Questions and Answers of Modern Electrodynamics

The half-space z > 0 has uniform magnetization M = −Mẑ. The half-space z < 0 has uniform magnetization M = + Mẑ. Find the magnetic field B at every point in space using (a) The
The correct form of the interaction energy between two current carrying circuits was much debated in the 1870s by Maxwell and the German theoretical physicists Carl Neumann, Wilhelm Weber, and
The figure shows two square, current-carrying loops with side length a and center-to-center separation c. The currents I1 and I2 circulate in the same direction.(a) Evaluate Neumann’s integral and
A straight wire carries a current I1 down the symmetry axis of a toroidal solenoid with a rectangular cross section of area ab. The solenoid has inner radius R and is composed of N turns of a wire
A circular loop of radius R carries a current I1. A straight wire with a current I2 in the plane of the loop passes a distance d from the loop center.(a) Use an energy method to show that the force
Consider a collection of current loops whereUsing the notation of the text for variables that are to be held constant in partial derivatives, prove that the magnetic force exerted on the ith loop
(a) Use an energy method to determine how much current must be passed through the coil if it is to remain unstretched when a mass m is hung from its bottom end. Neglect stray fields at the ends of
A filamentary wire carries a current along the positive z-axis in the presence of a constant magnetic field B = B0 x̂ + B'ẑ.(a) Find the straight line in space along which the magnitude of the
Two infinite, straight, parallel wires, each carrying current I in the same direction, are coincident with the lines (1, 0, z) and (−1, 0, z). In addition, there is a large external field B0 = B0
The LevitronTM is a toy in which a spinning magnetic top floats stably in air above a magnetized base. The potential energy of this system is V̂B(r) = Mgz − m · B(r) if we model the top as a
A superconducting sphere (radius R) placed in a uniform magnetic field B spontaneously generates currents on its surface which produce a dipole magnetic field. The field is equivalent to that
Two identical point dipoles m = mẑ sit rigidly at (±a, 0, 0). A third point dipole M is free to rotate at its fixed position (0, y, z). Find the Cartesian components of M which correspond to
The diagram shows three small magnetic dipoles at the vertices of an equilateral triangle. Moments mB and mC point permanently along the internal angle bisectors. Moment mA is free to rotate in the
Two point dipoles m1 and m2 on the x-axis are separated by a distance R and misaligned from the positive x-axis by small angles α and β as shown below. A uniform agnetic field B points along the
The chemical diagnostic tool of nuclear magnetic resonance uses a static magnetic field B0 = B0ẑ and a small-amplitude radio-frequency magnetic field B1(t) to orient and manipulate nuclear spins
Let α(r) be an arbitrary scalar function. A magnetic field which satisfies ∇ × B = αB is called force-free because the Lorentz force density j(r) × B(r) vanishes everywhere. There is some
Two identical, current-carrying rectangular loops are oriented at right angles, one in the vertical x-y plane, one in the horizontal x-z plane. The horizontal loop moves infinitesimally slowly from z
Two origin-centered circular rings have radii a and b a and Ib. A narrow insulating rod coincident with their common diameter permits the smaller ring to rotate freely inside the larger ring. Show
Two finite-length, concentric, cylindrical solenoids carry current in the same direction. The outer solenoid is very slightly longer and has a very slightly larger radius than the inner solenoid, but
Two long, parallel wires of length L, separation d, and cross sectional radius a are connected by a U-turn at each end to form a closed circuit. Insert a battery with potential difference V at one
Let a steady current I flow up the y-axis and let the initial position and velocity of a particle with mass m and charge q be r0 = (x0, 0, 0) and v0 = (0, ν0, 0).(a) Show that the motion of the
An axial electric field E = Eẑ and a radial magnetic field B = Bp̂ coexist in the volume V between two short cylindrical shells concentric with the z-axis. Suppose V is filled with xenon gas and,
A particle with mass m and charge q moves non-relativistically in static fields E(r) and B(r). Show that a re-scaling of the magnetic field and the time is sufficient for a particle with mass M and
Roll up an ohmic sheet to form an infinitely long, origin-centered cylinder of radius a. Cut a narrow slot along the length of the cylinder and insert a line source of EMF so the electrostatic
Show that a current density with vector potential A(r) = f (r)r has zero magnetic dipole moment.
