Questions and Answers of Modern Electrodynamics

In the absence of free charge or free current, the Maxwell equations in optically active matter are(a) Let P = (ϵ − 0)E and M = (μ − 1 0 − μ −1)B, but do not introduce D or H. Assume plane
Waves in Matter in the ϕ = 0 Gauge(a) Find a gauge function that makes ϕ(r, t) = 0 a valid choice of gauge.(b) Derive the (generalized) inhomogeneous wave equation in matter (μ,) satisfied by the
For propagation along the z-axis, a medium supports left circular polarization with index of refraction nL and right circular polarization with index of refraction nR. If a plane wave propagating
A point source of light is embedded near the flat surface of a dielectric with index of refraction n. Treat the emitted light as a collection of plane waves (light rays) that propagate is otropically
(a) Explain why the matching conditions for the normal components of D and B are not needed to derive the Fresnel equations.(b) Derive the matching conditions for the components of the Poynting
A plane wave in vacuum with wave vector kI reflects from a nonmagnetic sample into a plane wave with wave vector kR. At x ray wavelengths, the index of refraction of essentially all matter is very
Let k1 = (k1x , 0, k1z) be the wave vector of a plane wave incident on the x-y plane which separates simple medium 1 from simple medium 2.(a) Use (i) The Maxwell matching conditions for each
A plane wave E = ˆyAexp[i(kz + ωt)] propagating in vacuum in the −z-direction impinges at normal incidence on the front face of a (transversely infinite) slab with thickness d, index of
Derive the Fresnel transmission amplitude formulae for non-magnetic matter: E₁ ΤΕ = 2 cos 0₁ sin 0₂ sin(0₁ + 0₂) and E₁ TM = 2 cos 0₁ sin 0₂ sin(01+0₂) cos(0₁-0₂)
An optical fiber consist of a solid rod of material with index of refraction nf cladded by a cylindrical shell of material with index ncƒ. Find the largest angle θ so that a wave incident from a
The optical properties of a remarkable class of materials called topological insulators (TI) are captured by constitutive relations which involve the fine structure constant, α = (e2/ − hc) / (4π
A piece of glass in the shape of a rhombic prism can be used to convert linearly polarized light into circularly polarized light and vice versa. The effect is based on the phase change of totally
Consider a TE (s-polarized) plane wave incident at angle θ1 onto a good conductor with skin depth δ(ω) from a transparent dielectric with index of refraction n1. Both materials are non-magnetic.
A plane wave is incident on a flat interface between two transparent, non-magnetic media. Let γI be the angle between the incident electric field vector and the plane of incidence. The corresponding
Show that the time-averaged rate at which power flows through a unit surface area of an ohmic conductor is exactly equal to the time averaged rate of Joule heating (per unit surface area) in the bulk
Consider planewave refraction from a non-conducting medium (ε , μ) into a conducting medium (ε , μ , σ). Ohmic loss requires that the refracted wave vector k2 be complex. The figure below shows
Airy’s problem is the transmission of a monochromatic plane wave through a transparent film (ε , μ) of thickness d. The text solved this problem by summing an infinite number of single-interface
A plane wave with electric field Einc(x, z) = ŷE0 exp[ik(z sin θ − x cos θ) − iωt] is incident on a perfect conductor which occupies the half-spacex < 0. Find the pressure exerted on
An electromagnetic wave with wave vector k = q + iκ propagates in simple matter with index of refraction n. Prove that the phase velocity of this wave is always less than c/n.
