b) A particle is subjected to a constant force F, beginning at time t = 0 and lasting until time T . Find the most general solution a(t) to the equation of motion in each of the three periods: (i) t 0; (ii) 0 t T ; (iii) t > T . (c) Impose the continuity condition (a) at t = 0 and t = T . Show that you can either eliminate the runaway in region (iii) or avoid preacceleration in region (i), but not both. (d) If you choose to eliminate the runaway, what is the acceleration as a function of time, in each interval? How about the velocity? (The latter must, of course, be continuous at t = 0 and t = T .) Assume the particle was originally at rest: v(−∞) = 0. (e) Plot a(t) and v(t), both for an uncharged particle and for a (nonrunaway) charged particle, subject to this force. Reference equation 11.80 ... Get solution

20. Deduce Eq. 11.100 from Eq. 11.99. Here are three methods: (a) Use the Abraham-Lorentz formula to determine the radiation reaction on each end of the dumbbell; add this to the interaction term (Eq. 11.99). (b) Method (a) has the defect that it uses the Abraham-Lorentz formula—the very thing that we were trying to derive. To avoid this, let F(q) be the total d-independent part of the self-force on a charge q. Then ... where Fint is the interaction part (Eq. 11.99), and F(q/2) is the self-force on each end. Now, F(q) must be proportional to q2, since the field is proportional to q and the force is qE. So F(q/2) = (1/4)F(q). Take it from there. (c) Smear out the charge along a strip of length L oriented perpendicular to the motion (the charge density, then, is λ = q/L); find the cumulative interaction force for all pairs of segments, using Eq. 11.99 (with the correspondence q/2 → λ dy1, at one end and q/2 → λ dy2 at the other). Make sure you don’t count the same pair twice. Equation 11.100 ... Equation 11.99 ... Get solution

21. An electric dipole rotates at constant angular velocity ω in the x y plane. (The charges, ±q, are at ... the magnitude of the dipole moment is p = 2qR.) (a) Find the interaction term in the self-torque (analogous to Eq. 11.99). Assume the motion is nonrelativistic (ωR ≪ c). (b) Use the method of Prob. 11.20(a) to obtain the total radiation reaction torque on this system. (c) Check that this result is consistent with the power radiated (Eq. 11.60). Reference equation 11.60 ... Reference equation 11.99 ... Reference prob 11.20 Deduce Eq. 11.100 from Eq. 11.99. Here are three methods: (a) Use the Abraham-Lorentz formula to determine the radiation reaction on each end of the dumbbell; add this to the interaction term (Eq. 11.99). Get solution

22. A particle of mass m and charge q is attached to a spring with force constant k, hanging from the ceiling (Fig. 11.18). Its equilibrium position is a distance h above the floor. It is pulled down a distance d below equilibrium and released, at time t = 0. (a) Under the usual assumptions (d ≪ λ ≪ h), calculate the intensity of the radiation hitting the floor, as a function of the distance R from the point directly below q. [Note: The intensity here is the average power per unit area of floor.] ... At what R is the radiation most intense? Neglect the radiative damping of the oscillator. (b) As a check on your formula, assume the floor is of infinite extent, and calculate the average energy per unit time striking the entire floor. Is it what you’d expect? (c) Because it is losing energy in the form of radiation, the amplitude of the oscillation will gradually decrease. After what time τ has the amplitude been reduced to d/e? (Assume the fraction of the total energy lost in one cycle is very small.) Get solution

23. A radio tower rises to height h above flat horizontal ground. At the top is a magnetic dipole antenna, of radius b, with its axis vertical. FM station KRUD broadcasts from this antenna at (angular) frequency ω, with a total radiated power P (that’s averaged, of course, over a full cycle). Neighbors have complained about problems they attribute to excessive radiation from the tower—interference with their stereo systems, mechanical garage doors opening and closing mysteriously, and a variety of suspicious medical problems. But the city engineer who measured the radiation level at the base of the tower found it to be well below the accepted standard. You have been hired by the Neighborhood Association to assess the engineer’s report. (a) In terms of the variables given (not all of which may be relevant), find the formula for the intensity of the radiation at ground level, a distance R from the base of the tower. You may assume that b ≪ c/ ω ≪ h. [Note:We are interested only in the magnitude of the radiation, not in its direction—when measurements are taken, the detector will be aimed directly at the antenna.] (b) How far from the base of the tower should the engineer have made the measurement? What is the formula for the intensity at this location? (c) KRUD’s actual power output is 35 kilowatts, its frequency is 90 MHz, the antenna’s radius is 6 cm, and the height of the tower is 200 m. The city’s radioemission limit is 200 microwatts/cm2. Is KRUD in compliance? Get solution

