Questions and Answers of Electrodynamics

A long solenoid with radius a and n tums per unit length carries a time-dependent current 1(t) in the Ф direction. Find the electric field (magnitude and direction) at a distance s from the axis
An alternating current I = I0 cos (wt) flows down a long straight wire, and returns along a coaxial conducting tube of radius a. (a) In what direction does the induced electric field point
A long solenoid of radius a, carrying n tums per unit length, is looped by a wire with resistance R, as shown in Fig. 7.27. (a) If the current in the solenoid is increasing at a constant rate (dl/dt
A square loop, side a, resistance R, lies a distance s from an infinite straight wire that carries current I (Fig. 7.28). Now someone cuts the wire, so that I drops to zero. In what direction does
A toroidal coil has a rectangular cross section, with inner radius a, outer radius a + w, and height h. It carries a total of N tightly wound turns, and the current is increasing at a constant rate
A small loop of wire (radius a) lies a distance z above the center of a large loop (radius b), as shown in Fig. 7.36. The planes of the two loops are parallel, and perpendicular to the common
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
Find the self-inductance per unit length of a long solenoid, of radius R, carrying n tums per unit length.
Try to compute the self-inductance of the "hairpin" loop shown in Fig. 7.37. (Neglect the contribution from the ends; most of the flux comes from the long straight section.) You'll mn into a snag
An alternating current I0 cos(wt) (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
A capacitor C is charged up to a potential V and connected to an inductor L. as shown schematically in Fig. 7.38. At time t = 0 the switch S is closed. Find the current in the circuit as a function
Find the energy stored in a section of length l of a long solenoid (radius R. current I, n tums per unit length),(a) Using Eq. 7.29 (you found L in Prob. 7.22);(b) Using Eq. 7.30 (we worked out A in
Calculate the energy stored in the toroidal coil of Ex. 7.11, by applying Eq. 7.34. Use the answer to check Eq. 7.27.
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
Suppose the circuit in Fig. 7.40 has been connected for a long time when suddenly, at time t = 0, switch S is thrown, bypassing the battery. (a) What is the current at any subsequent time t?? (b)
Two tiny wire loops, with areas a1 and a2, are situated a displacement π apart (Fig. 7.41). (a) Find their mutual inductance. (b) Suppose a current I1 is flowing in loop 1, and we propose to turn on
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
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
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?)
Suppose (the theta function is defined in Prob. 1.45b). Show that these fields satisfy all of Maxwell's equations, and determine p and J. Descrlbe the physical situation that gives rise to these
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.
Suppose a magnetic monopole qm passes through a resistanceless loop of wire with self-inductance L. What current is induced in the loop?
Sea water at frequency v = 4 x 108 Hz has permittivity ε = 81ε0, permeability µ = µ0, and resistivity p = 0.23 Ω ∙ m. What is the ratio of conduction current to displacement current?
Two very large metal plates are held a distance d apart, one at potential zero, the other at potential V0 (Fig. 7.48). A metal sphere of radius a (a
Two long, straight copper pipes, each of radius a, are held a distance 2d apart (see Fig. 7.49). One is at potential V0, the other at ?? V0. The space surrounding the pipes is filled with weakly
A common textbook problem asks you to calculate the resistance of a cone-shaped object, of resistivity p, with length L, radius a at one end, and radius b at the other (Fig. 7.50). The two ends are
A rare case in which the electrostatic field E for a circuit can actually be calculated is the following Imagine an infinitely long cylindrical sheet, of uniform resistivity and radius a. A slot
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
A familiar demonstration of superconductivity (Problem. 7.42) is the levitation of a magnet over a piece of superconducting matedhal. This phenomenon can be analyzed using the method of images. 19
If a magnetic dipole levitating above an infinite superconducting plane (Problem 7.43) is free to rotate, what orientation will it adopt, and how high above the surface will it float?
A perfectly conducting spherical shell of radius a rotates about the z axis with angular velocity to, in a uniform magnetic field B = B0z,. Calculate the emf developed between the "north pole" and
Refer to Prob. 7.11 (and use the result of Prob. 5.40, if it helps):(a) Does the square ring fall faster in the orientation shown (Fig. 7.19), or when rotated 45o about an axis coming out of the
(a) Use the analogy between Faraday's law and Ampere's law, together with the Biot-Savart law, to show that for Faraday-induced electric fields. (b) Referring to Prob. 5.50a, show that where A is the
Electrons undergoing cyclotron motion can be speeded up by increasing the magnetic field; the accompanying electric field will impart tangential acceleration. This is the principle of the betatron.