A corner reflector has two semi-infinite, perfect-conductor surfaces joined at a common edge with a right angle between the two surfaces. Prove that a right (left) circularly polarized plane wave
(a) Derive the generalizedwave equation satisfied by E(r, t) in non magnetic matter when the permittivity is a function of position, (r). Specialize the equation to the case when (r) = (z) and E(r,
A plane electromagnetic wave EI cos(kIz + ωIt) is incident on a perfectly reflecting mirror (solid line) that moves with constant velocity v = νẑ. The reflected plane wave is ER cos(kRz −
A vacuum wave E0(r, t) = x̂E0 exp[i(kyy + kzz − ωt)] strikes a perfectly conducting surface.(a) Write down the total electric field E which exists above the surface when the latter is
An external magnetic field B0 can cause the straight-line path of a laser beam to deflect inside a non-simple material where the constitutive relations are B = μH and D = εE − iγB × E. To see
Drude’s conductivity formula fails when the frequency ω is low and the mean time τ between electron collisions is large. If v̅ is a characteristic electron speed, one says that the normal skin
Let ϵ(ω) ϵ / 0 = 1 − ω2p/ω2 be the dielectric function of the half-space z > 0. The half-space z (a) Relate κ to qІІ in each medium.(b) Use ∇ · D = 0 and the matching conditions for
A non-magnetic dielectric consists of N atoms per unit volume. Model the polarization of this system as the dipole moment per unit volume P = −Ner, where r(t) is the displacement of each electron
Let a transverse electromagnetic wave H = x̂Hx exp i(ky − ωt) propagate in a linear magnetic medium exposed to a static magnetic field B = Bz ẑ. If γ and τ are constants, experiment shows
Let ϵ(ω) / ϵ0 = 1 − ω2p /ω2 be the dielectric function of a plasma where ωp is the plasma frequency. In a typical laboratory or astrophysical environment, any attempt to create a voltage
The figure below shows a sample of “artificial matter” composed of infinite, parallel, filamentary wires. Each row of wires carries current in the opposite direction from the rows just above and
(a) Consider a medium composed of N one-dimensional, undamped Lorentz oscillators per unit volume. Show by explicit calculation that the time average ofis equal to the time average of the sum of the
(a) Δ(x) is an acceptable representation of a delta function if Δ(0) diverges and it “filters” any smooth test function f (x):Show that these properties are satisfied byLet the real and
Let the dielectric function (ω) = ϵ0n2(ω) characterize a macroscopic sphere of matter composed of N electrons. If the wavelength of the incident field is large compared to the sphere radius a, it
An electromagnetic wave E = δE exp(−iωt) can induce a net magnetization in a metal. To see this, let the density and velocity of the electrons at a typical point be n = + δn exp(−iωt) and v
The Lorentz-model dielectric function satisfies the f-sum rule (see Application 18.5),Show this explicitly for the case when the damping constant is small. Ja do o Im ê(w) = J €ow. 2 P
Consider the Lorentz-type index of refractionThe damping constant" T > 0 and ƒ is called the oscillator strength. Assume |ƒ|(a) Produce an argument based on monochromatic plane wave propagation
If the photon had a mass M, the dispersion relation for electromagnetic waves in vacuum would be A limit on can be determined by measuring the difference in arrival times of the highest- and
A long transmission line consists of two identical wires embedded in a medium with permittivity ϵ and permeability μ. Let the wire separation d be large compared to the wire radius a. Calculate the
The dielectric function of the ionosphere is ε(ω)ε / ε0 = 1 − Ω2/ω2, where is a constant. Explain why a radio operator, exploiting the reflection of radio waves from the ionosphere,
Show that plane wave propagation does not occur at all frequencies in a medium where the current density j is proportional to the vector potential: μ0j = −k20A.
Arectangular wave guide with a constant cross section and perfectly conducting walls contains a curved section as sketched below. Also indicated is a local Cartesian coordinate system where the
Derive the in homogeneous wave equation satisfied by the electric field E(r, t) in a system where ρ(r, t) = 0 but j(r, t) ≠ = 0. Show that this equation has a plane wave solution E = E0 exp[i(k ·
(a) Suppose that ∇2⊥ ψp = λpψp and ∇2⊥ ψq = λqψq in a two dimensional domain A where either ψ|C = 0 or ∂ψ/∂n|C = 0 on the perimeter C of A. Use one of Green’s identities to
Consider time-harmonic solutions to the Maxwell equations in vacuum where the fields are independent of the azimuthal angle φ. TEM solutions of this type also have no radial component to the fields:
The TM and TE modes of a hollow-tube waveguide are determined by the two-dimensional Helmholtz equation [∇2⊥ + γ2]ψ = 0 with boundary conditions ψ|S = 0 and ∂ψ/∂n|S = 0, respectively.
