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engineering
modern electrodynamics

Modern Electrodynamics 1st Edition Andrew Zangwill - Solutions

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 vector potential in the ϕ = 0 gauge. There is no free charge or free current anywhere.(c) Find 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 through this medium has E(z = 0, t) = x̂E exp(−iωt), find the values of z where the wave is
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 away from the source. Find the fraction of light rays that can refract out of the dielectric into
(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 vector at a flat interface between two transparent media.
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 slightly less than 1. Typically, n ≈ 1 − δ with δ ∼ 10−5. This makes the phenomenon of total
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 component of the electric field and (ii) The fact that plane waves in a simple medium are transverse to
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 refraction n, and magnetic permeability μ0. The back face of the slab (z = 0) is coated with a very thin
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 medium with index na remains in the solid rod by repeated total internal reflection from the cladding
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 ϵ0). With α0 = α √ ϵ0/μ0, the relations are(a) Begin with the Maxwell equations in matter
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 internally reflected light.(a) Show that the “s” or ⊥ Fresnel reflection amplitude for total
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. Show that the phase of the reflected wave with respect to the incident wave is approximately
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 angles for the reflected and refracted field vectors are γR and γT.(a) Use the Fresnel equations
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 of the conductor.
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 a proposed refraction geometry where k2 = q + iκ.(a) Explain why κ points in the +z-direction and
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 Fresnel reflections and transmissions. Here, we specialize to normal incidence and use the matching
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 the conductor by (a) Evaluating the Lorentz force on the currents generated on the surface of
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 incident on the reflector as indicated in the diagram reflects back toward the source as a right (left)
(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, t) = ˆxE(z, t ).(b) Let E(z, t) = E(z) exp(−iωt) and let (z) = ε0[1 + α cos(2k0z)]. Show that
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 − ωRt).(a) Use conservation of momentum to show that the force exerted on an area A of the mirror iswhere
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 defined by z = 0.(b) To the field E found in part (a), add a linear combination of plane waves E'such that
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 this, let a linearly polarized, monochromatic plane wave with electric field strength E enter the
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 effect becomes anomalous when the mean distance between collisions ℓ = v̅τ exceeds the skin depth
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 E to deduce that(c) Derive the dispersion relationand find the physically allowed solution ω(q) 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 from its nucleus. The dynamics of each electron is modeled as a damped harmonic oscillator:(a) Do not
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 that the induced magnetization obeys(a) The first term on the right describes precession of the
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 drop V (t) = V cos ωt across the plasma generates a region of vacuum (called the “sheath”) on
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 below it. Each row is also displaced (vertically and horizontally) by a distance a/2 from the rows
(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 electric, kinetic, and potential energy densities of the medium.(b) Consider a Drude medium with
(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 imaginary parts of X̂(ω) = X'(ω) + iX"(ω) satisfy the Kramers-Kr¨onig relations. Substitute one
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 is legitimate to use a quasistatic approximation. This problem equates two expressions for the
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 = v̅ + δv exp(−iωt ), where n̅ is the mean density of the electrons and v̅ = 0 is the mean
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 that ƒ > 0 describes an absorbing medium (like a conventional dielectric) which extracts energy
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 lowest-frequency components of a wave packet received from an astrophysical source that emits EM
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 capacitance per unit length C and the inductance per unit length . Confirm the general relation C =
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, nearly always receives signals with ω > Ωfrom distant broadcasting stations, but only occasionally
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 z-axis and y-axis remain tangent and normal to the walls, respectively.(a) The scalar function Φ
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 · r − ωt)] for a system of non-interacting electrons (number density n) that respond to an
(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 show that(b) Let Ep(r, t) = Ep(x, y)ei(hpz−ωt) and Bp(r, t) = Bp(x, y)ei(hpz−ωt) be the fields
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: Er = Br = 0.(a) Show that the conditions stated above decouple the Maxwell curl equations into two
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 same equation and boundary conditions apply when ψ(x, y) is the wave function of a free particle
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 concentric circles. The other takes the form of metal wires arranged like the spokes of a wheel. One of
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 field E = x̂E0 exp[iω(z/c − t )] incident on the slab fromz < 0. Use n(ω) to write formulae
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 infinite length and a width equal to the radius R of the waveguide.(a) Show that the baffle increases
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 amplitudes to form the scalar functionDraw the outline of a two-dimensional resonant cavity which
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 that propagate in this medium.
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 rod of radius a > R. The net charge of the entire structure is zero. Assume that no charge
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 occupies the vacuum space above the conductor.(b) Calculate the time-averaged Poynting vector
(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 cross section. Show that KTM = νpσTMẑ where νp is the phase velocity of the wave propagating