24. As a model for electric quadrupole radiation, consider two oppositely oriented oscillating electric dipoles, separated by a distance d, as shown in Fig. 11.19. Use the results of Sect. 11.1.2 for the potentials of each dipole, but note that they are not located at the origin. Keeping only the terms of first order in d: ... (a) Find the scalar and vector potentials. (b) Find the electric and magnetic fields. (c) Find the Poynting vector and the power radiated. Sketch the intensity profile as a function of θ. Get solution

25. As you know, the magnetic north pole of the earth does not coincide with the geographic north pole—in fact, it’s off by about 11? Relative to the fixed axis of rotation, therefore, the magnetic dipole moment of the earth is changing with time, and the earth must be giving off magnetic dipole radiation. (a) Find the formula for the total power radiated, in terms of the following parameters: Ψ (the angle between the geographic and magnetic north poles), M (the magnitude of the earth’s magnetic dipole moment), and ω (the angular velocity of rotation of the earth). [Hint: refer to Prob. 11.4 or Prob. 11.11.] (b) Using the fact that the earth’s magnetic field is about half a gauss at the equator, estimate the magnetic dipole moment M of the earth. (c) Find the power radiated. (d) Pulsars are thought to be rotating neutron stars, with a typical radius of 10 km, a rotational period of 10−3 s, and a surface magnetic field of 108 T. What sort of radiated power would you expect from such a star?20 Reference Prob. 11.4 A rotating electric dipole can be thought of as the superposition of two oscillating dipoles, one along the x axis and the other along the y axis (Fig. 11.7), with the latter out of phase by 90˚: ... Using the principle of superposition and Eqs. 11.18 and 11.19 (perhaps in the form suggested by Prob. 11.2), find the fields of a rotating dipole. Also find the Poynting vector and the intensity of the radiation. Sketch the intensity profile as a function of the polar angle θ, and calculate the total power radiated. Does the answer seem reasonable? (Note that power, being quadratic in the fields, does not satisfy the superposition principle. In this instance, however, it seems to. How do you account for this?) Reference Prob. 11.11 A current I (t) flows around the circular ring in Fig. 11.8. Derive the general formula for the power radiated (analogous to Eq. 11.60), expressing your answer in terms of the magnetic dipole moment, m(t), of the loop. Figure 11.8 ... Equation 11.60 ... Get solution

26. An ideal electric dipole is situated at the origin; its dipole moment points in the ... direction, and is quadratic in time: ... Where ... is a constant. (a) Use the method of Section 11.1.2 to determine the (exact) electric and magnetic fields, for all r > 0 (there’s also a delta- unction term at the origin, but we’re not concerned with that). (b) Calculate the power, P(r, t), passing through a sphere of radius r . (c) Find the total power radiated (Eq. 11.2), and check that your answer is consistent with Eq. 11.60.21 Reference equation 11.60 ... Reference equation 11.2 ... Get solution

27. In Section 11.2.1 we calculated the energy per unit time radiated by a (nonrelativistic) point charge—the Larmor formula. In the same spirit: (a) Calculate the momentum per unit time radiated. (b) Calculate the angular momentum per unit time radiated. Get solution

28. Suppose the (electrically neutral) yz plane carries a time-dependent but uniform surface current ... (a) Find the electric and magnetic fields at a height x above the plane if (i) a constant current is turned on at t = 0: ... (ii) a linearly increasing current is turned on at t = 0: ... (b) Show that the retarded vector potential can be written in the form ... and from this determine E and B. (c) Show that the total power radiated per unit area of surface is ... Explain what you mean by “radiation,” in this case, given that the source is not localized.22 Get solution

29. Use the duality transformation (Prob. 7.64) to construct the electric and magnetic fields of a magnetic monopole qm in arbitrary motion, and find the “Larmor formula” for the power radiated Reference prob 7.64 iv class="question"> (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 α = 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. Get solution

