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 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?
> 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
> (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
> (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 current: I (t) = q0δ(t ).
> Confirm that the retarded potentials satisfy the Lorenz gauge condition. [Hint: First show that where ∇denotes derivatives with respect to r, and ∇( denotes derivatives with respect to rr. Next, noting t
> Show that the differential equations for V and A (Eqs. 10.4 and 10.5) can be written in the more symmetrical form where
> Write down the (real) electric and magnetic fields for a monochromatic plane wave of amplitude E0, frequency ω, and phase angle zero that is (a) traveling in the negative x direction and polarized in the z direction; (b) traveling in the direction from
> Equation 9.36 describes the most general linearly polarized wave on a string. Linear (or “plane”) polarization (so called because the displacement is parallel to a fixed vector nˆ) results from the combination of horizontally and vertically polarized wave
> Suppose string 2 is embedded in a viscous medium (such as molasses), which imposes a drag force that is proportional to its (transverse) speed: / (a) Derive the modified wave equation describing the motion of the string. (b) Solve this equation, assuming
> (a) Formulate an appropriate boundary condition, to replace Eq. 9.27, for the case of two strings under tension T joined by a knot of mass m. (b) Find the amplitude and phase of the reflected and transmitted waves for the case where the knot has a mass m
> Suppose you send an incident wave of specified shape, gI (z-v1t), down string number 1. It gives rise to a reflected wave, hR(z+v1t), and a transmitted wave, gT (z-v2t). By imposing the boundary conditions 9.26 and 9.27, find h R and gT .
> Consider the resonant cavity produced by closing off the two ends of a rectangular wave guide, at z=0 and at z=d, making a perfectly conducting empty box. Show that the resonant frequencies for both TE and TM modes are given by for integers l, m, and n.
> Evaluate the following integrals:
> Obtain Eq. 9.20 directly from the wave equation, by separation of variables.
> According to Snell’s law, when light passes from an optically dense medium into a less dense one (n1 > n2) the propagation vector k bends away from the normal (Fig. 9.28). In particular, if the light is incident at the critical angle
> Light from an aquarium (Fig. 9.27) goes from water / through a plane of glass / into air (n = 1). Assuming it’s a monochromatic plane wave and that it strikes the glass at normal incidence, find the minimum and maximum
> A microwave antenna radiating at 10 GHz is to be protected from the environment by a plastic shield of dielectric constant 2.5. What is the minimum thickness of this shielding that will allow perfect transmission (assuming normal incidence)? [Hint: Use E
> Light of (angular) frequency ω passes from medium 1, through a slab (thickness d) of medium 2, and into medium 3 (for instance, from water through glass into air, as in Fig. 9.27). Show that the transmission coefficient for normal incidence is given by /
> Suppose / (This is, incidentally, the simplest possible spherical wave. For notational convenience, let (kr − ωt) ≡ u in your calculations.) (a) Show that E obeys all four of Maxwell’s equations, in vacuum, and find the associated magnetic field. (b) Calc
> A plane wave traveling through vacuum in the z direction encounters a perfect conductor occupying the region z(( 0, and reflects back: / (a) Find the accompanying magnetic field (in the region z < 0). (b) Assuming B=0 inside the conductor, find the current
> The “inversion theorem” for Fourier transforms states that Use this to determine A˜(k), in Eq. 9.20, in terms of f (z, 0) and f˙(z, 0).
> (a) Show directly that Eqs. 9.197 satisfy Maxwell’s equations (Eq. 9.177) and the boundary conditions (Eq. 9.175). (b) Find the charge density, λ(z, t), and the current, I (z, t), on the inner conductor.
> Work out the theory of TM modes for a rectangular wave guide. In particular, find the longitudinal electric field, the cutoff frequencies, and the wave and group velocities. Find the ratio of the lowest TM cutoff frequency to the lowest TE cutoff frequency
> (a) Find the divergence of the function v = s(2 + sin2 φ) sˆ + s sin φ cos φ φˆ+ 3z zˆ. (b) Test the divergence theorem for this function, using the quarter-cylinder (radius 2,
> Confirm that the energy in the TEmn mode travels at the group velocity. [Hint: Find the time averaged Poynting vector S and the energy density u (use Prob. 9.12 if you wish). Integrate over the cross section of the wave guide to get the energy per unit ti
> Use Eq. 9.19 to determine A3 and δ3 in terms of A1, A2, δ1, and δ2.
> Consider a rectangular wave guide with dimensions 2.28 cm*1.01 cm. What TE modes will propagate in this wave guide, if the driving frequency is 1.70* 1010 Hz? Suppose you wanted to excite only one TE mode; what range of frequencies could you use? What ar
> Show that the mode TE00 cannot occur in a rectangular wave guide. [Hint: In this case ω/c=k, so Eqs. 9.180 are indeterminate, and you must go back to Eq. 9.179. Show that Bz is a constant, and hence—applying Faraday’s law in integral form to a cross sect
> (a) Derive Eqs. 9.179, and from these obtain Eqs. 9.180. (b) Put Eq. 9.180 into Maxwell’s equations (i) and (ii) to obtain Eq. 9.181. Check that you get the same results using (i) and (iv) of Eq. 9.179.
> Starting with Eq. 9.170, calculate the group velocity, assuming there is only one resonance, at ω0. Use a computer to graph y ≡ vg/c as a function of x ≡ (ω/ω0)2, from x = 0 to 2, (a) for γ = 0, and (b) for γ = (0.1)ω0. Let (Nq2)/(2mε0ω02) = 0.003. Note
> Find the width of the anomalous dispersion region for the case of a single resonance at frequency ω0. Assume γ
> If you take the model in Ex. 4.1 at face value, what natural frequency do you get? Put in the actual numbers. Where, in the electromagnetic spectrum, does this lie, assuming the radius of the atom is 0.5 Å? Find the coefficients of refraction and dispersi
> (a) Shallow water is nondispersive; waves travel at a speed that is proportional to the square root of the depth. In deep water, however, the waves can’t “feel” all the way down to the bottomâ&#
> Calculate the reflection coefficient for light at an air-to-silver inter- face (μ1 = μ2 = μ0, ε1 = ε0,σ = 6 × 107(Ω · m)−1), at optical frequencies (ω = 4 × 1015/s).
> Express the cylindrical unit vectors sˆ, φˆ, zˆ in terms of xˆ, yˆ, zˆ (that is, derive Eq. 1.75). “Invert” your formulas to get xˆ, yˆ, zˆ in terms of sˆ, φˆ, zˆ (and φ).
> (a) Calculate the (time-averaged) energy density of an electromagnetic plane wave in a conducting medium (Eq. 9.138). Show that the magnetic contribution always dominates. / (b) Show that the intensity is /
> (a) Show that the skin depth in a poor conductor / (independent of frequency). Find the skin depth (in meters) for (pure) water. (Use the static values of ε, μ, and σ ; your answers will be valid, then, only at relatively low frequencies.) (b) Show that
> Show that the standing wave f (z, t) A sin(kz) cos(kvt ) satisfies the wave equation, and express it as the sum of a wave traveling to the left and a wave traveling to the right (Eq. 9.6).
> (a) Suppose you imbedded some free charge in a piece of glass. About how long would it take for the charge to flow to the surface? (b) Silver is an excellent conductor, but it’s expensive. Suppose you were designing a microwave experiment to operate at a
> The index of refraction of diamond is 2.42. Construct the graph analogous to Fig. 9.16 for the air/diamond interface. (Assume μ1= μ2 =μ0.) In particular, calculate (a) the amplitudes at normal incidence, (b) Brewster’s angle, and (c) the “crossover” a
