The magnetic field of an infinite straight wire carrying a steady cur- rent I can be obtained from the displacement current term in the Ampère/Maxwell law, as follows: Picture the current as consisting of a uniform line charge λ moving along the z axis at speed v (so that I= λv), with a tiny gap of length ε, which reaches the origin at time t=0. In the next instant (up to t ε/v) there is no real current passing through a circular Amperian loop in the xy plane, but there is a displacement current, due to the “missing” charge in the gap. (a) Use Coulomb’s law to calculate the z component of the electric field, for points in the xy plane a distance s from the origin, due to a segment of wire with uniform density −λ extending from z1 = vt − ε to z2 = vt . (b) Determine the flux of this electric field through a circle of radius a in the xy plane. (c) Find the displacement current through this circle. Show that Id is equal to I , in the limit as the gap width (ε) goes to zero.35
> (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
> Calculate the power (energy per unit time) transported down the cables of Ex. 7.13 and Prob. 7.62, assuming the two conductors are held at potential difference V, and carry current I (down one and back up the other).
> An infinite number of different surfaces can be fit to a given boundary line, and yet, in defining the magnetic flux through a loop, / I never specified the particular surface to be used. Justify this apparent oversight.
> A square loop of wire (side a) lies on a table, a distance s from a very long straight wire, which carries a current I , as shown in Fig. 7.18. (a) Find the flux of B through the loop. (b) If someone now pulls the loop directly away from t
> Express the unit vectors rˆ, θˆ, φˆ in terms of xˆ, yˆ, zˆ (that is, derive Eq. 1.64). Check your answers several ways (rˆ · rˆ =? 1, θˆ · φˆ =? 0, rˆ × θˆ =? φˆ,.. .). Also work out the inverse formulas, giving xˆ, yˆ, zˆ in terms of rˆ, θˆ,
> A metal bar of mass m slides frictionlessly on two parallel conducting rails a distance l apart (Fig. 7.17). A resistor R is connected across the rails, and a uniform magnetic field B, pointing into the page, fills the entir
> (a) Show that Maxwell’s equations with magnetic charge (Eq. 7.44) are invariant under the duality transformation where / and α is an arbitrary rotation angle in “E/B-space.” Charge and c
> Prove Alfven’s theorem: In a perfectly conducting fluid (say, a gas of free electrons), the magnetic flux through any closed loop moving with the fluid is constant in time. (The magnetic &ium
> A certain transmission line is constructed from two thin metal “rib- bons,” of width w, a very small distance h
> Suppose J(r) is constant in time but ρ(r, t) is not—conditions that might prevail, for instance, during the charging of a capacitor. (a) Show that the charge density at any particular point is a linear function of time: Ï
> A rectangular loop of wire is situated so that one end (height h) is between the plates of a parallel-plate capacitor (Fig. 7.9), oriented parallel to the field E. The other end is way outside, where the field is essentially
> An infinite wire runs along the z axis; it carries a current I (z) that is a function of z (but not of t ), and a charge density λ(t) that is a function of t (but not of z). (a) By examining the charge flowing into a segment dz in a time dt , show that dλ/
> A transformer (Prob. 7.57) takes an input AC voltage of amplitude V1, and delivers an output voltage of amplitude V2, which is determined by the turns ratio (V2/ V1 N2/N1). If N2 > N1, the output voltage is greater than the input voltage. Why doesn’t thi
> Two coils are wrapped around a cylindrical form in such a way that the same flux passes through every turn of both coils. (In practice this is achieved by inserting an iron core through the cylinder; this has the effect of concentrating th
> Find formulas for r,θ,φ in terms of x, y, z (the inverse, in other words, of Eq. 1.62).