30. Assuming you exclude the runaway solution in Prob. 11.19, calculate (a) the work done by the external force, (b) the final kinetic energy (assume the initial kinetic energy was zero), (c) the total energy radiated. Check that energy is conserved in this process. Reference prob 11.19 With the inclusion of the radiation reaction force (Eq. 11.80), Newton’s second law for a charged particle becomes ... where F is the external force acting on the particle. (a) In contrast to the case of an uncharged particle (a = F/m), acceleration (like position and velocity) must now be a continuous function of time, even if the force changes abruptly. (Physically, the radiation reaction damps out any rapid change in a.) Prove that a is continuous at any time t, by integrating the equation of motion above from ... and taking the limit ... (b) A particle is subjected to a constant force F, beginning at time t = 0 and lasting until time T . Find the most general solution a(t) to the equation of motion in each of the three periods: (i) t 0; (ii) 0 t T ; (iii) t > T . (c) Impose the continuity condition (a) at t = 0 and t = T . Show that you can either eliminate the runaway in region (iii) or avoid preacceleration in region (i), but not both. (d) If you choose to eliminate the runaway, what is the acceleration as a function of time, in each interval? How about the velocity? (The latter must, of course, be continuous at t = 0 and t = T .) Assume the particle was originally at rest: v(−∞) = 0. (e) Plot a(t) and v(t), both for an uncharged particle and for a (nonrunaway) charged particle, subject to this force. Get solution

31. (a) Repeat Prob. 11.19, but this time let the external force be a Dirac delta function: F(t) = kδ(t) (for some constant k).25 [Note that the acceleration is now discontinuous at t = 0 (though the velocity must still be continuous); use the method of Prob. 11.19 a) to show that ; Δa = −k/mτ . In this problem there are only two intervals to consider: (i) t 0, and (ii) t > 0.] (b) As in Prob. 11.30, check that energy is conserved in this process. Reference prob 11.19 With the inclusion of the radiation reaction force (Eq. 11.80), Newton’s second law for a charged particle becomes ... where F is the external force acting on the particle. (a) In contrast to the case of an uncharged particle (a = F/m), acceleration (like position and velocity) must now be a continuous function of time, even if the force changes abruptly. (Physically, the radiation reaction damps out any rapid change in a.) Prove that a is continuous at any time t, by integrating the equation of motion above from ... and taking the limit ... (b) A particle is subjected to a constant force F, beginning at time t = 0 and lasting until time T . Find the most general solution a(t) to the equation of motion in each of the three periods: (i) t 0; (ii) 0 t T ; (iii) t > T . (c) Impose the continuity condition (a) at t = 0 and t = T . Show that you can either eliminate the runaway in region (iii) or avoid preacceleration in region (i), but not both. (d) If you choose to eliminate the runaway, what is the acceleration as a function of time, in each interval? How about the velocity? (The latter must, of course, be continuous at t = 0 and t = T .) Assume the particle was originally at rest: v(−∞) = 0. (e) Plot a(t) and v(t), both for an uncharged particle and for a (nonrunaway) charged particle, subject to this force. Reference prob 11.30 Assuming you exclude the runaway solution in Prob. 11.19, calculate (a) the work done by the external force, (b) the final kinetic energy (assume the initial kinetic energy was zero), (c) the total energy radiated. Check that energy is conserved in this process. Get solution

