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 / after the signal from a, and the star has meanwhile advanced a distance / across the celestial sphere; by “apparent speed,” I mean /.) 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.
> Check that Eq. 2.29 satisfies Poisson’s equation, by applying the Laplacian and using Eq. 1.102.
> Use Eq. 2.29 to calculate the potential inside a uniformly charged solid sphere of radius R and total charge q. Compare your answer to Prob. 2.21.
> Find the potential on the axis of a uniformly charged solid cylinder, a distance z from the center. The length of the cylinder is L, its radius is R, and the charge density is ρ. Use your result to calculate the electric field at this point. (Assume that
> A conical surface (an empty ice-cream cone) carries a uniform surface charge σ. The height of the cone is h, as is the radius of the top. Find the potential difference between points a (the vertex) and b (the center of the top).
> Using Eqs. 2.27 and 2.30, find the potential at a distance z above the center of the charge distributions in Fig. 2.34. In each case, compute E=-∇V, and compare your answers with Ex. 2.1, Ex. 2.2, and Prob. 2.6, respectiv
> For the configuration of Prob. 2.16, find the potential difference between a point on the axis and a point on the outer cylinder. Note that it is not necessary to commit yourself to a particular reference point, if you use Eq. 2.22.
> For the charge configuration of Prob. 2.15, find the potential at the center, using infinity as your reference point.
> (a) Prove that the two-dimensional rotation matrix (Eq. 1.29) preserves dot products. (That is, show that (b) What constraints must the elements (Rij ) of the three-dimensional rotation matrix (Eq. 1.30) satisfy, in order to preserve the length of A (fo
> Find the potential a distance s from an infinitely long straight wire that carries a uniform line charge λ. Compute the gradient of your potential, and check that it yields the correct field.
> Find the potential inside and outside a uniformly charged solid sphere whose radius is R and whose total charge is q. Use infinity as your reference point. Compute the gradient of V in each region, and check that it yields the correct field. Sketch V (r).
> One of these is an impossible electrostatic field. Which one? (a) E = k[xy xˆ + 2yz yˆ + 3xz zˆ]; (b) E = k[y2 xˆ + (2xy + z2) yˆ + 2yz zˆ]. Here k is a constant with the appropriate units. For the possible one, find the potential, using the origin as your
> Find the electric field (magnitude and direction) a distance z above the midpoint between equal and opposite charges (±q), a distance d apart (same as Example 2.1, except that the charge at x = +d/2 is −q).
> Calculate ∇×E directly from Eq. 2.8, by the method of Sect. 2.2.2. Refer to Prob. 1.63 if you get stuck.
> Two spheres, each of radius R and carrying uniform volume charge densities +ρ and -ρ, respectively, are placed so that they partially overlap (Fig. 2.28). Call the vector from the positive center to the negative center d. Show that
> An infinite plane slab, of thickness 2d, carries a uniform volume charge density ρ (Fig. 2.27). Find the electric field, as a function of y, where y = 0 at the center. Plot E versus y, calling E positive when it p
> A long coaxial cable (Fig. 2.26) carries a uniform volume charge density ρ on the inner cylinder (radius a), and a uniform surface charge density on the outer cylindrical shell (radius b). This surface charge is negative and is of just the rig
> A thick spherical shell carries charge density / (Fig. 2.25). Find the electric field in the three regions: (i) r (ii) a (iii) r > b. Plot |E| as a function of r , for the case b = 2a.
> Find the electric field inside a sphere that carries a charge density proportional to the distance from the origin, ρ=kr , for some constant k. [Hint: This charge density is not uniform, and you must integrate to get the enclosed charge.]
> Find the separation vector r from the source point (2,8,7) to the field point (4,6,8). Determine its magnitude (r), and construct the unit vector rˆ.
> Find the electric field a distance s from an infinitely long straight wire that carries a uniform line charge λ. Compare Eq. 2.9.
> Use Gauss’s law to find the electric field inside a uniformly charged solid sphere (charge density ρ). Compare your answer to Prob. 2.8.
> Use Gauss’s law to find the electric field inside and outside a spherical shell of radius R that carries a uniform surface charge density σ. Compare your answer to Prob. 2.7.