> (a) Use the Neumann formula (Eq. 7.23) to calculate the mutual inductance of the configuration in Fig. 7.37, assuming a is very small /Compare your answer to Prob. 7.22. (b) For the general case (not assuming a is small), show that where
> In the discussion of motional emf (Sect. 7.1.3) I assumed that the wire loop (Fig. 7.10) has a resistance R; the current generated is then I= v Bh/R. But what if the wire is made out of perfectly conducting material, so that R is zero? In that case, the
> A circular wire loop (radius r , resistance R) encloses a region of uniform magnetic field, B, perpendicular to its plane. The field (occupying the shaded region in Fig. 7.56) increases linearly with time (B= α
> The current in a long solenoid is increasing linearly with time, so the flux is proportional to t : Ф =αt . Two voltmeters are connected to diametrically opposite points ( A and B), together with resistors (R1 and
> An atomic electron (charge q) circles about the nucleus (charge Q) in an orbit of radius r ; the centripetal acceleration is provided, of course, by the Coulomb attraction of opposite charges. Now a small magnetic field dB is slowly turned on, perpendicu
> An infinite wire carrying a constant current / direction is moving in the y direction at a constant speed v. Find the electric field, in the quasistatic approximation, at the instant the wire coincides with the z axis (Fig. 7.54). /
> Electrons undergoing cyclotron motion can be sped up by increasing the magnetic field; the accompanying electric field will impart tangential acceleration. This is the principle of the betatron. One would like to keep the radius of the orbit constant dur
> A battery of emf ε and internal resistance r is hooked up to a variable “load” resistance, R. If you want to deliver the maximum possible power to the load, what resistance R should you choose? (You can’t change E and r , of course.)
> (a) Referring to Prob. 5.52(a) and Eq. 7.18, show that for Faraday-induced electric fields. Check this result by taking the divergence and curl of both sides. (b) A spherical shell of radius R carries a uniform surface charge σ . It spins abou
> Refer to Prob. 7.11 (and use the result of Prob. 5.42): How long does is take a falling circular ring (radius a, mass m, resistance R) to cross the bottom of the magnetic field B, at its (changing) terminal velocity?
> (a) Show that / (b) Show that
> A perfectly conducting spherical shell of radius a rotates about the z axis with angular velocity ω, in a uniform magnetic field Calculate the emf developed between the “north pole” and the e
> If a magnetic dipole levitating above an infinite superconducting plane (Prob. 7.45) is free to rotate, what orientation will it adopt, and how high above the surface will it float?
> A familiar demonstration of superconductivity (Prob. 7.44) is the levitation of a magnet over a piece of superconducting material. This phenomenon can be analyzed using the method of images.31 Treat the magnet as a perfect dipole m, a height z above the
> In a perfect conductor, the conductivity is infinite, so E=0 (Eq. 7.3), and any net charge resides on the surface (just as it does for an imperfect conductor, in electrostatics). (a) Show that the magnetic field is constant (∂B/∂t=0), inside a perfect cond
> The magnetic field outside a long straight wire carrying a steady current I is The electric field inside the wire is uniform: where ρ is the resistivity and a is the radius (see Exs. 7.1 and 7.3). Question: What i
> A rare case in which the electrostatic field E for a circuit can actually be calculated is the following:28 Imagine an infinitely long cylindrical sheet, of uniform resistivity and radius a. A slot (corresponding to the batt
> Two long, straight copper pipes, each of radius a, are held a distance 2d apart (see Fig. 7.50). One is at potential V0, the other at V0. The space surrounding the pipes is filled with weakly conducting material of conductivity σ
> Sea water at frequency / Hz has permittivity / permeability μ=μ0, and resistivity ρ=0.23 Ω m. What is the ratio of conduction current to displacement current? [Hint: Consider a parallel-plate capacitor immersed in sea water and driven by a voltage V0
> Suppose the conductivity of the material separating the cylinders in Ex. 7.2 is not uniform; specifically, σ(s)=k/s, for some constant k. Find the resistance between the cylinders. [Hint: Because σ is a function of position, Eq. 7.5 does not hold, the cha
> Suppose a magnetic monopole qm passes through a resistanceless loop of wire with self-inductance L. What current is induced in the loop?27
> Check Corollary 1 by using the same function and boundary line as in Ex. 1.11, but integrating over the five faces of the cube in Fig. 1.35. The back of the cube is open.
> Assuming that “Coulomb’s law” for magnetic charges (qm ) reads work out the force law for a monopole qm moving with velocity v through electric and magnetic fields E and B.26
> Suppose / (The theta function is defined in Prob. 1.46b). Show that these fields satisfy all of Maxwell’s equations, and determine ρ and J. Describe the physical situation that gives rise to these fields.