32. A charged particle, traveling in from −∞along the x axis, encounters a rectangular potential energy barrier ... Show that, because of the radiation reaction, it is possible for the particle to tunnel through the barrier—that is, even if the incident kinetic energy is less than U0, the particle can pass through.26 [Hint: Your task is to solve the equation ... subject to the force ... Refer to Probs. 11.19 and 11.31, but notice that this time the force is a specified function of x, not t. There are three regions to consider: (i) x 0, (ii) 0 x L, (iii) x > L. Find the general solution for a(t), v(t), and x(t) in each region, exclude the runaway in region (iii), and impose the appropriate boundary conditions at x = 0 and x = L. Show that the final velocity (v f ) is related to the time T spent traversing the barrier by the equation ... and the initial velocity (at x = −∞) is ... To simplify these results (since all we’re looking for is a specific example), suppose the final kinetic energy is half the barrier height. Show that in this case ... In particular, if you choose ... the initial kinetic energy is (8/9)U0, and the particle makes it through, even though it didn’t have sufficient energy to get over the barrier!] Reference prob 11.19With the inclusion of the radiation reaction force (Eq. 11.80), Newton’s second law for a charged particle becomes ... where F is the external force acting on the particle. (a) In contrast to the case of an uncharged particle (a = F/m), acceleration (like position and velocity) must now be a continuous function of time, even if the force changes abruptly. (Physically, the radiation reaction damps out any rapid change in a.) Prove that a is continuous at any time t, by integrating the equation of motion above from ... and taking the limit ... (b) A particle is subjected to a constant force F, beginning at time t = 0 and lasting until time T . Find the most general solution a(t) to the equation of motion in each of the three periods: (i) t 0; (ii) 0 t T ; (iii) t > T . (c) Impose the continuity condition (a) at t = 0 and t = T . Show that you can either eliminate the runaway in region (iii) or avoid preacceleration in region (i), but not both. (d) If you choose to eliminate the runaway, what is the acceleration as a function of time, in each interval? How about the velocity? (The latter must, of course, be continuous at t = 0 and t = T .) Assume the particle was originally at rest: v(−∞) = 0. (e) Plot a(t) and v(t), both for an uncharged particle and for a (nonrunaway) charged particle, subject to this force. Reference prob 11.31 (a) Repeat Prob. 11.19, but this time let the external force be a Dirac delta function: F(t) = kδ(t) (for some constant k).25 [Note that the acceleration is now discontinuous at t = 0 (though the velocity must still be continuous); use the method of Prob. 11.19 a) to show that ; Δa = −k/mτ . In this problem there are only two intervals to consider: (i) t 0, and (ii) t > 0.] (b) As in Prob. 11.30, check that energy is conserved in this process. Get solution

33. (a) Find the radiation reaction force on a particle moving with arbitrary velocity in a straight line, by reconstructing the argument in Sect. 11.2.3 without assuming v(tr ) = 0. (b) Show that this result is consistent (in the sense of Eq. 11.78) with the power radiated by such a particle (Eq. 11.75). Equation 11.78 ... Equation 11.75 ... Get solution

34. a) Does a particle in hyperbolic motion (Eq. 10.52) radiate? (Use the exact formula (Eq. 11.75) to calculate the power radiated.) (b) Does a particle in hyperbolic motion experience a radiation reaction? (Use the exact formula (Prob. 11.33) to determine the reaction force.) [Comment: These famous questions carry important implications for the principle of equivalence.27] Reference equation 11.75 ... Reference equation 10.52 ... Reference prob 11.33 Get solution

35. Use the result of Prob. 10.34 to determine the power radiated by an ideal electric dipole, p(t), at the origin. Check that your answer is consistent with Eq. 11.22, in the case of sinusoidal time dependence, and with Prob. 11.26, in the case of quadratic time dependence. the result of Prob. 10.34 ... Reference equation 11.22 ... Reference problem 11.26 Get solution

### Chapter #10 Solutions - Introduction to Electrodynamics - David J Griffiths - 4th Edition

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

### Chapter #9 Solutions - Introduction to Electrodynamics - David J Griffiths - 4th Edition

1. By explicit differentiation, check that the functions f1, f2, and f3 in the text satisfy the wave equation. Show that f4 and f5 do not. Get solution

2. Show that the standing wave f (z, t) = A sin(kz) cos(kvt) satisfies the wave equation, and express it as the sum of a wave traveling to the left and a wave traveling to the right (Eq. 9.6). Equation 9.6 f ( z , t ) = g(z − vt) + h(z + vt). Get solution

3. Use Eq. 9.19 to determine A3 and δ3 in terms of A1, A2, δ1, and δ2. Equation 9.19 ... Get solution

4. Obtain Eq. 9.20 directly from the wave equation, by separation of variables. Equation 9.20 ... Get solution

5. Suppose you send an incident wave of specified shape, gI (z − v1t), down string number 1. It gives rise to a reflected wave, hR(z + v1t), and a transmitted wave, gT (z − v2t). By imposing the boundary conditions 9.26 and 9.27, find hR and gT . Get solution