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 each of the six perfectly conducting walls of a resonant cavity defined by 0 ≤ x ≤ a, 0 ≤ y ≤
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 the cut-off frequency for these modes.(b) Write explicit formulae for the transverse fields for the
(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. Prove that the change in ω2 is only second-order in δE or δB. This implies that any choice of E(r)
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 which propagate around the tube circumference and thus do not depend on the longitudinalcoordinate
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 TE10 mode (only) in the +z-direction toward waveguide 2. Find the amplitude of the various modes
(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 implies that a general TE wave can be derived from an “electric vector potential” Ã, where
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 resonant frequencies numerically, label them so that ω1 ≤ ω2 ≤ ω3 ≤ · · ·, and construct a
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 exit hole. If ν is large, we can use the impulse approximation to compute the momentum “kick”
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 guide and find (but do not try to solve) the transcendental equations that determine the mode
(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 central axis according to n(ρ) = n0[1 − α2ρ2]. Ensure that the foregoing applies and show that 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 the field H(r, t) that accompanies E(r, t).(b) Show that two types of solutions exist, one where E
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 oscillators. Identify the electric and magnetic contributions explicitly. Assume that the vector
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 the zero-order Hankel function of the first kind. Go(p, p') = HD (k\p - p'\),
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 cubits and a half shall be the length thereof, and a cubit and a half the breadth thereof, and a cubit
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 of the time-harmonic dipole moments p(t) and m(t ) induced in the object by the relevant cavity mode
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 zone of) such storms reveal Poynting fluxes in the p̂ direction (with respect to the ẑ-direction
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 current I (t) = I0 cos ωt everywhere inside the wire.(a) Solve an appropriate Helmholtz equation
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 electrons, each moving with uniform speed v. What force does this observer report is felt by an electron in
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 electric field is static everywhere on the symmetry axis.
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 incident from the vacuum is E = E0 x̂ exp[iω(y/c − t )].(b) Repeat part (a) if the incident field
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 crystal and that the real, dimensionless matrix elements satisfy m > m' > 0.(a) Show that ω(k,
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 ) = χ0 θ(t) sin(ω0t).Delta functions appear naturally in some of these if you use a convergence factor
(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 regarded as a special case of a homogeneous medium that obeys Ohm’s law with a conductivity tensor
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 formwhereThe family of curves R(ρ, z) = const. = R0 may be interpreted as electrical field lines.(a) Show
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) × exp(−iωt ).(a) If V is a volume bounded by a surface S, prove the Lorentz reciprocity theorem:(b)
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, because it depends on j⊥ (r , t − |r − r '|/c), but it is not. Use the properties of the transverse
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 4лEо d p² pp p² H + dt 2R3 CR² + :}, c²R ret + Pret C3
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 scalar and vector potentials.(c) Calculate E(r, t) and B(r, t) from the Lorenz gauge scalar and vector
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 limit when the radiation wavelength is very large compared to the length of the antenna. Such an
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 G2(x, y, t) = ∫∞ −∞ dz G(x, y, z, t) is a solution of the inhomogeneous wave equation
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 wavelength which is much larger than the mean separation between the electrons. Find the radiated power per
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). w² iwpoj (rw) = V x [V x f(r)] - -f(r).
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 the same frequency ω but have different phases δk. That is,(a) Show that the angular distribution
(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, explain why not. If so, give an example.
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 below.(a) In the Rayleigh limit when k0h 0α(ẑ · E0)ẑ. Assume an induced surface current density,and
(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 for backscattering in the physical optics approximation is(b) Specialize to a flat, rectangular plate
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 center.(a) Calculate the electric dipole moment p(t). Is there electric dipole radiation?(b) Calculate
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 an observation point r has the formThe vector ∝ = p0 if the source has a time-dependent
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 an argument for the time delay between the two. Make a polar plot centered on the antenna and regard
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) so thatsolve Maxwell’s equations in free space. The wave vector k = x̂ kx + ẑ kz is
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 dependence of the radiation fields on the variable ρ (cylindrical coordinates) which measures 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 which bounds V. The text used a general conservation law to establish thatDerive this expression
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 t = 0 has the value p0 = qs x̂.(a) Show that the electric field in the radiation zone isExplain
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 magnetic moment m radiates energy, the (polar) orientation angle of the moment decays to zero as (t) cos
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 z-axis. The time-dependence of the current iswhere τ > 2a/c.(a) Show that the radiation vector
(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 electric and magnetic fields approach their expected values as t →∞. I(t) 0 R. Z
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 angular distribution of power in the x-y plane at several representative times between t = 0 and t =
Let Einc = e0E0 exp[i(kz − ωt)] be the electric field of a planewave propagating in a homogeneous dielectric medium. The wave vector k = nk0 = nω/c, where n is the index of refractionof the medium. Suppose that the number density of scatters increases from N to N + δN in a thin layer of the

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