Questions and Answers of Modern Electrodynamics

Positive charges Q and Q' are placed on opposite sides of a grounded sphere of radius R at distances of 2R and 4R, respectively, from the sphere center. Show that Q' is repelled from the sphere if Q'
A point electric dipole with moment p sits at the center of a grounded, conducting, spherical shell of radius R. Use the method of images to show that the electric field inside the shell is the sum
The plane z = 0 is grounded except for an finite area S0 which is held at potential ϕ0. Show that the electrostatic potential away from the plane is p(x, y, z) = Polz| 2π So d²r' r-r'³*
Maintain the plane z = 0 at potential V and introduce a grounded conductor somewhere into the space z > 0. Use the “magic rule” for the Dirichlet Green function to find the charge density
Find the free-space Green functionG(d)0 (r, r') in d = 1, 2, 3 space dimensions by the method of eigenfunction expansion. For d = 2, you will need (i) an integral representation of J0(x); (ii) the
The free-space Green function in two dimensions (potential of a line charge) is:Use the method of direct integration to reduce the two-dimensional equationto a one-dimensional equation and establish
An empty beer can is bounded by the surfaces z = 0, z = h, and ρ = R. By slamming it against his forehead, a frustrated football fan dents the can into the shape shown below. Our interest is the
Write δ(r − r') = δ(r⊥ − r'⊥)δ(z − z') and use direct integration to derive Weyl’s formula for the free-space Green function in three dimensions, Go(r, r')
A cosmic string is a one-dimensional object with an extraordinarily large linear mass density (μ ∼ 1022 kg/m) which (in some theories) formed during the initial cool-down of the Universe after the
Show that can be used as the complex potential for an array of equally spaced, parallel, charged lines in the y = 0 plane. Let n be an integer and let x = na and x = (n + 1/2 )a be the positions of
Let ˆn be the normal to an equipotential surface at a point P. The principal radii of curvature of the surface at P are R1 and R2. A formula due to George Green relates normal derivatives (∂/∂n
The Poisson integral formulagives the potential at any point r inside a sphere if we specify the potential ¯ϕ (rS) at every point on the surface of the sphere. Derive this formula by summing the
The z-axis runs down the center of an infinitely long heating duct with a square cross section. For a real metal duct (not a perfect conductor), the electrostatic potential φ(x, y) varies linearly
Four identical positive point charges sit at (a, a), (−a, a), (−a,−a), and (a,−a) in the z = 0 plane. Very near the origin, the electrostatic potential can be written in the form(a) Deduce
Use the orthogonality properties of the spherical harmonics to prove the following identities for a function φ(r) which satisfies Laplace’s equation in and on an origin-centered spherical surface
The parallel plates of a microchannel plate electron multiplier are segmented into conducting strips of width b so the potential can be fixed on the strips at staggered values. We model this using
The square region defined by−a ≤ x ≤ a and−a ≤ y ≤ a in the z = 0 plane is a conductor held at potential φ = V . The rest of the z = 0 plane is a conductor held at potential φ = 0. The
Two flat conductor plates (infinite in the x- and y-directions) occupy the planes z = ±d. The x > 0 portion of both plates is held at φ = +φ0. The x < 0 portion of both plates is held at φ
Confirm Poisson formula (derived in Section 6.3) for the case when the volume V is a rectangular slab which is infinite in the x and y directions and occupies the interval −t ≤ z ≤ t otherwise.