The figure below shows two circular conducting tubes in cross section. Each tube has a thin metal screen inserted at one point along its length. One screen takes the form of metal wires bent into
An infinite slab of material with index of refraction n(ω) and group velocity νg < 0 occupies the space 0 < z < a. The rest of space is vacuum.(a) Consider a plane wave with electric
The figure below shows the circular cross section of an infinitely long metallic waveguide with an infinitesimally thin, metallic baffle inserted into its otherwise hollow interior. The baffle has
The two-dimensional vectors km shown below are inclined at angles θm = mπ/3 with respect to the positive x-axis. The vectors share a common magnitude |km|= k.Superpose six waves with alternating
The polarization of a medium obeys P = γ ∇ ×E.(a) Find the propagation equation for the electric field E(r, t) in this medium.(b) Find the dispersion relation and polarization of the plane waves
An electromagnetic oscillator is formed when charge sloshes back and forth between two identical, perfectly conducting spheres of radius R connected by a very thin, very long, perfectly conducting
A monochromatic plane wave in vacuum (x > 0) with Ez > 0 and Ex < 0 strikes a perfect conductor (x < 0) at an angle of incidence θ.(a) Show that, in steady state, a non-uniform TM wave
(a) Calculate the induced surface charge density σTM and the longitudinal surface current density KTM associated with the propagation of a TM mode in a perfectly conducting waveguide with a uniform
In connection with the law of conservation of linear momentum, we showed that the electromagnetic force on a volume V can be written in the formUse this formula to find the time-averaged force on
A perfectly conducting waveguide has a cross section in the shape of a semicircle with radius R.(a) Find the longitudinal fields Ez and Bz for the TM and TE modes, respectively. Find also
(a) Show that the frequency of any mode (E,B) of a resonant cavity with volume V can be computed from(b) Suppose E → E + δE or B → B + δB, where δE and δB satisfy the boundary conditions.
Consider a hollow conducting tube with a circular cross section of radius R and infinite length.(a) Find monochromatic TE (Ez = 0) and TM (Hz = 0) solutions of the Maxwell equations inside the tube
Two rectangular waveguides with different major side lengths (a1 < a2) along the x-axis and equal minor sides (b1 = b2) along the y-axis are butt-joined in the z = 0 plane.Waveguide 1 propagates a
(a) Show that a general TM wave in a hollow-tube waveguide can be derived from a longitudinal vector potential A(r, t) = ẑA(r⊥) exp(ihz − ωt) which satisfies the wave equation.(b) Duality
A perfectly conducting resonant cavity has the shape of a rectangular box where the length, width, and height are chosen as three unequal irrational numbers. Evaluate (at least) the first 105
A particle with charge q and velocity v = νẑ enters a perfectly conducting radio-frequency resonant cavity (length L) through a tiny entrance hole and then exits the cavity through an equally tiny
An infinitely long coaxial waveguide is formed in the vacuum volume between two concentric, perfectly conducting cylinders with radii b and a > b.(a) Find E and B for the TE and TM modes of this
(a) Clearly state the conditions required for the electric field in a medium with a spatially varying index of refraction to satisfy the equation(b) Let the index decrease radially away from a
A medium with index of refraction n1 occupies the half-space y 0 is filled with a medium with index of refraction n2. A propagating electric field bound to the y = 0 plane can have the form(a) Find
Prove the assertion made in the text that the total electromagnetic energy of a lossless resonant cavity can be put in the form of the total mechanical energy of a collection of undamped harmonic
Confirm by direct substitution into the defining equation that the free-space Green function for the Helmholtz equation in two-dimensional plane polar coordinates ρ = (ρ,φ) iswhere H(1) 0 (x) is
Find the numerical value of the lowest resonant frequency ω0 and the exact half-width " of the resonant cavity God instructed Moses to build in Exodus 25:10-11: “Make an ark of acacia-wood: two
A small polarizable and magnetizable object inserted into a resonant cavity produces a shift in each resonance frequency by an amount δω/ω. Derive an expression for this frequency shift in terms
A lighting strike associated with a thunderstorm acts very much like a broadband antenna. Explain why data from airplane-borne electric and magnetic field sensors flown immediately above (in the near
An infinitely long straight wire on the z-axis has a circular cross section and obeys j(ω) = σ0E(ω) for all ρ ≤ a. After initial transients, one finds the charge density ρ(r, t) ≡ 0 and the
A cylindrical column of electrons has uniform charge density ρ0 and radius a. (a) Find the force on an electron at a radius r < a.(b) A moving observer sees the column as a beam of
N identical, equally-spaced point particles, each with charge q, move in a circle of radius a. Each particle moves with the same constant speed v around the ring. Show that the Li´enard-Wiechert
The regiony 0 is filled with material where μ = μ0 and Dij = ϵ ijEj. Let α, β, and γ be real numbers and take the dielectric matrix as(a) Write out the electric field everywhere if a wave