6. (a) Formulate an appropriate boundary condition, to replace Eq. 9.27, for the case of two strings under tension T joined by a knot of mass m. (b) Find the amplitude and phase of the reflected and transmitted waves for the case where the knot has a mass m and the second string is massless. Reference equation 9.27 ... Get solution

7. Suppose string 2 is embedded in a viscous medium (such as molasses), which imposes a drag force that is proportional to its (transverse) speed: ... (a) Derive the modified wave equation describing the motion of the string. (b) Solve this equation, assuming the string vibrates at the incident frequency ω. That is, look for solutions of the form... (c) Show that the waves are attenuated (that is, their amplitude decreases with increasing z). Find the characteristic penetration distance, at which the amplitude is 1/e of its original value, in terms of γ, T,μ, and ω. (d) If a wave of amplitude AI , phase δI = 0, and frequency ω is incident from the left (string 1), find the reflected wave’s amplitude and phase. Get solution

8. Equation 9.36 describes the most general linearly polarized wave on a string. Linear (or “plane”) polarization (so called because the displacement is parallel to a fixed vector ...) results from the combination of horizontally and vertically polarized waves of the same phase (Eq. 9.39). If the two components are of equal amplitude, but out of phase by 90° (say, δv = 0, δh = 90?), the result is a circularly polarized wave. In that case: (a) At a fixed point z, show that the string moves in a circle about the z axis. Does it go clockwise or counterclockwise, as you look down the axis toward the origin? How would you construct a wave circling the other way? (In optics, the clockwise case is called right circular polarization, and the counterclockwise, left circular polarization.)3 (b) Sketch the string at time t = 0. (c) How would you shake the string in order to produce a circularly polarized wave? Equation 9.36 ... Get solution

9. Write down the (real) electric and magnetic fields for a monochromatic plane wave of amplitude E0, frequency ω, and phase angle zero that is (a) traveling in the negative x direction and polarized in the z direction; (b) traveling in the direction from the origin to the point (1, 1, 1), with polarization parallel to the xz plane. In each case, sketch the wave, and give the explicit Cartesian components of k and ... Get solution

10. The intensity of sunlight hitting the earth is about 1300 W/m2. If sunlight strikes a perfect absorber, what pressure does it exert? How about a perfect reflector? What fraction of atmospheric pressure does this amount to? Get solution

11. Consider a particle of charge q and mass m, free to move in the xy plane in response to an electromagnetic wave propagating in the z direction (Eq. 9.48—might as well set δ = 0). (a) Ignoring the magnetic force, find the velocity of the particle, as a function of time. (Assume the average velocity is zero.) (b) Now calculate the resulting magnetic force on the particle. (c) Show that the (time) average magnetic force is zero. The problem with this naive model for the pressure of light is that the velocity is 90 ? out of phase with the fields. For energy to be absorbed, there’s got to be some resistance to the motion of the charges. Suppose we include a force of the form −γmv, for some damping constant γ . (d) Repeat part (a) (ignore the exponentially damped transient). Repeat part (b), and find the average magnetic force on the particle.9 ... Get solution

12. In the complex notation there is a clever device for finding the time average of a product. Suppose f (r, t) = A cos (k · r – ω + δa) and g(r, t) = B cos (k · r − ωt + δb). Show that ... where the star denotes complex conjugation. [Note that this only works if the two waves have the same k and ω, but they need not have the same amplitude or phase.] For example, ... Get solution

13. Find all elements of the Maxwell stress tensor for a monochromatic plane wave traveling in the z direction and linearly polarized in the x direction (Eq. 9.48). Does your answer make sense? (Remember that ...represents the momentum flux density.) How is the momentum flux density related to the energy density, in this case? Reference equation 9.48 ... Get solution

14. Calculate the exact reflection and transmission coefficients, without assuming μ1 = μ2 = μ0. Confirm that R + T = 1. Get solution

15. In writing Eqs. 9.76 and 9.77, I tacitly assumed that the reflected and transmitted waves have the same polarization as the incident wave—along the x direction. Prove that this must be so. [Hint: Let the polarization vectors of the transmitted and reflected waves be ... and prove from the boundary conditions that θT = θR = 0.] Reference equation 9.76 ... Reference equation 9.77 ... Get solution