The figure shows an infinitely long and deep slot formed by two grounded conductor plates at x = 0 and x = a and a conductor plate at z = 0 held at a potential φ0. Find the potential inside the slot
A capacitor is formed by the infinite grounded plane z = 0 and an infinite, solid, conducting cone with interior angle π/4 held at potential V. A tiny insulating spot at the cone vertex (the origin
Find the volume charge density ρ and surface charge density σ which must be placed in and on a sphere of radius R to produce a field inside the sphere of: E = • 1³8 + 1/8 (1² - 138 - 1/0
A spherical conducting shell centered at the origin has radius R1 and is maintained at potential V1. A second spherical conducting shell maintained at potential V2 has radius R2 > R1 but is
The figure below shows an infinitely long cylindrical shell from which a finite angular range has been removed. Let the shell be a conductor raised to a potential corresponding to a charge per unit
Two semi-infinite, hollow cylinders of radius R are coaxial with the z-axis. Apart from an insulating ring of thickness d → 0, the two cylinders abut one another at z = 0 and are held at potentials
A conducting sphere with radius R and charge Q sits at the origin of coordinates. The space outside the sphere above the z = 0 plane has dielectric constant κ1. The space outside the sphere below
A set of known constants αn parameterizes the potential in a volume r < a asLet ˆz point along θ = 0 and insert a solid conducting sphere of radius R < a at the origin. Show that the force
A spherical shell of radius R is divided into three conducting segments by two very thin air gaps located at latitudes θ0 and π − θ0. The center segment is grounded. The upper and lower segments
The figure below is a cross section of an infinite, conducting cylindrical shell. Two infinitesimally thin strips of insulating material divide the cylinder into two segments. One segment is held at
Let the z-axis be the symmetry axis for an infinite number of identical rings, each with charge Q and radius R. There is one ring in each of the planes z = 0, z = ±b, z = ±2b, etc. Exploit the
Let V (z) be the potential on the axis of an axially symmetric electrostatic potential in vacuum. Show that the potential at any point in space is V(p, z) = 1 π Л fd² 0 de V(z+ip cos ().
Consider a parallel-plate capacitor with circular plates of radius a separated by a distance 2L.A paper published in 1983 proposed a solution for the potential for this situation of the formWhere
Two wedge-shaped dielectrics meet along the ray φ = 0. The opposite edge of each wedge is held at a fixed potential by a metal plate. The system is invariant to translations perpendicular to the
The x > 0 half of a conducting plane at z = 0 is held at zero potential. The x < 0 half of the plane is held at potential V . A tiny gap at x = 0 prevents electrical contact between the two
Give a physical realization of the electrostatic boundary value problem whose solution is provided by the complex potential. f(w) = i V₁ + V₂ 2 + V₁ - V₂ 2 R+iw In [R+ R-iw
An infinitely long conducting cylinder (radius a) oriented along the z-axis is exposed to a uniform electric field E0 ˆy.(a) Consider the conformal map g(w) = w + a2/w, where g = u + iv and w = x +
A spherical charge distribution ρ1(r) has total charge Q1 and a second, non-overlapping spherical charge distribution ρ2(r) has total charge Q2. The distance between the centers of the two
A model hydrogen atom is composed of a point nucleus with charge +|e| and an electron charge distributionShow that the ionization energy (the energy to remove the electronic charge and disperse it to
This problem exploits the ring and disk electric fields calculated Example 2.1.(a) Find E(r) inside and outside a uniformly charged spherical shell by superposing the electric fields produced by a
A surprisingly realistic microscopic model for the charge density of a semi-infinite metal (with z = 0 as its macroscopic surface) consists of a positive charge distributionAnd a negative charge
The Potential of a Charged Line Segment The line segment from P to P' in the diagram below carries a uniform charge per unit length λ. The vector a is coincident with the segment. The vectors c and
Identical point electric dipoles are placed at the vertices of the regular polyhedra shown below. All the dipoles are parallel but the direction they point is arbitrary. Show that the electric field
Two coplanar dipoles are oriented as shown in the figure below.Find the equilibrium value of the angle θ' if the angle θ is fixed. P Zo 0 P '0'
Find the electric dipole moment of:(a) A ring with charge per unit length λ = λ0 cos ϕ where ϕ is the angular variable in cylindrical coordinates.(b) A sphere with charge per unit areas σ = σ0
(a) Show that the potential due to a double-layer surface S with a dipole density τ (rS ) ˆn is where dΩ is the differential element of solid angle as viewed from r.(b) Use this result to derive
The text used Poisson’s equation to show that the charge density of a point electric dipole with moment p located at the point r0 is ρD(r) = −p · ∇δ(r − r0).(a) Derive the given formula
Use the electric stress tensor formalism to prove that no isolated charge distribution ρ(r) can exert a net force on itself. Distinguish the cases when ρ(r) has a net charge and when it does not.