Show that the formula for the magnetic dipole moment derived in Example 11.1,is consistent with the spherical multipole expansion of the vector potential derived in Section 11.4, Example 11.1 Let
If such a thing existed, the magnetic field of a point particle with magnetic charge g at rest at the origin would be Bmono(r) = (μ0gr/(4πr3). Show that the magnetic field of a point magnetic
A filamentary current loop traverses eight edges of a cube with side length 2b as shown below.(a) Find the magnetic dipole moment m of this structure.(b) Do you expect a negligible or a
It is true (but not obvious) which any vector field V(r) which satisfies ∇ · V(r) = 0 can be written uniquely in the formwhere L = −ir × ∇ is the angular momentum operator and ψ(r) and γ
(a) Let I be the current carried by a wire bent into a planar loop. Place the origin of coordinates at an observation point P in the plane of the loop. Show that the magnitude of the magnetic field
A voltage difference V0 causes a steady current to flow from the top conductor to the bottom conductor (in the sketch below) through an ohmic medium with conductivity σ. Find an approximate
Use Green’s second identity to prove that GD(r, r') = GD(r', r).
The text writes two expressions for the magnetic field of point magnetic dipole at the origin:ψ0(r) is the magnetic scalar potential of a point dipole. Prove that the delta function content of these
Find the magnetic moment of a planar spiral with inner radius a and outer radius b composed of N turns of a filamentary wire that carries a steady current I. a b
The text produced a spherical multipole expansion for the magnetic scalar potential ψ(r) based on the identityA Cartesian expansion for the scalar potential can be developed from the same starting
A charge Q is uniformly distributed over the surface of a sphere of radius R. The sphere spins at a constant angular frequency ω.(a) Show that the current density of this configuration can be
Show that the first non-zero term in an interior Cartesian multipole expansion of the vector potential can be written in the form A(r) = (μ0/4π)G × r where G is a constant vector. Show that the
A semi-infinite solenoid (conce ntric with the negative z-axis) has cross sectional area A = πR2, n turns per unit length of a wire with current I , and magnetic moment per unit length m = nIA. When
(a) Find the vector potential inside and outside a solenoid that generates a magnetic field B = Bẑ inside an infinite cylinder of radius R. Work in the Coulomb gauge. (b) The A haronov-Bohm
Let j (r) be an arbitrary current distribution.(a) Show that the components of the magnetic dipole moment m = 1/ 2 ∫ d3 r r × j are invariant to a rigid shift of the origin of coordinates.(b) Show
A current distribution produces the vector potentialWhat is the magnetic moment associated with this current distribution? 0 A(r, 0, 0) = o Ao sin $ - exp(-λr). 4π r
A quantum particle with charge q, mass m, and momentum p in a magnetic field B(r) = ∇ × A(r) has velocity ν(r) = p/m − (q/m)A(r). This means that a charge distribution ρ(r) generates a
A compact disk with radius R and uniform surface charge density σ rotates with angular speed ω. Find the magnetic dipole moment m when the axis of rotation is(a) The symmetry axis of the disk;(b)
(a) Show by direct calculation that the Coulomb gauge condition ∇ · A = 0 applies to(b) Find the choice of gauge where a valid representation of the vector potential is A(r): Ho 4π d³r' j(r) |rr|
A cylindrical solenoid with length L and cross sectional area A = πR2 is formed by wrapping n turns per unit length of a wire that carries a current I . Estimate the magnitude of the magnetic field
The text describes a Helmholtz coil as two parallel, coaxial, and circular current loops of radius R separated by a distance R. Each loop carries a current I in the same direction.(a) Use the
A current I0 flows up the z-axis from z = z1 to z = z2 as shown below.(a) Use the Biot-Savart law to show that the magnetic field in the z = 0 plane is(b) Symmetry and the Coulomb gauge vector
Consider a charge distribution ρ(r) in rigid, uniform motion with velocity υ.(a) Show that the magnetic field produced by this system is B(r) = (ν/c2) × E, where E(r) is the electric field
The magnetic scalar potential in a volume V is ψ(x, y, z) = (C/2) ln(x2 + y2). Find a vector potential A = Ax x̂ + Ay ŷ which produces the same magnetic field.
