Questions and Answers of Electrodynamics

Find the fields, and the charge and current distributions, corresponding to
Suppose V = 0 and A = A0 sin(kx – wt) y, where A0, w, and k are constants. Find E and B, and check that they satisfy Maxwell's equations in vacuum. What condition must you impose on w and k?
Use the gauge function λ = – (1/4πε0) (qt/r ) to transform the potentials in Prob. 10.3, and comment on the result.
Which of the potentials in Ex. 10.1, Prob. 10.3, and Prob. 10.4 are in the Coulomb gauge? Which are in the Lorentz gauge? (Notice that these gauges are not mutually exclusive.)
I showed that it is always possible to pick a vector potential whose divergence is zero (Coulomb gauge). Show that it is always possible to choose ∆ ∙ A = – µ0ε0 (∂V/∂t), as required for
Confirm that the retarded potentials satisfy the Lorentz gauge condition.
(a) Suppose the wire in Ex. 10.2 carries a linearly increasing current I(t) = kt, for t > 0. Find the electric and magnetic fields generated.(b) Do the same for the case of a sudden burst of
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
Problem 10.11 Suppose J(r) is constant in time, so (Prob. 7.55) p(r, t) = p(r, 0) + p(r, 0)t. Show that that is, Coulomb's law holds, with the charge density evaluated at the non-retarded time.
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 clarity, I
A particle of charge q moves in a circle of radius a at constant angular velocity w. (Assume that the circle lies in the xy plane, centered at the origin, and at time t = 0 the charge is at (a, 0),
Show that the scalar potential of a point charge moving with constant velocity (Eq. 10.42) can be written equivalently as where R = r - vt is the vector from the present (.t) position of the particle
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
Determine the Lienard-Wiechert potentials for a charge in hyperbolic motion (Eq. 10.45). Assume the point r is on the x axis and to the right of the charge.
Derive Eq. 10.63. First show that
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 What are the fields on the axis to the left of the
(a) Use Eq. 10.68 to calculate the electric field a distance d from an infinite straight wire carrying a uniform line charge Z, moving at a constant speed v down the wire.(b) Use Eq. 10.69 to find
For the configuration in Prob. 10.13, 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
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 w. Find the (exact)
Check that the potentials of a point charge moving at constant velocity (Eqs. 10.42 and 10.43) satisfy the Lorentz gauge condition (Eq. 10.12).
One particle, of charge ql, is held at rest at the origin. Another particle, of charge q2, approaches along the x axis, in hyperbolic motion: x(t) = √b2 + (ct)2; it reaches the closest point, b, at
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 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 =
Check that the retarded potentials of an oscillating dipole (Eqs. 11.12 and 11.17) satisfy the Lorentz gauge condition. Do not use approximation 3.
Equation 11.14 can be expressed in "coordinate-free" form by writing P0 cos θ = P0 ∙ r. Do so, and likewise for Eqs. 11.17, 11.18. 11.19, and 11.21.
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
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 90o: p
Calculate the electric and magnetic fields of an oscillating magnetic dipole without using approximation 3. [Do they look familiar? Compare Prob. 9.33.] Find the Poynting vector, and show that the
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.
Use the "duality" transformation of Prob. 7.60, together with the fields of an oscillating electric dipole (Eqs. 11.18 and I 1.19), to determine the fields that would be produced by an oscillating
Apply Eqs. 11.59 and 11.60 to the rotating dipole of Prob. 11.4. Explain any apparent discrepancies with your previous 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 Ф, where λ0 is constant and Ф is the usual azimuthal angle. The
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?
As a model for electric quadrupole radiation, consider two oppositely oriented oscillating electric dipoles, separated by a distance d, as shown in Fig. 11.10. Use the results of Sect. 11.1.2 for the
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
(a) Suppose an electron decelerated at a constant rate a from some initial velocity v0 down to zero. What fraction of its initial kinetic energy is lost to radiation? (The rest is absorbed by
In Bohr's theory of hydrogen, the electron in its ground state was supposed to travel in a circle of radius 5 x 10–11 m, held in orbit by the Coulomb attraction of the proton. According to
Find the angle θmax at which the maximum radiation is emitted, in Ex. 11.3 (see Fig. l l.14). Show that for ultrarelativistic speeds (v close to c), θ max ?? ??(1 ?? β)/2. What is the intensity of
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
(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
(a) Assuming (implausibly) that γ is entirely attributable to radiation damping (Eq. 11.84), show that for optical dispersion the damping is "small" (γ << w0). Assume that the relevant
Deduce Eq. 11.100 from Eq. 11.99, as follows: (a) Use the Abraham-Lorentz formula to determine the radiation reaction on each end of the dumbbell; add this to the interaction term (Eq.
