Generalize the laws of relativistic electrodynamics (Eqs.

Problem 73P 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

Problem 1P 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.

Problem 2P 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

Problem 4P 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?

Problem 3P (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

Problem 6P 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 ? 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 ?t after the signal from a, and the star has meanwhile advanced a distance ?s across the celestial sphere; by “apparent speed,” I mean ?s/ ?t.) What angle ? gives the maximum apparent speed? Show that the apparent speed can be much greater than c, even if v itself is less than c.

Problem 7P 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.

Problem 8P 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?

Problem 5P 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?

Problem 9P 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.)

Problem 11P A record turntable of radius R rotates at angular velocity ? (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 ? and R? According to the rules of ordinary geometry, it has to be ?. What’s going on here?9

Problem 12P 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

Problem 13P 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

Problem 10P 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?

Problem 15P 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?

Problem 16P 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).

Problem 17P 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

Problem 18P (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

Problem 14P (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 ? 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?

Problem 20P (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.

Problem 21P 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.

Problem 19P The parallel between rotations and Lorentz transformations is even more striking if we introduce the rapidity: (a) Express the Lorentz transformation matrix ? (Eq. 12.24) in terms of ?, 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

Problem 22P (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.

Problem 23P 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).

Problem 25P 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 ?x and ?y of the proper velocity. (c) Find the zeroth component of the 4-velocity, ?0. 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

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

Problem 26P Find the invariant product of the 4-velocity with itself, timelike, spacelike, or lightlike?

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

Problem 29P (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 = m?. 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

Problem 27P Prove that the divergence of a curl is always zero. Check it for function v a in Prob. 1.15.

Problem 31P 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.

Problem 32P 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, m? and m? (assume m? = 0).

Problem 33P 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?

Problem 30P If a particle’s kinetic energy is n times its rest energy, what is its speed?

Problem 34P 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.

Problem 36P 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?

Problem 37P 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.

Problem 38P 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.

Problem 35P 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.]

Problem 41P 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.]

Problem 40P Show that where ? is the angle between u and F.

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

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

Problem 45P (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.]

Problem 46P 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

Problem 47P (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?

Problem 43P A parallel-plate capacitor, at rest in S 0 and tilted at a 45? angle to the x0 axis, carries charge densities }?0 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?

Problem 49P Work out the remaining five parts to Eq. 12.118.

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

Problem 48P An electromagnetic plane wave of (angular) frequency ? 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 ?, 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?

Problem 51P 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?

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

Problem 55P Work out, and interpret physically, the ? = 0 component of the electromagnetic force law, Eq. 12.128. Equation 12.128

Problem 54P Show that the second equation in Eq. 12.127 can be expressed in terms of the field tensor F??as follows: Equation 12.127

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

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

Problem 57P Show that the potential representation (Eq. 12.133) automatically satisfies ?G??/?x? = 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 F??as follows:

Problem 58P 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

Problem 60P Calculate the threshold (minimum) momentum the pion must have in order for the process ?+ p ? K + ? to occur. The proton p is initially at rest. Use m? 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-of momentum 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.

Problem 61P 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 ?, the scattering angle, and v, the speed, in the center-of-momentum frame.

Problem 59P Inertial system moves at constant velocity as usual. Find the Lorentz transformation matrix ? (Eq. 12.25).

Problem 62P 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).

Problem 63P 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

Problem 64P 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

Problem 65P 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 ?, moving at velocity so that K = ?v, and the magnetic dipole consists of a uniform line charge ?, circulating at speed v (same v) around a square loop of side l, as shown, so that m = ?vl2. 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

Problem 66P 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?

Problem 67P 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

Problem 68P “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

Problem 69P 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.

Problem 71P Use the Larmor formula (Eq. 11.70) and special relativity to derive the Liénard formula (Eq. 11.73). Equation 11.70 Equation 11.73

Problem 72P 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

Problem 70P (a) Construct a tensor D?? (analogous to F??) out of D and H. Use it to express Maxwell’s equations inside matter in terms of the free current density J ? f . (b) Construct the dual tensor H?? (analogous to G??). (c) Minkowski proposed the relativistic constitutive relations for linear media: Where is the proper30 permittivity, ? is the proper permeability, and ?? 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 = ?H,