Biot and Savart derived their eponymous formula using a currentcarrying wire bent as shown below. Find B(r) in the plane of the wire at a distance d from the bend along the axis of symmetry.
Show that the normal derivative of the Coulomb gauge vector potential suffers a jump discontinuity at a surface endowed with a current density K(rS).
Use the solid angle representation of the magnetic scalar potential ψ(r) to find B(r) everywhere for an infinite, straight line of current I . State carefully the surface you have chosen to
Let B(x, z) be the magnetic field produced by a surface current density K(y, z) = K(z) ŷ confined to the x = x0 plane.(a) Show that the Biot-Savart law for this situation reduces to a one
The figure below shows an infinitely long current filament wound in the form of a circular helix with radius R and pitch l , i.e., is the distance along the z-axis occupied by one wind of the helix.
A charge Q is uniformly distributed over the surface of a sphere of radius R. The sphere spins at a constant angular frequency with ω = ωẑ. Use B = −∇ψ to find the magnetic field everywhere.
Find the surface current density K(θ,φ) on the surface of sphere of radius a which will produce a magnetic field inside the sphere of B<(x, y, z) = (B0 a)(xx̂ − yŷ). Express your answer in
A circular loop with radius R and current I lies in the x-y plane centered on the z-axis. The magnetic field on the symmetry axis is R² B(z) = μ₁¹ (R² + z²)³/2². z In cylindrical
(a) Use superposition and the magnetic field on the symmetry axis of a current ring to find the magnetic field at the midpoint of the symmetry axis of a cylindrical solenoid. The solenoid has radius
The diagram below shows that the resistance between the terminals A and B is determined by a motif of three resistors R2 − R1 − R2 repeated an infinite number of times. Determine the ratio
A current I starts at z = −∞and flows up the z-axis as a linear filament until its hits an origin-centered sphere of radius R. The current spreads out uniformly over the surface of the sphere and
The z-axis coincides with the symmetry axis of a flat disk of radius R in the x-y plane. Sketch and justify in words the pattern of currents that must flow in the disk to produce the magnetic field
The conductivity of the Earth’s atmosphere increases with height due to ionization by solar radiation. At an altitude of about H = 50 km, the atmosphere can be considered practically an ideal
(a) Use the Biot-Savart law to find B(r) everywhere for a current sheet at x = 0 with K = Kẑ.(b) Check your answer to part (a) by superposing the magnetic field from an infinite number of straight
(a) Two rings of radius R, coaxial with the z-axis, are separated by a distance 2b and carry a current I in the same direction. Make explicit use of the formula for the magnetic field of a single
(a) Consider a semi-infinite and tightlywound solenoid with a circular cross section. Prove that the magnetic flux which passes out through the open end of the solenoid is exactly one-half the flux
A uniform surface current K = Kẑ confined to a strip of width b carries a total current I . Find the magnetic field at a point in the plane of the strip that lies a perpendicular distance a from
Current flows on the surface of a spherical shell with radius R and conductivity σ. The potential is specified on two rings as ϕ(θ = α) = V cos nφ and ϕ(θ = π − α) = −V cos nφ. Show
A spherical shell with radius a has conductivity σ in the angular range α1 < θ < π − α2. Otherwise, the shell is perfectly conducting and a potential difference V is maintained between
A potential difference V drives a steady current I through an ohmic wire with conductivity σ and a constant circular cross section. One portion of the wire has the shape of a circular arc with inner
The diagram below illustrates a reciprocal principle satisfied by an ohmic sample of any shape.The principle asserts that if the impressed currents satisfy IA = IB, the measured voltages satisfy
The diagram below shows an ohmic film with conductivity σ, thickness d, infinite length, and semi-infinite width. A total current I enters the film at the point A through a line contact (modeled as
The annulus shown below is cut from a planar metal sheet with thickness t and conductivity σ.(a) Let V be the voltage between the edge CD and the edge FA. Solve Laplace’s equation to find the
A square plate of copper metal can be used as a crude variable resistor by making suitable choices of the places to attach leads that carry current to and from the plate.(a) Current enters at A and
Steady current flows in the x-direction in an infinite, two-dimensional strip defined by |y| < L. The current density j is constant everywhere in the strip and the conductivity varies in space
A current I flows up the z-axis and is intercepted by an origincentered sphere with radius R and conductivity σ. The current enters and exits the sphere through small conducting electrodes which