> A charge q sits at the back corner of a cube, as shown in Fig. 2.17. What is the flux of E through the shaded side?
> (a) Twelve equal charges, q, are situated at the corners of a regular 12-sided polygon (for instance, one on each numeral of a clock face). What is the net force on a test charge Q at the center? (b) Suppose one of the 12 q’s is removed (the one at “6 o’
> In case you’re not persuaded that ∇2(1/r) = −4πδ3(r) (Eq. 1.102 with rr = 0 for simplicity), try replacing r by and watching what happens Specially, let / To demonstrate th
> (a) Find the divergence of the function First compute it directly, as in Eq. 1.84. Test your result using the divergence theorem, as in Eq. 1.85. Is there a delta function at the origin, as there was for / What is the general formula for the divergence
> The integral / is sometimes called the vector area of the surface S. If S happens to be flat, then |a| is the ordinary (scalar) area, obviously. (a) Find the vector area of a hemispherical bowl of radius R. (b) Show that a = 0 for any clo
> Although the gradient, divergence, and curl theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that: (a) / where c is a constant, in the divergence theorem; use the product
> Here are two cute checks of the fundamental theorems: (a) Combine Corollary 2 to the gradient theorem with Stokes’ theorem (v=∇T, in this case). Show that the result is consistent with what you already knew about second derivatives. (b) Combine Corollary
> Prove that [A × (B × C)]+ [B × (C × A)]+ [C × (A × B)]= 0. Under what conditions does A × (B × C) = (A × B) × C?
> Check the divergence theorem for the function v = r 2 sin θ rˆ + 4r 2 cos θ θˆ + r 2 tan θ φˆ, using the volume of the “ice-cream coneâ
> 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 lig
> A rocket ship leaves earth at a speed of / 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
> Generalize the laws of relativistic electrodynamics (Eqs. 12.127 and 12.128) to include magnetic charge. [Refer to Sect. 7.3.4.]
> 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
> Use the Larmor formula (Eq. 11.70) and special relativity to derive the Liénard formula (Eq. 11.73).
> (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 / / (b) Construct the dual tensor Hμν (analogous to Gμν ). [Answer: H 01 ≡ Hx , H 12 ≡−cDz , etc.] (c
> In a laboratory experiment, a muon is observed to travel 800 m before disintegrating. A graduate student looks up the lifetime of a muon / and concludes that its speed was / Faster than light! Identify the student’s error, and find the actual speed of th
> 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 transformi
> Check Stokes’ theorem for the function / using the triangular surface shown in Fig. 1.51. [Answer: a2]
> Derive” the Lorentz force law, as follows: Let charge q be at rest in S¯, so F¯ = qE¯ , and let S¯ move with velocity v = v xˆ with respect to S. Use the transformation rules (Eqs. 12.67 and 12.109) to rewrite F¯ in terms of F, and E¯ in terms of E and
> 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 electro
> 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 S¯, moving relative to S with velocity v given by / the fields E
> A stationary magnetic dipole, m = m zˆ, is situated above an infinite uniform surface current, K = K xˆ (Fig. 12.44). (a) Find the torque on the dipole, using Eq. 6.1. (b) Suppose that the surface current consists of
> An ideal magnetic dipole moment m is located at the origin of an inertial system S¯ that moves with speed v in the x direction with respect to inertial system S. In S¯ the vector potential is / (Eq. 5.85), and the scalar potentia
> 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)
> 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 ). / /
> 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, a
> Calculate the threshold (minimum) momentum the pion must have in order for the process π+ p (K+Σ to occur. The proton p is initially at rest. / (all in MeV). [Hint: To formulate the threshold condition, examine the collision in the center-of- momentu
> Compute the line integral of v = (r cos2 θ) rˆ − (r cos θ sin θ) θˆ + 3r φˆ around the path shown in Fig. 1.50 (the points are labeled by th
> Inertial system S¯ moves at constant velocity v = βc(cos φ xˆ + sin φ yˆ) with respect to S. Their axes are parallel to one another, and their origins coincide at t=
> Show that the Liénard-Wiechert potentials (Eqs. 10.46 and 10.47) can be expressed in relativistic notation as Where /
> Show that the potential representation (Eq. 12.133) automatically satisfies ∂Gμν /∂ xν = 0. [Suggestion: Use Prob. 12.54.]