A particle of mass m and charge q is attached to a spring with force constant k, hanging from the ceiling (Fig. 11.19). Its equilibrium position is a distance h above the floor. It is pulled down a
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
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 11o. Relative to the fixed axis of rotation therefore, the magnetic
Suppose the (electrically neutral) y z plane carries a time-dependent but uniform surface current K(t) z. (a) Find the electric and magnetic fields at a height x above the plane if (i) A constant
When a charged particle approaches (or leaves) a conducting surface, radiation is emitted, associated with the changing electric dipole moment of the charge and its image. If the particle has mass m
Use the duality transformation (Prob. 7.60) to construct the electric and magnetic fields of a magnetic monopole qm in arbitrary motion, and find the "Larmor formula" for the power radiated.
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
(a) Repeat Prob. 11.19, but this time let the external force be a Dirac delta function: F(t) = kS(t) (for some constant k). 17 [Note that the acceleration is now discontinuous at t = 0 (though the
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
(a) Find the radiation reaction force on a particle moving with arbitrary velocity in a straight line, by reconstructing the argument in Sect. 11.2.3 without assuming v(tr) = 0.(b) Show that this
(a) Does a particle in hyperbolic motion (Eq. 10.45) radiate? (Use the exact formula (Eq. 11.75) to calculate the power radiated.)(b) Does a particle in hyperbolic motion experience aradiation
Use Galileo's velocity addition rule. Let S be an inertial reference system.(a) Suppose that S moves with constant velocity relative to S. Show that S is also an inertial reference system.(b)
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,
(a) What's the percent error introduced when you use Galileo's role, 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
As the outlaws escape in their getaway car, which goes ?c, the police officer fires a bullet from the pursuit car, which only goes ?c (Figure 12.3). The muzzle velocity of the bullet (relative to the
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
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
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 x 10??6 s) and concludes that its speed was faster than
A rocket ship leaves earth at a speed of 5/3c. 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
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
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
A record turntable of radius R rotates at angular velocity w (Fig. 12.15). The circumference is presumably Lorentz-contracted, but the radius (being perpendicular to the velocity) is not. What's the
Solve Eqs 12.18 for x, y, z, t in terms of x, y, z, t, and check that you recover Eqs. 12.19.
Sophie Zabar, clairvoyants, 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,
(a) In Ex. 12.6 we found how velocities in the x direction transform when you go from S to S. Derive the analogous formulas for velocities in the y and z directions. (b) A spotlight is mounted on 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
The twin paradox revisited. On their 21st birthday, one twin gets on a moving sidewalk, which carries her out to star X at speed 4/5c; her twin brother stays home. When the traveling twin gets to
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
(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
The parallel between rotations and Lorentz transformations is even more striking if we introduce the rapidity: θ = tanh–1 (v/c).(a) Express the Lorentz transformation matrix A (Eq. 12.24) in terms
(a) Event A happens at point (xA = 5, yA = 3, zA = 0) and at time tA given by ctA = 15; event B occurs at (10, 8, 0) and ct B = 5, both in system S.(i) What is the invariant interval between A and
The coordinates of event A are (xA, 0, 0), tA, and the coordinates of event B are (xB, 0, 0), t B. Assuming the interval between them is space like, find the velocity of the system in which they are
(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
Inertial system S moves in the x direction at speed 3/5-c 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
(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
A car is traveling along the 45o line in S (Fig. 12.25), at (ordinary) speed (2/??5)c. (a) Find the components ux and uy of the (ordinary) velocity. (b) Find the components ηx and ηy of the proper
Find the invariant product of the 4-velocity with itself, ημ ημ.
Consider a particle in hyperbolic motion, (a) Find the proper time r as a function of t, assuming the clocks are set so that τ = 0 when ?t = 0. (b) Find x and v (ordinary velocity) as functions of
(a) Repeat Prob. 12.2 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
If a particle's kinetic energy is n times its rest energy, what is its speed?
Suppose you have a collection of particles, all moving in the x direction, with energies E1, E2, E3, ... and moment a P1, P2, P3, ... Find the velocity of the center of momentum frame, in which the
Find the velocity of the muon in Ex. 12.8.
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
A neutral pion of (rest) mass m and (relativistic) momentum p = 3/4mc decays into two photons. One of the photons is emitted in the same direction as the original pion, and the other in the opposite
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
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
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
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.
Define proper acceleration in the obvious way: (a) Find α0 and of in terms of u and a (the ordinary acceleration). (b) Express αμ αμ in terms of u and a. (c) Show that ημ αμ = 0. (d) Write
Show that where is the anlge between u and F.
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
Why can't the electric field in Fig. 12.35b have a z component? After all, the magnetic field does.
A parallel-plate capacitor, at rest in S 0 and tilted at a 45o 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
(a) Check that Gauss's law, f E ∙ da = (1/ε0) Qenc is obeyed by the field of a point charge in uniform motion, by integrating over a sphere of radius R centered on the charge.(b) Find the Poynting
(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
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.
(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
An electromagnetic plane wave of (angular) frequency w 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

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