The diagram shows two identical, charge-neutral, origin centered disks. One disk lies in the x-z plane. The other is tipped away from the first by an α angle around the z-axis. The charge density of
Place a point electric dipole p = p ˆz at the origin and release a point charge q (initially at rest) from the point (x0, y0, 0) in the x-y plane away from the origin. Show that the particle moves
Show that dW = −E(r) · dp is the work increment required to assemble a point electric dipole with moment dp at the point r beginning with charge dispersed at infinity.
A soap bubble (an insulating, spherical shell of radius R) is uniformly coated with polar molecules so that a dipole double layer with τ = τ ˆr forms on its surface. Find the potential at every
The z-axis is the symmetry axis for an origin-centered ring with charge Q and radius a which lies in the x-y plane. A coplanar and concentric ring with radius b > a has charge −Q. Calculate the
Molecules adsorbed on the surface of a solid crystal surface at low temperature typically arrange themselves into a periodic arrangement, e.g., one molecule lies at the center of each a × a square
The low-energy Born approximation to the amplitude for electron scattering from a neutron is proportional to the volume integral of the potential energy of interaction between the electron and the
Find the primitive, Cartesian monopole, dipole, and quadrupole moments for each of the following charge distributions. Use the geometrical center of each as the origin.(a) Two charges +q at two
The z-axis is the symmetry axis of a disk of radius R which lies in the x-y plane and carries a uniform charge per unit area σ. Let Q be the total charge on the disk.(a) Evaluate the exterior
(a) Place two charges +q at two diagonal corners of a square (±a,±a, 0) and two minus charges −q at the two other diagonals of the square (±a,∓a, 0). Evaluate the primitive quadrupole moment
(a) Let ϕ(R, θ, φ) be specified values of the electrostatic potential on the surface of a sphere. Show that the general form of an exterior, spherical multipole expansion implies thatb) The eight
(a) Show that the charge density of a point quadrupole is ρ(r) = Qij ∇i ∇j δ(r − r0).(b) Show that the force on a point quadrupole in a field E(r) is Qij ∂i∂jE(r0).(c) Show that the
Six point charges form an ideal hexagon in the z = 0 plane as shown below. The absolute values of the charges are the same, but the signs of any two adjacent charges are opposite.(a) What is the
How does the leading contribution to the electrostatic interaction energy between two nitrogen molecules depend on the distance R between them?
A charge distribution is contained entirely inside a black box. Measurements of the electrostatic potential outside the box reveal that all of the exterior multipole moments for ℓ = 1, 2, . . . are
(a) Evaluate the exterior spherical multipole moments for a shell of radius R which carries a surface charge density σ(θ, φ) = σ0 sin θ cos φ.(b) Write φ(r < R,θ,ϕ) in the form φ(x, y,
An asymptotic (long-distance) electrostatic potential has the form(a) Use a traceless Cartesian multipole expansion to show that no localized charge distribution exists which can produce an
Let V be a charge-free volume of space. Use an interior spherical multipole expansion to show that the average value of the electrostatic potential φ(r) over the surface of any spherical sub-volume
Two infinite conducting planes are held at zero potential at z = −d and z = d. An infinite sheet with uniform charge per unit area σ is interposed between them at an arbitrary point.(a) Find the
A solid conductor has a vacuum cavity of arbitrary shape scooped out of its interior. Use Earnshaw’s theorem to prove that E = 0 inside the cavity.
A spherical conducting shell with radius b is concentric with and encloses a conducting ball with radius a. Compute the capacitance C = Q/Δφ when(a) the shell is grounded and the ball has charge
A point charge q lies a distancer > R from the center of an uncharged, conducting sphere of radius R. Express the induced surface charge density in the formwhere θ is the polar angle measured
A research paper published in the journal Applied Physics Letters describes experiments performed with three identical spherical conductors suspended from above by insulating wires so a (fictitious)
A metal ball with radius R1 has charge Q. A second metal ball with radius R2 has zero charge. Now connect the balls together using a fine conducting wire. Assume that the balls are separated by a
A capacitor is formed from three very long, concentric, conducting, cylindrical shells with radii a < b < c. Find the capacitance per unit length of this structure if a fine wire connects the
Three concentric spherical metallic shells with radiic > b > a have charges ec, eb, and ea , respectively. Find the change in potential of the outermost shell when the innermost shell is
The text derived Green’s reciprocity theorem for a set of conductors as a special case of a more general result. For conductors with charges and potentials (qk, ϕK) and the same set of conductors
Four identical conducting balls are attached to insulating supports that sit on the floor as shown below. One ball has charge Q; its support is fixed in space. The other three balls are uncharged but
(a) What is the self-capacitance (in farads) of the Earth? How much energy is required to add one electron to the (neutral) Earth?(b) What is the self-capacitance (in farads) of a conducting
The square region defined by −a ≤ x ≤ a and −a ≤ y ≤ a in the plane z = 0 is a conductor held at potential ϕ = V. The rest of the plane z = 0 is a conductor held at potential ϕ = 0. The
A conducting disk of radius R held at potential V sits in the x-y plane centered on the z-axis.(a) Use the charge density for this system calculated in the text to find the potential everywhere on
Let C be the capacitance of capacitor formed from two identical, flat conductor plates separated by a distance d. The plates have area A and arbitrary shape. When d << √ A, we know that the
A grounded metal plate is partially inserted into a parallel-plate capacitor with potential difference ϕ2 − ϕ1 > 0 as shown in the diagram below. Find the elements of the capacitance matrix.