Show that the lines of current density j obey a “law of refraction” at the flat boundary between two ohmic media with conductivities σ1 and σ2. Use the geometry shown below. 0₁ 02 0₁ 10₂
The electrodes of a spherical capacitor have radii a and b > a. The inner electrode is grounded; the outer electrode is held at potential V. In vacuum diode mode, the thermionic current which
The diagram shows a wire connected to the Earth (conductivity σE) through a perfectly conducting sphere of radius a which is half-buried in the Earth. The layer of earth immediately adjacent to the
Consider the vacuum diode problem treated in the text with the space between the plates filled with a poor conductor with dielectric permittivity ϵ. For matter of this kind, v = ũE, where the
An infinite, two-dimensional network has a honeycomb structure with one hexagon edge removed. Otherwise, the resistance of every hexagon edge is r. Find the resistance of the network when a current I
A battery maintains a potential difference V between the two halves of the cover of a tank (L×∞×h) filled with salty water. Find the current density j(x, y, z) induced in the water. N -L/2
A thin membrane with conductivity σ and thickness δ separates two regions with conductivity σ.Assume uniform current flow in the z-direction in the figure above. When δ is small, it makes sense
Two highly conducting spheres with radii a1 and a2 are used to inject and extract current from points deep inside a tank of weakly conducting fluid. Show that the resistance between the spheres
(a) Derive an integral expression for the charge density σ(φ, z) induced on the outer surface of a conducting tube of radius R when a point charge q is placed at a perpendicular distance s > R
(a) A long straight rod with cross sectional area A and conductivity σ accelerates parallel to its length with acceleration a. Write down the Drude-like equation of motion for the average velocity v
The Dirichlet Green function for any finite volume V can always be written in the form(a) Use the physical meaning of the Dirichlet Green function to prove that(b) Use Earnshaw’s theorem to prove
A wire with conductivity σ carries a steady current I. Confirm the statement made in the text that a charge Q ∼ = ϵ0 I/σ accumulates on the wire’s surface in the immediate neighborhood of
(a) Use the completeness relation,and the method of direct integration to show that(b) Show that G(r, r') above is identical to the image solution for this problem. ΣYEM (F)YM (P)
Let b be the perpendicular distance between an infinite line with uniform charge per unit length λ and the center of an infinite conducting cylinder with radius R = b/2.(a) Show that the charge
An infinitely long cylindrical conductor carries a constant current with density jz(r).(a) Despite Ohm’s law, compute the radial electric field Er (r) that ensures that the radial component of the
A steady current is produced by a collection of moving charges confined to a volume V . Prove that the rate at which work is done on these moving charges by the electric field produced by a static
(a) Use completeness relations to represent δ(x − x')δ(y − y') and then the method of direct integration for the inhomogeneous differential equation which remains to find the interior Dirichlet
(a) Let Φ(ρ,Ф) be a solution of Laplace’s equation in a cylindrical region ρ < R. Show that the function Ψ(ρ,φ) = Φ(R2/ρ,φ) is a solution of Laplace’s equation in the region ρ >
In 1910, Debye suggested that the work function W of a metal could be computed as the work performed against the electrostatic image force when an electron is removed from the interior of a finite
The text showed that the attractive force F between an origin-centered, grounded, conducting sphere of radius R and a point charge located at a point s > R on the positive z-axis varies as 1/s3
Two semi-infinite and grounded conducting planes meet at a right angle as seen edge-on in the diagram. Find the charge induced on each plane when a point charge Q is introduced as shown. 20
The diagram below shows a rod of length L and net charge Q (distributed uniformly over its length) oriented parallel to a grounded infinite conducting plane at the distance d from the plane.(a)
An infinite slab with dielectric constant κ = ε/ε0 lies between z = a and z = b = a + c. A point charge q sits at the origin of coordinates. Let β = (κ − 1)/(κ + 1) and use solutions of
Suppose that a collection of image point charges q1, q2, . . . , qN is used to find the force on a point charge q at position rq due to the presence of a conductor held at potential ϕC. Let UA be
A point charge q is placed at a distance 2R from the center of an isolated, conducting sphere of radius R. The force on q is observed to be zero at this position. Now move the charge to a distance 3R

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