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 time and per unit length carried by the wave, and take their ratio.]
> 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
> (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,
> 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
> Analyze the case of polarization perpendicular to the plane of incidence (i.e. electric fields in the y direction, in Fig. 9.15). Impose the boundary conditions (Eq. 9.101), and obtain the Fresnel equations for / Sketch / as functions of θI , for the ca
> Suppose Aeiax + Beibx = Ceicx , for some nonzero constants A, B, C , a, b, c, and for all x . Prove that a = b = c and A + B = C .
> In writing Eqs. 9.76 and 9.77, I tacitly assumed that the reflected and transmitted waves have the same polarization as the incident wave—along the x direction. Prove that this must be so. [Hint: Let the polarization vect
> Calculate the exact reflection and transmission coefficients, without assuming μ1 = μ2 = μ0. Confirm that R + T = 1
> Find all elements of the Maxwell stress tensor for a monochromatic plane wave traveling in the z direction and linearly polarized in the x direction (Eq. 9.48). Does your answer make sense? (Remember that /represents the momentum flux density.) Ho
> Compute the gradient and Laplacian of the function T=r (cos θ sin θ cos φ). Check the Laplacian by converting T to Cartesian coordinates and using Eq. 1.42. Test the gradient theorem for this function, using the path
> In the complex notation there is a clever device for finding the time average of a product. Suppose f (r, t) = A cos (k · r − ωt + δa) and g(r, t) = B cos (k r ωt + Î
> Consider a particle of charge q and mass m, free to move in the xy plane in response to an electromagnetic wave propagating in the z direction (Eq. 9.48—might as well set δ = 0). (a) Ignoring the magnetic force, find the velocity of the particle, as a fun
> The intensity of sunlight hitting the earth is about 1300 W/m2. If sunlight strikes a perfect absorber, what pressure does it exert? How about a perfect reflector? What fraction of atmospheric pressure does this amount to?
> By explicit differentiation, check that the functions f1, f2, and f3 in the text satisfy the wave equation. Show that f4 and f5 do not.
> Two concentric spherical shells carry uniformly distributed charges +Q (at radius a) and −Q (at radius b > a). They are immersed in a uniform magnetic field B = B0 zˆ. (a) Find the angular momentum of the fields (with respect to the center). (b) Now the ma
> In Ex. 8.4, suppose that instead of turning off the magnetic field (by reducing I ) we turn off the electric field, by connecting a weakly10 conducting radial spoke between the cylinders. (We’ll have to cut a slot in the solenoid, so the cylinders can stil
> Consider an infinite parallel-plate capacitor, with the lower plate (at z = −d/2) carrying surface charge density −σ , and the upper plate (at z = +d/2) carrying charge density +σ .
> A charged parallel-plate capacitor (with uniform electric field E = E zˆ) is placed in a uniform magnetic field B = B xˆ, as shown in Fig. 8.6. (a) Find the electromagnetic momentum in the space between the plates. (b) Now a resisti
> Imagine two parallel infinite sheets, carrying uniform surface charge σ (on the sheet at z =d) and -σ (at z=0). They are moving in the y direction at constant speed v (as in Problem 5.17). (a) What is the electromagnetic momentum in a region of area A? (
> (a) Consider two equal point charges q, separated by a distance 2a. Construct the plane equidistant from the two charges. By integrating Maxwell’s stress tensor over this plane, determine the force of one charge on the other. (b) Do the same for charges
> Compute the divergence of the function v = (r cos θ) rˆ + (r sin θ) θˆ + (r sin θ cos φ) φˆ. Check the divergence theorem for this function, using
> Use the cross product to find the components of the unit vector n perpendicular to the shaded plane in Fig. 1.11.
> Calculate the force of magnetic attraction between the northern and southern hemispheres of a uniformly charged spinning spherical shell, with radius R, angular velocity ω, and surface charge density σ . [This is the same as Prob. 5.44, but this time use
> A circular disk of radius R and mass M carries n point charges (q), attached at regular intervals around its rim. At time t=0 the disk lies in the xy plane, with its center at the origin, and is rotating about the z axis with angular velocity ω0, when it
> (a) Carry through the argument in Sect. 8.1.2, starting with Eq. 8.6, but using J f in place of J. Show that the Poynting vector becomes S = E × H, (8.46) and the rate of change of the energy density in the fields is For linea
> A point charge q is a distance a > R from the axis of an infinite solenoid (radius R, n turns per unit length, current I ). Find the linear momentum and the angular momentum (with respect to the origin) in the fields. (Pu
> Because the cylinders in Ex. 8.4 are left rotating (at angular velocities ωa and ωb, say), there is actually a residual magnetic field, and hence angular momentum in the fields, even after the current in the solenoid has been extinguished. If the cylinders
> Consider an ideal stationary magnetic dipole m in a static electric field E. Show that the fields carry momentum / [Hint: There are several ways to do this. The simplest method is to start with and use integration by parts
> Consider the charging capacitor in Prob. 7.34. (a) Find the electric and magnetic fields in the gap, as functions of the distance s from the axis and the time t . (Assume the charge is zero at t = 0.) (b) Find the energy density uem and the Poynting vecto
> Suppose you had an electric charge qe and a magnetic monopole qm . The field of the electric charge is / (of course), and the field of the magnetic monopole is Find the total angular momentum stored in the fi
> Work out the formulas for u, S, g, and /in the presence of magnetic charge. [Hint: Start with the generalized Maxwell equations (7.44) and Lorentz force law (Eq. 8.44), and follow the derivations in Sections 8.1.2, 8.2.2, and 8.2.3.]
> Picture the electron as a uniformly charged spherical shell, with charge e and radius R, spinning at angular velocity ω. (a) Calculate the total energy contained in the electromagnetic fields. (b) Calculate the total angular momentum contained in the field
> (a) Check the divergence theorem for the function / using as your volume the sphere of radius R, centered at the origin. (b) Do the same for / (If the answer surprises you, look back at Prob. 1.16.)
> A sphere of radius R carries a uniform polarization P and a uniform magnetization M (not necessarily in the same direction). Find the electromagnetic momentum of this configuration. [Answer: (4/9)πμ0 R3(M × P)]
> A point charge q is located at the center of a toroidal coil of rectangular cross section, inner radius a, outer radius a+w, and height h, which carries a total of N tightly-wound turns and current I . (a) Find the electromagnetic momentum p of this confi
> An infinitely long cylindrical tube, of radius a, moves at constant speed v along its axis. It carries a net charge per unit length λ, uniformly distributed over its surface. Surrounding it, at radius b, is another cylinder, moving with the same velocity
> A very long solenoid of radius a, with n turns per unit length, carries a current Is . Coaxial with the solenoid, at radius b ((a, is a circular ring of wire, with resistance R. When the current in the solenoid is (gradually) decreased, a current Ir is i
> Derive Eq. 8.43. [Hint: Use the method of Section 7.2.4, building the two currents up from zero to their final values.]
> Derive Eq. 8.39. [Hint: Treat the lower loop as a magnetic dipole.]
> Imagine an iron sphere of radius R that carries a charge Q and a uniform magnetization M = M zˆ. The sphere is initially at rest. (a) Compute the angular momentum stored in the electromagnetic fields. (b) Suppose the sphere is g