16. Suppose Aeiax + Beibx = Ceicx , for some nonzero constants A, B, C, a, b, c, and for all x. Prove that a = b = c and A + B = C. Get solution

17. Analyze the case of polarization perpendicular to the plane of incidence (i.e. electric fields in the y direction, in Fig. 9.15). Impose the boundary conditions (Eq. 9.101), and obtain the Fresnel equations for ... Sketch ... as functions of θI , for the case β = n2/n1 = 1.5. (Note that for this β the reflected wave is always 180° out of phase.) Show that there is no Brewster’s angle for any n1 and n2: ... is never zero (unless, of course, n1 = n2 and μ1 = μ2, in which case the two media are optically indistinguishable). Confirm that your Fresnel equations reduce to the proper forms at normal incidence. Compute the reflection and transmission coefficients, and check that they add up to 1. ... Get solution

18. The index of refraction of diamond is 2.42. Construct the graph analogous to Fig. 9.16 for the air/diamond interface. (Assume μ1 = μ2 = μ0.) In particular, calculate (a) the amplitudes at normal incidence, (b) Brewster’s angle, and (c) the “crossover” angle, at which the reflected and transmitted amplitudes are equal. Figure 9.16 ... Get solution

19. (a) Suppose you imbedded some free charge in a piece of glass. About how long would it take for the charge to flow to the surface? (b) Silver is an excellent conductor, but it’s expensive. Suppose you were designing a microwave experiment to operate at a frequency of 1010 Hz. How thick would you make the silver coatings? (c) Find the wavelength and propagation speed in copper for radio waves at 1 MHz. Compare the corresponding values in air (or vacuum). Get solution

20. (a) Show that the skin depth in a poor conductor ...(independent of frequency). Find the skin depth (in meters) for (pure) water. (Use the static values of ? , μ, and σ; your answers will be valid, then, only at relatively low frequencies.) (b) Show that the skin depth in a good conductor...(where λ is the wavelength in the conductor). Find the skin depth (in nanometers) for a typical metal (σ ≈ 107(Ω m) −1) in the visible range (ω ≈ 1015/s), assuming ? ≈ ? 0 and μ ≈ μ0. Why are metals opaque? (c) Show that in a good conductor the magnetic field lags the electric field by 45?, and find the ratio of their amplitudes. For a numerical example, use the “typical metal” in part (b). Get solution

21. (a) Calculate the (time-averaged) energy density of an electromagnetic plane wave in a conducting medium (Eq. 9.138). Show that the magnetic contribution always dominates. (b) Show that the intensity is... Equation 9.138 ... Get solution

22. Calculate the reflection coefficient for light at an air-to-silver interface...at optical frequencies (ω = 4 × 1015/s). Get solution

23. a) Shallow water is nondispersive; waves travel at a speed that is proportional to the square root of the depth. In deep water, however, the waves can’t “feel” all the way down to the bottom—they behave as though the depth were proportional to λ. (Actually, the distinction between “shallow” and “deep” itself depends on the wavelength: If the depth is less than λ, the water is “shallow”; if it is substantially greater than λ, the water is “deep.”) Show that the wave velocity of deep water waves is twice the group velocity. (b) In quantum mechanics, a free particle of mass m traveling in the x direction is described by the wave function ... where p is the momentum, and E = p2/2m is the kinetic energy. Calculate the group velocity and the wave velocity. Which one corresponds to the classical speed of the particle? Note that the wave velocity is half the group velocity. Get solution

24. If you take the model in Ex. 4.1 at face value, what natural frequency do you get? Put in the actual numbers. Where, in the electromagnetic spectrum, does this lie, assuming the radius of the atom is 0.5 Å? Find the coefficients of refraction and dispersion, and compare them with the measured values for hydrogen at 0 ? C and atmospheric pressure: A = 1.36 × 10−4, B = 7.7 × 10−15m2 . Get solution

25. Find the width of the anomalous dispersion region for the case of a single resonance at frequency ω0. Assume γ ≪ ω0. Show that the index of refraction assumes its maximum and minimum values at points where the absorption coefficient is at half-maximum. Get solution

26. Starting with Eq. 9.170, calculate the group velocity, assuming there is only one resonance, at ω0. Use a computer to graph y ≡ vg/c as a function of x ≡ (ω/ω0)2;, from x = 0 to 2, (a) for γ = 0, and (b) for γ = (0.1)ω0. Let (Nq2)/(2m ? 0 ω 2 0) = 0 . 003. Note that the group velocity can exceed c. Get solution