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
The current in a long solenoid is increasing linearly with time, so that the flux is proportional to t: Ф = at. Two voltmeters are connected to diametrically opposite points (A and B), together with
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 = vBh/R. But what if the wire is made out of perfectly
(a) Use the Neumann formula (Eq. 7.22) to calculate the mutual inductance of the configuration in Fig. 7.36, assuming a is very small (a (b) For the general case (not assuming a is small) show that
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;
A transformer (Prob. 7.53) takes an input AC voltage of amplitude V1, and delivers an output voltage of amplitude V2, which is dete nnined by the turns ratio (V2/V1 = N2/N1). If N2 > N1 the
Suppose J(r) is constant in time but p (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
The magnetic field of an infinite straight wire carrying a steady current I can be obtained from the displacement current term in the Ampere/Maxwell law, as follows: Picture the current as consisting
The magnetic field outside a long straight wire carrying a steady current 1 is (of course) The electric field inside the wire is uniform: wherepis the resistivity and a is the radius (see Exs. 7.1
Suppose J(r) is constant in time but p (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
(a) Show that Maxwell's equations with magnetic charge (Eq.7.43) are invariant under the duality transformation where c ?? 1/ ??ε0?0 and a is an arbitrary rotation angle in "E/B-space." Charge and
Calculate the power (energy per unit time) transported down the cables of Ex. 7.13 and Prob. 7.5 8, assuming the two conductors are held at potential difference V, and carry current I (down one and
Consider the charging capacitor in Prob. 7.31.(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 =
Calculate the force of magnetic attraction between the northern and southern hemispheres of a uniformly charged spinning spherical shell, with radius R, angular velocity to, and surface charge
(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
Consider an infinite parallel-plate capacitor, with the lower plate (at z = ?? d/2) carrying the charge density -a, and the upper plate (at z = +d/2) carrying the charge density (a) Determine all
A charged parallel-plate capacitor (with uniform electric field E = E z) is placed in a uniform magnetic field B = B x, as shown in Fig. 8.6. 3(a) Find the electromagnetic momentum in the space
In Ex. 8.4, suppose that instead of turning off the magnetic field (by reducing 1) we turn off the electric field, by connecting a weakly 6 conducting radial spoke between the cylinders. (We'll have
Imagine an iron sphere of radius R that carries a charge Q and a uniform magnetization M = Mz. The sphere is initially at rest.(a) Compute the angular momentum stored in the electromagnetic
A very long solen6id of radius a, with n turns per unit length, carries a current Is. Coaxial with the soienoid, at radius b >> a, is a circular ring of wire, with resistance R. When the
A sphere of radius R cames a uniform polarization P and a uniform magnetization M (not necessarily in the same direction). Find the electromagnetic momentum of this configuration.
Picture the electron as a uniformly charged spherical shell, with charge e and radius R, spinning at angular velocity ro.(a) Calculate the total energy contained in the electromagnetic fields.(b)
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
Paul De Young, of Hope College, points out that because the cylinders in Ex. 8.4 are left rotating (at angular velocities wa and wb, say), there is actually a residual magnetic field, and hence
A point charge q is a distance a > R from the axis of an infinite solenoid (radius R, n turns per unit length, current 1). Find the linear momentum and the angular momentum in the fields. (Put q
(a) Carry through the argument in Sect. 8.1.2, starting with Eq. 8.6, but using Jf in place of J. Show that the Poynting vector becomes S = E x H, and the rate of change of the energy density in the
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.
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).
Use Eq. 9.19 to determine A3 and δ3 in terms of A1, A2, δ1, and δ2.
Obtain Eq. 9.20 directly from the wave equation, by separation of variables.
Suppose you send an incident wave of specified shape, g1 (z – V1t), down string number 1. It gives rise to a reflected wave, h R (z + v1t), and a transmitted wave, gT (z – v2t). By imposing the
(1). 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.(2). Find the amplitude and phase of the reflected and
(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
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 n) results from the
Write down the (real) electric and magnetic fields for a monochromatic plane wave of amplitude E0, frequency co, and phase angle zero that is(a) Traveling in the negative x direction and polarized in
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
In the complex notation there is a clever device for finding the time average of a product. Suppose f(r, t) = Acos (k ?? r ?? wt + δa) and g(r, t) = B cos (k ?? r ?? wt + δb). Show that (fg) =
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
Calculate the exact reflection and transmission coefficients, without assuming µ1 = µ2 = µ0. Confirm that R + T = 1.
In writing Eqs 9.76 and 9.77,1 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.
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 q- B = C.
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 9.101, and obtain the Fresnel
The index of refraction of diamond is 2.42. Construct the graph analogous to Fig. 9.16 for the air/diamond interface. (Assume ?1 = ?2 = ?0.) In particular, calculate (a) The amplitudes at normal
(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
(a) Show that the skin depth in a poor conductor (σ << wε) is (2/σ)√ε/µ (independent of frequency). Find the skin depth (in meters) for (pure) water.(b) Show that the skin depth in a
(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
Calculate the reflection coefficient for light at an air-to-silver interface (µ1 = µ2 = µ0, ε1 = ε0, σ = 6 x 107 (Ω ∙ m–l), at optical frequencies (w = 4 x 1015/s).
(a) Shallow water is non dispersive; the 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
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
Find the width of the anomalous dispersion region for the case of a single resonance at frequency w0. Assume γ << w0. Show that the index of refraction assumes its maximum and minimum values
Assuming negligible damping (γj = 0), calculate the group velocity (vg = dw/dk) of the waves described by Eqs. 9.166 and 9.169. Show that vg < c, even when v > c.
(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.
Show that the mode TE00 cannot occur in a rectangular wave guide.
Consider a rectangular wave guide with dimensions 2.28 cm x 1.01 cm. What TE modes will propagate in this wave guide, if the driving frequency is 1.70 x 1010 Hz? Suppose you wanted to excite only one
Confirm that the energy in the TEmn mode travels at the group velocity.
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
(a) Show directly that Eqs. 9.197 satisfy Maxwell's equations (9.177) and the boundary conditions 9.175.(b) Find the charge density, λ(z, t), and the current, I(z, t), on the inner conductor.
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).
Suppose (This is, incidentally, the simplest possible spherical wave. For notational convenience, let (kr – wt) ≡ u in your calculations.)(a) Show that E obeys all four of Maxwell's equations, in
Light of (angular) frequency ro 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
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
Light from an aquarium (Fig. 9.27) goes from water (n = ¾) through a plane of glass (n = 3/2) into air (n = 1). Assuming it's a monochromatic plane wave and that it strikes the glass at normal
According to Snell's law, when light passes from an optically dense medium into a less dense one (n! > n2) the propagation vector k bends away from the normal (Fig. 9.28). In particular, if the
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
Show that the differential equations for V and A (Eqs. 10.4 and 10.5) can be written in the more symmetrical form where
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 =

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