A point dipole p is placed at r = r0 outside a grounded conducting sphere of radius R. Use Green’s reciprocity (and a comparison system with zero volume charge density) to find the charge drawn up
Let d be the separation between two infinite, parallel, perfectly conducting plates. The lower surface of the upper plate has charge per unit area σ1 > 0. The upper surface of the lower plate has
A point electric dipole with moment p is placed at the center of a hollow spherical cavity scooped out of an infinite conducting medium.(a) Find the surface charge density induced on the surface of
Confirm the assertion made in the text that the inverse relation between the matrix of capacitive coefficients and the matrix of potential coefficients implies the equivalence of these two
A non-conducting square has a fixed surface charge distribution. Make a rectangle with the same area and total charge by cutting off a slice from one side of the square and gluing it onto an adjacent
Two pyramid-shaped conductors each carry a net charge Q.(a) Transfer charge δQ from pyramid 2 to pyramid 1. Derive a condition on the coefficients of potential Pij which guarantees that this
A long, straight wire has length L and a circular cross section with area πa2. Arrange two such wires so they are parallel and separated by a distance d. You may assume that L >> d >> a
A battery maintains the potential difference V between the spheres of a spherical capacitor with capacitance C. Move the center of the inner sphere away from the center of the outer sphere by an
(a) A spherical metal shell is charged to an electrostatic potential V . Cut this shell in half and pull the halves infinitesimally apart. Find the force with which one hemisphere of the shell repels
A conducting shell of radius R has total charge Q. If sawed in half, the two halves of the shell will fly apart. This can be prevented by placing a point charge Q' at the center of the shell.(a) What
Two spheres with radius R have uniform but equal and opposite charge densities ±ρ. The centers of the two spheres fail to coincide by an infinitesimal displacement vector δ. Show by direct
Find a polarization P(r) which produces a polarization charge density in the form of an origin-centered sphere with radius R and uniform volume charge density ρP.
Find the total electrostatic energy of a ball with radius R and uniform polarization P.
The polarization in all of space has the form P = PΘ(r − R)ˆr, where P and R are constants. Find the polarization charge density and the electric field everywhere.
A cube is polarized uniformly parallel to one of its edges. Show that the electric field at the center of the cube is E(0) = −P/3ε0. Compare with E(0) for a uniformly polarized sphere.
The electrostatic polarization inside an origin-centered sphere is P(r) = P(r). (a) Show that φ(r) outside the sphere is equal to the potential of a point electric dipole at the origin with a
A polarizable sphere of radius R is filled with free charge with uniform density ρc. The dielectric constant of the sphere is κ.(a) Find the polarization P(r).(b) Confirm explicitly that the total
An air-gap capacitor with parallel-plate area A discharges by the electrical breakdown of the air between its parallel plates (separation d) when the voltage between its plates exceeds V0. Lay a slab
A semiconductor with permittivity ε occupies the space z ≥ 0. One “dopes” such a semiconductor by implanting neutral, foreign atoms with uniform density ND in the near-surface region 0 ≤ z
(a) The entire volume between two concentric spherical shells is filled with a material with uniform polarization P. Find E(r) everywhere. (b) The entire volume inside a sphere of radius R is

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