(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: 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.
> 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
> 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 ob
> 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.
> 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
> 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
> 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, a
> 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:
> (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 θÂ
> For Theorem 2, show that (d) ⇒ (a), (a) ⇒ (c), (c) ⇒ (b), (b) ⇒ (c), and (c) ⇒ (a).
> 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 a
> Solve Eqs. 12.18 for x, y, z, t in terms of / and check that you recover Eqs. 12.19.
> 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
> A sailboat is manufactured so that the mast leans at an angle θ¯ 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?
> Let S be an inertial reference system. Use Galileo’s velocity addition rule. (a) Suppose that / moves with constant velocity relative to . Show that / is also an inertial reference system. [Hint: Use the definition in
> Apply Eqs. 11.59 and 11.60 to the rotating dipole of Prob. 11.4. Explain any apparent discrepancies with your previous answer.
> 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) Q0e−t/RC . (a) What fraction of its initial energy (Q02/2C ) does it radiate away? (b) If C=1pF, R=1000 Ω
> 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 ma
> 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. [Answer: 3 × 105 (b/λ)4 ξ]
> 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 u
> For Theorem 1, show that (d) ⇒ (a), (a) ⇒ (c), (c) ⇒ (b), (b) ⇒ (c), and (c) ⇒ (a).
> 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 90â—¦: Using the principle of superposition a
> 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
> (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
> (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 / (b) Show that this result is consistent (in the sense of Eq. 11.78) with the power
> 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
> (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
> 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 p
> 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, wher
> 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.23
> Suppose the (electrically neutral) yz plane carries a time-dependent but uniform surface current K(t) zˆ. (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: (
> (a) Let / Calculate the divergence and curl of F1 and F2. Which one can be written as the gradient of a scalar? Find a scalar potential that does the job. Which one can be written as the curl of a vector? Find a suitable vector potential. (b) Show that
> Prove the BAC-CAB rule by writing out both sides in component form.
> 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 u
> An ideal electric dipole is situated at the origin; its dipole moment points in the zˆ direction, and is quadratic in time: where p¨0 is a constant. (a) Use the method of Section 11.1.2 to determine the (exact) electric and magnetic
> 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 ea
> 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
> 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 averag
> 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
> An electric dipole rotates at constant angular velocity ω in the xy plane. 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 (Ï
> 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 Ab
> Equation 11.14 can be expressed in “coordinate-free” form by writing p0 cos θ = p0 · rˆ. Do so, and likewise for Eqs. 11.17, 11.18. 11.19, and 11.21.
> 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), accel
> Evaluate the integral / (where is a sphere of radius R, centered at the origin) by two different methods, as in Ex. 1.16.
> 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
> (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
> 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),
> 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 propor
> 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 s
> 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
> 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?