> 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 Jμ=0, has the form of an invariant product of two vectors. Th
> Work out, and interpret physically, the μ=0 component of the electromagnetic force law, Eq. 12.128.
> Show that the second equation in Eq. 12.127 can be expressed in terms of the field tensor Fμν as follows:
> Obtain the continuity equation (Eq. 12.126) directly from Maxwell’s equations (Eq. 12.127).
> 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).
> 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
> Prove that the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation (that is: if tμν is symmetric, show that t¯μν is also symmetric, and likewise for antisymmetric).
> Compute the line integral of v = 6 xˆ + yz2 yˆ + (3y + z) zˆ along the triangular path shown in Fig. 1.49. Check your answer using Stokes’ theorem. [Answer: 8/3]
> 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?
> Work out the remaining five parts to Eq. 12.118.
> 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
> (a) Show that (E · B) is relativistically invariant. (b) Show that (E 2 − c2 B2) 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
> 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, doin
> (a) Charge qA is at rest at the origin in system ; 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 syst
> 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 , which moves with speed v in the x direction with resp
> A parallel-plate capacitor, at rest in S0 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 S0. (a) F
> Why can’t the electric field in Fig. 12.35b have a z component? After all, the magnetic field does.
> 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.]
> Check Stokes’ theorem using the function / (a and b are constants) and the circular path of radius R, centered at the origin in the xy plane. [Answer: π R2(b − a)]
> Show that / where θ is the angle between u and F.
> As the outlaws escape in their getaway car, which goes / the police officer fires a bullet from the pursuit car, which only goes / The muzzle velocity of the bullet (relative to the gun) is / Does the bullet reach its target (a) according to Galileo, (b)
> Define proper acceleration in the obvious way: / (a) Find α0 and α in terms of u and a (the ordinary acceleration). (b) Express αμαμ in terms of u and a. (c) Show that ημαμ = 0. (d) Write the Minkowski version of Newton’s second law, Eq. 12.68, in terms
> 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.
> 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.
> 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◦ t
> 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
> A neutral pion of (rest) mass m and (relativistic) momentum / 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.
> 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?
> Find the velocity of the muon in Ex. 12.8.
> Check the divergence theorem for the function v = r 2 cos θ rˆ + r 2 cos φ θˆ − r 2 cos θ sin φ φˆ, using as your volume one octant of the sphere of radius R (Fig. 1.48). Make sure you include the entire surface. [Answer: π R4/4]
> 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.
> If a particle’s kinetic energy is n times its rest energy, what is its speed?
> (a) What’s the percent error introduced when you use Galileo’s rule, instead of Einstein’s, with vAB = 5 mi/h and vBC = 60 mi/h? (b) Suppose you could run at half the speed of light down the corridor
> (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 S¯. Assume all motion is along the x axis. (b) Now do the
> 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 v
> 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 studi
> Find the invariant product of the 4-velocity with itself, ημημ. Is ημ timelike, spacelike, or lightlike?
> 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) Fin
> (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,
> Inertial system S¯ moves in the x direction at speed /relative to system S . (The x axis slides long the x axis, and the origins coincide at t= t¯ =0, as usual.) (a) On graph paper set up a Cartesian coordinate system with axes ct and x . Care- fully dr
> (a) Which of the vectors in Problem 1.15 can be expressed as the gradient of a scalar? Find a scalar function that does the job. (b) Which can be expressed as the curl of a vector? Find such a vector.
> (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 spacelike? (b) There’s an old limerick that runs as follows
> The coordinates of event A are (x A, 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.
> (a) Event A happens at point (x A = 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
> As an illustration of the principle of relativity in classical mechanics, consider the following generic collision: In inertial frame , particle A (mass m A, velocity uA) hits particle B (mass m B , velocity uB ). In the course of the collision some mass
> 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 respec
> (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 followe
