Questions and Answers of Modern Electrodynamics

A magnetic field B(z, t) = ŷ B0 cos ωt is applied just outside the lower (z = 0) surface of a semi-infinite slab of ohmic material which extends to z = ∞.(a) Find the quasi-magnetostatic
A steady current flows through seven wires as indicated in the figure. Find the asymptotic form of the vector potential using Cartesian coordinates and Cartesian unit vectors. (a, -a/2,0) I (-a,
Consider a two-loop circuit where the total magnetic energy is UB Prove that M2 ≤ L1L2. Do not simply quote a theorem from the theory of matrices. (L₁1²+2M1₁ 1₂+ L₂12)
A very long piece of magnetic recording tape has a length L, a width ω << L, and a thickness t << ω. The tape has a magnetization M(x) = x̂ M cos kx.(a) Solve Laplace’s equation for
An insulating sphere with radius R rotates with angular velocity ω = ωẑ. The total charge Q of the sphere is uniformly distributed over its surface.(a) Show that the magnetic field outside the
What magnetization M(r) imposed on an infinite piece of iron produces the largest magnetic field at a given point? Francis Bitter (a pioneer in the design of high-field magnets) posed this problem in
Einstein published the following argument in 1910. The solid lines with arrows in the figure below show the directions of current flow for a “can-of-current”. A constant current I flows up the
Find the self-inductance of a coil produced by winding a wire N >> 1 times around the surface of sphere of radius R such that the density of turns is uniform along the z-axis perpendicular to
Let N be the demagnetization tensor in the principal axis system of a prolate (a > b) ellipsoid of revolution uniformly magnetized along its symmetry axis.(a) Explain why only Nzz need be
The diagram shows a cylindrical hole of radius R drilled through a permanent magnet which is infinite in the two directions transverse to its thickness t. Find the magnetic field everywhere when R
The Moon has no magnetic field outside of itself, despite the observed permanent magnetization of rocks collected by lunar missions. One explanation supposes that the Moon once had a geodynamo (like
A uniform external field Bext = Bext x̂ is applied to an infinitely long cylindrical shell with inside radius a, outside radius b, and relative permeability κm = μ/μ0. The rest of space is
A straight wire with radius a and magnetic permeability μ carries a conduction current density j0 = j0 ẑ = ẑ /0 πa2.(a) Find the induced magnetization M and the volume and surface
(a) A cylindrical solenoid occupies the interval −L/2 ≤ z ≤ L/2, has radius R, and is wound tightly with N turns of a wire which carries current I. Use superposition and derive an expression
(a) An infinitely large film of insulating magnetic material has permeability μ and thickness h. A uniform external magnetic field B0 is oriented perpendicular to the plane of the film. Find
The diagram shows a filamentary loop with radius a, steady current I , and mass per unit length ρ. The loop is levitated against its weight at a height h above the north pole of a very long
The space x > 0 (x < 0) is occupied by a medium with magnetic permeability μ1 (μ2). A line current I points out of the paper in medium 1 at a distance a from the interface with medium 2.(a)
A point magnetic dipole is located at the center of a magnetizable sphere with radius R and permeability μ. Find H(r) everywhere.
An infinitely long and straight wire carries a steady current I in the +z-direction at a distance d from the surface of a perfect conductor that occupies the half-space x > 0.(a) Find the current
An infinite cylinder of radius R filled with matter with permeability μ1 is embedded in an infinite medium with permeability μ2. A wire carries a current Iƒ up the z-axis of the cylinder. Show by
In 1935, the brothers Fritz and Heinz London described superconductivity using a phenomenological constitutive equation where a length δ > 0 relates the current density to the Coulomb vector
A surface current density K = −Kx̂ flows in the half-plane (x > 0, z = 0). The current accumulates on the line x = 0 which bounds the half-plane.(a) Find E(r, t) and B(r, t) in the
A superconducting sphere of radius R is placed in an external magnetic field B0. Show that the current which develops on the surface of the superconductor has density μ0K = (3/2)r̂ × B0.
A long cylinder has a cross sectional area A of arbitrary shape that is constant over its length L >> √A. The cylinder is longitudinally magnetized with a uniform magnetic moment per unit
The diagram shows two ways to periodically arrange N identical permanent bar magnets. The magnetic moments of the magnets are all parallel in configuration 1. The moments are alternately parallel and
Let L0 be the self-inductance of a free current loop in vacuum. Show that the self-inductance changes to L = κmL0 when the vacuum is replaced by a magnetic medium with permeability μ = κmμ0.
