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 <
> 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
> 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 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?
> 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. [Answer: /
> 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 ab
> 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.
> Evaluate the following integrals: (a) / where a is a fixed vector, a is its magnitude, and the integral is over all space. (b) / where V is a cube of side 2, centered on the origin, and b= 4 yˆ + 3 zˆ. (c) / where V is a sphere of radius 6 about the orig
> Derive Eq. 10.23. [Hint: Start by dotting v into Eq. 10.17.]
> 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.
> A time-dependent point charge q(t) at the origin, ρ(r, t) = q(t)δ3(r), is fed by a current J(r, t) = −(1/4π)(q˙/r 2) rˆ, where q˙ ≡ dq/dt . (a) Check that charge is conserved, by confirming that the continuity equation is obeyed. (b) Find the scalar and v
> 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 wa
> 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.)
> Suppose / 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?
> 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 te
> 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
> 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 .
> 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.
> (a) Write an expression for the volume charge density ρ(r) of a point charge q at rr. Make sure that the volume integral of ρ equals q. (b) What is the volume charge density of an electric dipole, consisting of a point charge −q at the origin and a point
> 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 pote
> (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.
> 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?
> 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
> 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).
> 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 /
> 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
> 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 po
> 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
> (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.
> (a) Show that / [Hint: Use integration by parts.] (b) Let θ(x) be the step function: Show that dθ/dx = δ(x).
> 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
> 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?
> 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 Po
> Derive Eq. 10.70. First show that
> 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.16
> 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 horizo
> 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
> 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 potenti
> Suppose the current density changes slowly enough that we can (to good approximation) ignore all higher derivatives in the Taylor expansion J(tr ) = J(t) + (tr − t)J˙(t ) + ··· (for
> 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.
> Evaluate the following integrals:
> 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 (−∞ Calculate the retarded vector potential A at the center. Find the electric fi