Let the half-space z ≥ 0 be filled with a magnetic crystal where H = μ−1 · B. The inverse permeability matrix μ−1 is (rows and columns labeled by x, y, z)Assume that ε = ε0 inside the
Find the frequency-dependent susceptibility X̂ (ω) when the temporal susceptibility χ(t) of a medium is(a) χ(t ) = χ0 δ(t)(b) χ(t ) = χ0 θ(t)(c) χ(t ) = χ0 θ(t) exp(−t/τ)(d) χ(t ) =
(a) A homogeneous medium is characterized by a magnetic susceptibility χm. Use Faraday’s law to show that the magnetization current J = ∇ × M can be expressed in the formThis result can be
The text shows that the electric field of a point electric dipole at the origin with moment p(t) = p(t)ẑ produces an electric field (away from the source) in cylindrical coordinates of the
A current density j1(r) exp(−iωt ) produces fields E1(r) exp(−iωt) and B1(r) × exp(−iωt ). A second current density j2 exp(−iωt ) produces fields E2(r) exp(−iωt) and B2(r) ×
The scalar and vector potentials in the Coulomb gauge arewhere(a) The scalar potential ϕC(r, t) is not retarded because it depends on ρ(r' , t). The vector potential AC(r, t) looks retarded,
A point electric dipole with moment p(t) has a fixed position in space. Show that the rate at which energy flows through a spherical surface of radius R centered at the dipole is du dt = 21 3
A charge density ρ(r, t) = q(t )δ(r) where q(t ) = 0 for t τ.(a) Calculate E(r, t) and B(r, t) using symmetry and elementary methods.(b) Calculate E(r, t) and B(r, t) from the Coulomb gauge
Consider the time-harmonic dipole antenna with imposed current I (z) = I0 sin(kd − k|z|) discussed in Section 20.6.1 of the text.(a) Evaluate the time-averaged angular distribution of power in the
The Green function G(x, y, z, t > 0) = δ(t − r/c)/4πr is a solution of the inhomogeneous wave equation in three space dimensions with the source term −δ(x)δ(y)δ(z)δ(t).(a) Show that
A classical electron gas with number density n0 exhibits a Maxwell velocity distribution at temperature T . In the presence of a uniform magnetic field B0, the gas emits radiation at a
Suppose f(r) is a localized vector function and j(r, t) = j(r|ω) exp(−iωt) is a time-harmonic current density whereProve that j(r, t) does not radiate and find the physical meaning of f(r, t).
A current distribution consists of N identical sources. The kth source is identical to the first source except for a rigid translation by an amount Rk (k = 1, 2, . . . , N). The sources oscillate at
(a) Explain why a localized (and entirely classical) source of charge and current does not recoil when it emits dipole radiation.(b) Is recoil ever possible for a classical radiation source? If not,
A monochromatic plane wave scatters from a perfectly conducting wire where a h. Assume that both the propagation vector and the electric field of the incident wave lie in the y-z plane as shown
(a) Let S be the illuminated portion of a conductor. If r̂ is the local unit normal vector to S and k = k0 k̂ is the propagation direction of the backscattered wave, show that the cross section
Two identical point charges q are fixed to the ends of a rod of length 2 which rotates with constant angular velocity 1/2ω in the x-y plane about an axis perpendicular to the rod and through its
Let the origin of coordinates be centered on a compact, time-harmonic source of electromagnetic radiation. The time-averaged power radiated into a differential element of solid angle d Ω centered on
Let the current density in a linear antenna of length h be(a) Find Erad(r, t) for the current I (z, t) = Aδ(t). Your answer will have two terms. Determine the apparent origin of each term and give
Consider time-harmonic electromagnetic fields in the domain z ≥ 0 of the form(a) Let ŷ · ETE(x, z = 0, t = 0) = E̅y (x). Determine the scalar function ΛTE(kx) and the vector function TE(kx)
A very long (infinite) wire with a very small cross sectional area A is coincident with the z-axis. The wire carries a current I (t) = I exp(−iωt).(a) Use the Poynting vector to determine the
A collection of N charges lies inside a volume V. With respect to a fixed origin, the angular momentum of the charges and the electromagnetic fields they produce within V is(a) Let S be the surface
Two equal and opposite charges are attached to the ends of a rod of length s. The rod rotates counterclockwise in the x-y plane with angular speed ω = ck. The electric dipole moment of the system at
Apply a magnetic field B = Bẑ to a neutron with its spin (assign a “classical” magnitude S = − h/2 to the spin) oriented initially along the +y-axis. Show that, because the precessing
A square loop of wire in the x-y plane is centered at the origin with its edges (each of length 2a) parallel to the axes. Current flows counterclockwise around the loop as viewed from the positive
(a) Find the potentials and the fields produced by a current I (t) = I0 Θ(t) that turns on abruptly at t = 0 in a neutral, filamentary wire coincident with the entire z-axis.(b) Show that the
Two point electric dipoles are crossed in the x-y plane as shown below (at left). Both oscillate at frequency ω but with a π/2 phase difference.(a) Sketch the time evolution of the instantaneous

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