An infinite straight wire with a cross sectional area πa2 carries a low frequency current I(t) = I0 cos ωt. Without close inspection, it is impossible to tell if the wire is broken into
A tightly wound solenoid with n turns per unit length and cross sectional area A is bent into a flexible torus. Two external leads let the current flow in and out. Find the EMF that appears across
The text shows that a body with uniform polarization P and uniform velocity ν generates a magnetization M = P × ν. Confirm this by comparing the convection surface current density to the presumed
A time-independent surface current with density K flows in the x-y plane from infinity to the point r = 0 in a radially symmetric manner. As a result, charge accumulates at r = 0 at the rate dq/dt =
Consider a static and azimuthally symmetric charge distribution ρ(r) which produces no electric field outside itself except a single, pure, spherical electric multipole field of order ℓ. Show
A voltage V (t) = V0 sin ωt is applied between the plates of a circular capacitor filled with ohmic matter of conductivity σ. The radius R of the plates is very large compared to the plate
An ohmic ring with radius a, mass M, and total resistance R lies in the x-y plane. At t = 0, the center of the ring passes by the origin with velocity v = v0 x̂. How far does the ring travel
A distance d separates the infinitesimally thin and circular plates of a capacitor. The plates have radius R >> d and instantaneous charges ±Q(t) as they are slowly discharged by connection to
The diagram below shows a planar circuit composed of zero resistance wires, two resistors R1 and R2, and two voltmeters V1 and V2. A tightly wound solenoid with radius r produces a magnetic field
A tightly wound solenoid with radius R and length ℓ >>R is composed of N turns of conducting wire. The solenoid carries a slowly varying current I (t) and experiences a voltage drop V (t)
An annular disk has thickness t, inner radius R1, outer radius R2, and conductivity σ. Let a radial current I0 flow from the inner periphery to the outer periphery of the disk.(a) If n is the
A charge Q is distributed uniformly on a non-conducting ring of radius R and mass M. The ring is dropped from rest from a height h and falls to the ground through a non-uniform magnetic field B(r).
A positive point charge q moves with velocity v straight toward an infinitely large, grounded, ohmic plane.(a) Find the charge density σ(r) induced on the plane at the moment when the distance
The diagram shows a metal sphere (radius a) moving with speed v parallel to a straight wire which carries a current I . The distance between the wire and the center of the sphere is d >> a.(a)
The three parallel, ohmic wires shown below are driven at frequency ω by a common source of EMF. The wires have length L, separation b, and radius a where a << b << L. If δ is the skin
Confirm that the formulae below satisfy the four Maxwell equations in the quasi-electrostatic approximation. 7412100 [d³rs d³p P(r', t) |│r - r| E(r, t) = -V- B(r, t) = V x to f d ²³ 'd3,
A conducting wire frame with side lengths a and b lies at rest on a frictionless horizontal surface at a distance l from a long straight wire carrying a current I0 (see figure below). The mass of the
A small magnet (weight w) falls under gravity down the center of an infinitely long, vertical, conducting tube of radius a, wall thickness t << R, and conductivity σ. Let the tube be
Consider an ohmic wire with length L, radius a, and conductivity σ in the high-frequency limit of quasi-magnetostatics. Use a simple physical argument to show that (up to a dimensionless constant)
An early competitor of the Big Bang theory postulates the “continuous creation” of charged matter at a (very small) constant rate R at every point in space. In such a theory, the continuity
A longitudinal AC magnetic field B(t ) = ẑB0 cos ωt is driven through the interior of an ohmic tube with length L and radius R << L.(a) Find the low-frequency eddy-current density inside the
A slab of material with conductivity σ, electric permittivity , and magnetic permeability μ occupies the infinite volume between the planes y = ±d.(a) Find the steady-state magnetic field inside
The equation of motion for a magnetic dipole moment m which rotates about its center with an angular velocity Ω is dm / dt = Ω × m. Find the electric and magnetic fields associated with this
A wire loop with radius b in the x-y plane carries a time-harmonic current I0 cos ωt.(a) Find the value of I0 needed to levitate a small sphere with mass m, radius a, and conductivity σ at a