27. (a) Derive Eqs. 9.179, and from these obtain Eqs. 9.180. (b) Put Eq. 9.180 into Maxwell’s equations (i) and (ii) to obtain Eq. 9.181. Check that you get the same results using (i) and (iv) of Eq. 9.179. Reference equation 9.179 ... Reference equation 9.180 ... Reference equation 9.181 ... Get solution

28. Show that the mode TE00 cannot occur in a rectangular wave guide. [Hint: In this case ω/c = k, so Eqs. 9.180 are indeterminate, and you must go back to Eq. 9.179. Show that Bz is a constant, and hence—applying Faraday’s law in integral form to a cross section—that Bz = 0, so this would be a TEM mode.] Reference equation 9.179 ... Reference equation 9.180 ... Get solution

29. Consider a rectangular wave guide with dimensions 2.28 cm × 1.01 cm. What TE modes will propagate in this wave guide, if the driving frequency is 1.70 × 1010 Hz? Suppose you wanted to excite only one TE mode; what range of frequencies could you use? What are the corresponding wavelengths (in open space)? Get solution

30. Confirm that the energy in the TEmn mode travels at the group velocity. [Hint: Find the time averaged Poynting vector ...and the energy density ... (use Prob. 9.12 if you wish). Integrate over the cross section of the wave guide to get the energy per unit time and per unit length carried by the wave, and take their ratio.] Reference prob 9.12 In the complex notation there is a clever device for finding the time average of a product. Suppose f (r, t) = A cos (k · r – ω + δa) and g(r, t) = B cos (k · r − ωt + δb). Show that ... where the star denotes complex conjugation. [Note that this only works if the two waves have the same k and ω, but they need not have the same amplitude or phase.] For example, ... Get solution

31. Work out the theory of TM modes for a rectangular wave guide. In particular, find the longitudinal electric field, the cutoff frequencies, and the wave and group velocities. Find the ratio of the lowest TM cutoff frequency to the lowest TE cutoff frequency, for a given wave guide. [Caution: What is the lowest TM mode?] Get solution

32. (a) Show directly that Eqs. 9.197 satisfy Maxwell’s equations (Eq. 9.177) and the boundary conditions (Eq. 9.175). (b) Find the charge density, λ(z, t), and the current, I (z, t), on the inner conductor. Reference equation 9.197 ... Reference equation 9.197 ... Reference equation 9.195 ... Get solution

33. The “inversion theorem” for Fourier transforms states that ... Use this to determine ..., in Eq. 9.20, in terms of f (z, 0) and ...a Reference equation 9.20 ... Get solution

34. [The naive explanation for the pressure of light offered in Section 9.2.3 has its flaws, as you discovered if you worked Problem 9.11. Here’s another account, due originally to Planck.25] A plane wave traveling through vacuum in the z direction encounters a perfect conductor occupying the region z ≥ 0, and reflects back: ... (a) Find the accompanying magnetic field (in the region z 0). (b) Assuming B = 0 inside the conductor, find the current K on the surface z = 0, by invoking the appropriate boundary condition. (c) Find the magnetic force per unit area on the surface, and compare its time average with the expected radiation pressure (Eq. 9.64). Reference prob 9.11 Consider a particle of charge q and mass m, free to move in the xy plane in response to an electromagnetic wave propagating in the z direction (Eq. 9.48—might as well set δ = 0). (a) Ignoring the magnetic force, find the velocity of the particle, as a function of time. (Assume the average velocity is zero.) (b) Now calculate the resulting magnetic force on the particle. (c) Show that the (time) average magnetic force is zero. The problem with this naive model for the pressure of light is that the velocity is 90 ? out of phase with the fields. For energy to be absorbed, there’s got to be some resistance to the motion of the charges. Suppose we include a force of the form −γmv, for some damping constant γ . (d) Repeat part (a) (ignore the exponentially damped transient). Repeat part (b), and find the average magnetic force on the particle.9 ... Get solution