> Refer to Prob. 7.16, to which the correct answer was / (a) Find the displacement current density Jd . (b) Integrate it to get the total displacement current, (c) Compare Id and I . (What’s their ratio?) If the outer cylinder were, say,
> The preceding problem was an artificial model for the charging capacitor, designed to avoid complications associated with the current spreading out over the surface of the plates. For a more realistic model, imagine thin wires that connect
> A fat wire, radius a, carries a constant current I , uniformly distributed over its cross section. A narrow gap in the wire, of width w
> An infinite cylinder of radius R carries a uniform surface charge σ . We propose to set it spinning about its axis, at a final angular velocity ωf . How much work will this take, per unit length? Do it two ways, and compare your answers: (a) Find the magne
> Two tiny wire loops, with areas a1 and a2, are situated a displacement r apart (Fig. 7.42). (a) Find their mutual inductance. [Hint: Treat them as magnetic dipoles, and use Eq. 5.88.] Is your formula consistent with Eq. 7.24? (b) Suppose a current I1 is
> Suppose the circuit in Fig. 7.41 has been connected for a long time when suddenly, at time t = 0, switch S is thrown from A to B, bypassing the battery. (a) What is the current at any subsequent time t ? (b) What is the total energy delivered to the resi
> A long cable carries current in one direction uniformly distributed over its (circular) cross section. The current returns along the surface (there is a very thin insulating sheath separating the currents). Find the self-inductance per unit length.
> (a) Two metal objects are embedded in weakly conducting material of conductivity σ (Fig. 7.6). Show that the resistance between them is related to the capacitance of the arrangement by / (b) Suppose you connected a battery between 1 and 2, an
> Test Stokes’ theorem for the function v = (xy) xˆ + (2yz) yˆ + (3zx) zˆ, using the triangular shaded area of Fig. 1.34.
> Calculate the energy stored in the toroidal coil of Ex. 7.11, by applying Eq. 7.35. Use the answer to check Eq. 7.28.
> Find the energy stored in a section of length l of a long solenoid (radius R, current I , n turns per unit length), (a) using Eq. 7.30 (you found L in Prob. 7.24); (b) using Eq. 7.31 (we worked out A in Ex. 5.12); (c) using Eq. 7.35; (d) using Eq. 7.3
> A capacitor C is charged up to a voltage V and connected to an inductor L, as shown schematically in Fig. 7.39. At time t= 0, the switch S is closed. Find the current in the circuit as a function of time. How does your answer change if a resistor R is in
> An alternating current I (t)= I0 cos(ωt ) (amplitude 0.5 A, frequency 60 Hz) flows down a straight wire, which runs along the axis of a toroidal coil with rectangular cross section (inner radius 1 cm, outer radius 2 cm, height 1 cm, 1000 turns). The coil
> Try to compute the self-inductance of the “hairpin” loop shown in Fig. 7.38. (Neglect the contribution from the ends; most of the flux comes from the long straight section.) You’ll run
> Find the self-inductance per unit length of a long solenoid, of radius R, carrying n turns per unit length.
> A square loop of wire, of side a, lies midway between two long wires, 3a apart, and in the same plane. (Actually, the long wires are sides of a large rectangular loop, but the short ends are so far away that they can be neglected.) A clock- wise current
> A small loop of wire (radius a) is held a distance z above the center of a large loop (radius b), as shown in Fig. 7.37. The planes of the two loops are parallel, and perpendicular to the common axis. (a) Suppose current I flows in the big loop. Find the
> Imagine a uniform magnetic field, pointing in the z direction and filling all space / A positive charge is at rest, at the origin. Now somebody turns off the magnetic field, thereby inducing an electric field. In what direction does the charge move?16
> Where is ∂B/∂t nonzero, in Figure 7.21(b)? Exploit the analogy between Faraday’s law and Ampère’s law to sketch (qualitatively) the electric field
> Test the divergence theorem for the function v = (xy) xˆ + (2yz) yˆ+ (3zx) zˆ. Take as your volume the cube shown in Fig. 1.30, with sides of length 2.
> A capacitor C has been charged up to potential V0; at time t=0, it is connected to a resistor R, and begins to discharge (Fig. 7.5a). (a) Determine the charge on the capacitor as a function of time, Q(t). What is the current through the resistor, I (t)?