The side view below shows an iron magnet with two exposed pole faces moving slowly to the right at speed ν. A stationary conducting wire bent into a U-shape (P'Q'QP) is placed at an angle in the gap
An ohmic wire with resistance R is bent into a closed ring. Suppose a monopole with magnetic charge g approaches Earth from a distant galaxy, passes through the ring, and then continues on its way to
(a) Suppose ϕL and AL satisfy the Lorenz gauge constraint. What equation must Λ satisfy to ensure that A' = AL − ∇ and ϕ' = ϕL + ʌ̇ are Lorenz gauge potentials also?(b) What
The Helmholtz theorem guarantees that the vector potential can be decomposed in the form A = A⊥ + AІІ where ∇ · A⊥ = 0 and ∇ × A = 0. Show that a gauge invariant physical observable
Show that the Cartesian components of the transverse current density j⊥(r, t) used to define the Coulomb gauge vector potential can be written in terms of the total current density j(r, t) aswhere
A point charge q sits at (a, 0, 0), a point charge − q sits at (−a, 0, 0), and a uniform magnetic field B = Bẑ fills all of space. (a) Prove that the streamlines of the Poynting vector
An ohmic bar with mass m slides without friction on two parallel, perfectly conducting rails. A uniform magnetic field B points into the page as shown. Let R be the resistance of the bar over the
(a) Confirm that ϕ(r) = −r · E and A = −1/2 r × B are acceptable scalar and vector potentials, respectively, for a constant electric field E and a constant magnetic field B.(b) By direct
How do the normal and tangential components of the Poynting vector in matter behave at an interface between two simple media where no free current flows? Comment on the physical meaning of your
A flat and infinitely large sheet with uniform charge density σ moves with constant speed υ in a direction parallel to its surface. Confirm the differential form of Poynting’s theorem at every
A particle with charge q and mass m moves in static external fields E0(r) and B0(r). If ϕ0(rq) is the electrostatic potential at the position of the particle, and there is negligible radiation, the
A cable is made from two coaxial cylindrical shells. The outer shell has radius b and charge per unit length λ, and carries a longitudinal current I. The inner cylinder has radius a < b and
PEM in the Coulomb Gauge The Helmholtz theorem guarantees that any vector can be decomposed in the form v = v⊥ + vΙΙ where ∇ · v⊥ = 0 and ∇ × vΙΙ = 0. Use this fact,
An infinite cylindrical solenoid of radius R is wound tightly with n turns per unit length of a wire that carries a current I. A bead of mass m and charge q slides freely on a non-conducting circular
(a) Show that the electromagnetic angular momentum for static fields in the Coulomb gauge is(b) Evaluate LEM when a point electric dipole p sits at the origin in the presence of a static
(a) Show that the electromagnetic linear momentum PEM for static fields in all of space can be rewritten in terms of the current density j(r) and the electrostatic potential ϕ(r).(b) Point charges q
The figure belowshows a cutaway viewof an infinite cylindrical solenoid with radius R that creates a magnetic field B = B inside itself. An infinitely long cylinder of insulating material with radius
A point charge at rest sits in the magnetostatic field of a stationary permanent magnet with spin magnetization M(r).(a) Show that the field momentum is(b) Is there “hidden momentum” associated
Compute PEM for a point electric dipole with moment p located at the center of a hollow spherical shell (radius R) with uniform surface charge density σ which rotates at frequency ω. Do not assume
A point charge q sits at the origin. A magnetic field B(r) = B(x, y)ẑ fills all of space. Show that the field angular momentum is LEM = −(q2π) ϕB ẑ, where ϕB is the flux of B through
The text showed that the Coulomb-Lorentz force on a classical atom or molecule (a bounded collection of net neutral charges moving with center-of-mass velocity v) due to its electric dipole moment
Let f (ξ) be an arbitrary scalar function of the scalar variable ξ. We have learned that f (z − ct) is a traveling-wave solution of the one dimensional wave equation. In other words,We have also
Consider transverse plane waves in free space where E = E(z, t) and B = B(z, t).(a) Demand that the Poynting vector S = 0 and show that all such solutions must satisfy(b) The constancy of uEM implies
Let E(r, t) be a vector field that satisfiesUse the Helmholtz theorem to write a formula for B(r, t) in terms of E(r, t). Confirm that B(r, t) and E(r, t) satisfy all four Maxwell equations in free
(a) Let E = E0 cos(kz − ωt)x̂ + E0 cos(kz + ωt)x̂. Write E(z, t) in simpler form and find the associated magnetic field B(z, t).(b) For the fields in part (a), find the instantaneous and