35. Suppose ... (This is, incidentally, the simplest possible spherical wave. For notational convenience, let (kr − ωt) ≡ u in your calculations.) (a) Show that E obeys all four of Maxwell’s equations, in vacuum, and find the associated magnetic field. (b) Calculate the Poynting vector. Average S over a full cycle to get the intensity vector I. (Does it point in the expected direction? Does it fall off like r−2, as it should?) (c) Integrate I · da over a spherical surface to determine the total power radiated. Get solution

36. Light of (angular) frequency ω passes from medium 1, through a slab (thickness d) of medium 2, and into medium 3 (for instance, from water through glass into air, as in Fig. 9.27). Show that the transmission coefficient for normal incidence is given by ... [Hint: To the left, there is an incident wave and a reflected wave; to the right, there is a transmitted wave; inside the slab, there is a wave going to the right and a wave going to the left. Express each of these in terms of its complex amplitude, and relate the amplitudes by imposing suitable boundary conditions at the two interfaces. All three media are linear and homogeneous; assume μ1 = μ2 = μ3 = μ0.] Figure 9.27 ... Get solution

37. A microwave antenna radiating at 10 GHz is to be protected from the environment by a plastic shield of dielectric constant 2.5. What is the minimum thickness of this shielding that will allow perfect transmission (assuming normal incidence)? [Hint: Use Eq. 9.199.] Equation 9.199 ... Get solution

38. Light from an aquarium (Fig. 9.27) goes from water ... into air (n = 1). Assuming it’s a monochromatic plane wave and that it strikes the glass at normal incidence, find the minimum and maximum transmission coefficients (Eq. 9.199). You can see the fish clearly; how well can it see you? Equation 9.199 ... Figure 9.27 ... Get solution

39. According to Snell’s law, when light passes from an optically dense medium into a less dense one (n1 > n2) the propagation vector k bends away from the normal (Fig. 9.28). In particular, if the light is incident at the critical angle ... then θT = 90?, and the transmitted ray just grazes the surface. If θI exceeds θc, there is no refracted ray at all, only a reflected one (this is the phenomenon of total internal reflection, on which light pipes and fiber optics are based). But the fields are not zero in medium 2; what we get is a so-called evanescent wave, which is rapidly attenuated and transports no energy into medium 2.26 ... A quick way to construct the evanescent wave is simply to quote the results of Sect. 9.3.3, with kT = ωn2/c and ... the only change is that ... is now greater than 1, and ... is imaginary. (Obviously, θT can no longer be interpreted as an angle!) (a) Show that ... This is a wave propagating in the x direction (parallel to the interface!), and attenuated in the z direction. (b) Noting that α (Eq. 9.108) is now imaginary, use Eq. 9.109 to calculate the reflection coefficient for polarization parallel to the plane of incidence. [Notice that you get 100% reflection, which is better than at a conducting surface (see, for example, Prob. 9.22).] (c) Do the same for polarization perpendicular to the plane of incidence (use the results of Prob. 9.17). (d) In the case of polarization perpendicular to the plane of incidence, show that the (real) evanescent fields are ... (e) Check that the fields in (d) satisfy all of Maxwell’s equations (Eq. 9.67). (f) For the fields in (d), construct the Poynting vector, and show that, on average, no energy is transmitted in the z direction. Prob 9.22 Calculate the reflection coefficient for light at an air-to-silver interface...at optical frequencies (ω = 4 × 1015/s). Prob 9.17Analyze the case of polarization perpendicular to the plane of incidence (i.e. electric fields in the y direction, in Fig. 9.15). Impose the boundary conditions (Eq. 9.101), and obtain the Fresnel equations for ... Sketch ... as functions of θI , for the case β = n2/n1 = 1.5. (Note that for this β the reflected wave is always 180? out of phase.) Show that there is no Brewster’s angle for any n1 and n2: ... is never zero (unless, of course, n1 = n2 and μ1 = μ2, in which case the two media are optically indistinguishable). Confirm that your Fresnel equations reduce to the proper forms at normal incidence. Compute the reflection and transmission coefficients, and check that they add up to 1. ... Equation 9.67 ... Get solution

40. Consider the resonant cavity produced by closing off the two ends of a rectangular wave guide, at z = 0 and at z = d, making a perfectly conducting empty box. Show that the resonant frequencies for both TE and TM modes are given by ... for integers l,m, and n. Find the associated electric and magnetic fields. Get solution