(a) Let A(x) be a vector function of a scalar argument. Find the conditions that make A(k · r − ckt) a legitimate Coulomb gauge vector potential in empty space. Compute E(r, t) and B(r, t)
Let E = ŷ E0 exp[i(hz − ωt) − κx] be the electric field of a wave propagating in vacuum.(a) How are the real parameters h, κ, and ω related to one another?(b) Find the associated
(a) Show that the angular momentum of an electromagnetic field in empty space (no sources) can be written in the formNote any requirements that the fields must satisfy at infinity. The last term is
The diagram shows two electromagnetic beams intersecting at right angles. (EH, BH) propagates in the +x-direction. (EV,BV) propagates in the +y-direction. For simplicity, each beam is taken as a pure
Let ƒ(x) and ƒ̂(k) be Fourier transforms pairs soThe averages of the operator O with respect to the distributions f(x) and ƒ̂(k) are(a) Show that ĥ(k) = ik ƒ̂(k) if h(x) = df/dx.(b) Prove
It is difficult to visualize time-dependent electromagnetic fields in three dimensional space. However, for fields that propagate in the z-direction, it is interesting to focus on the locus of points
Let ε(t ) = Re E(0, t) = a1cos(ωt − δ1)ê1 + a2 cos(ωt − δ2)ê2. Except for the case of linear polarization, this vector sweeps out an ellipse as a function of time. Show
Let E = E1 + E2 be the electric field of the sum of two monochromatic plane waves propagating in the z-direction. One wave has frequency ω1 and is elliptically polarized. The other wave
The electric field of a plane wave is E(z, t) = x̂Acos (kz − ωt + δ1) + ŷB cos(kz − ωt + δ2). Show that the principal axes of the polarization ellipse for this field are rotated from the
Prove that any two antipodes on the Poincar´e sphere correspond to orthogonal states of polarization.
A(r, t) = (ax̂ + ibŷ)A0(ζ) exp(iζ) where ζ = k(z + ct) is a vector potential wave packet.(a) Find the (real) electric and magnetic fields in the approximation that the envelope function A(ζ)
The angular velocity ω(t)ẑ of the cylindrical shell shown below increases from zero and smoothly approaches the steady value ω0. The shell has infinitesimal thickness and carries a uniform charge
Show that the inequality UEM ≥ c |PEM| is a general property of electric and magnetic fields. Under what conditions are the two quantities equal?
The source of a magnetic field B0 is a steady current produced by a collection of moving charges confined to a volume V. These particles are exposed to an external electric field E0 produced by a
A superconductor has the property that its interior has B = 0 under all conditions. Let a sphere (radius R) of this kind sit in a uniform magnetic field B0.(a) Place a fictitious point magnetic
If f is a scalar function of one scalar variable, we know that f (z ± ct) are solutions of the one-dimensional wave equation.(a) Show that f (r ± ct) / r are solutions of the three-dimensional wave
Use a normalization slightly different from the text and write the complex electric field of an electromagnetic wave packet as(a) Show that the total linear momentum of the wave packet satisfies(b)
The scalar wave equation is (a) Find a general solution to this equation by separating variables in the form u(r)T (t).(b) Use Cartesian variables to separate the equation satisfied by u(r). Write
The functionis a beam-like solution to the scalar wave equation (in the paraxial approximation) whereThe Gouy phase, α(z), arises from the fact that the beam has a finite size in the transverse
Let u(x, y, z, t ) be a monochromatic, Gaussian beam solution of the scalar wave equation in the paraxial approximation. The beam propagates in the z-direction. Use a suitably chosen Lorenz gauge
Let A(ρ, z, t) = U(ρ, z, t) ẑ be a Lorenz gauge vector potential where U(ρ, z, t) = u(ρ, z) exp(−iωt) is an azimuthally symmetric, beam-like solution of the scalar wave equation which
(a) Superpose two scalar waves: a plane wave u1 exp(ikx) and a spherical wave (u2/r) exp(ik · r + δ). Show that the locus of points where constructive interference occurs defines a family of
Consider the outgoing spherical wave solutions of the threedimensional scalar wave equation in vacuum. Show that the phase velocity of the = 0 wave is c but that the phase velocity of the = 1 wave is
(a) Separate variables in cylindrical coordinates and find a general time-harmonic solution ψ(ρ,φ, z, t) of the scalar wave equation which propagates in the z-direction and remains finite on the z
A particle with charge q and mass m interacts with a circularly polarized plane wave in vacuum. The electric field of the wave is E(z, t) = Re {( + i ŷ)E0 exp[i(kz − ωt)]}.(a)Let v± = νx ±

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