Questions and Answers of Electrodynamics

Find the electric field at a height z above the center of a square sheet (side a) carrying a uniform surface charge σ. Check your result for the limiting cases a → ∞ and z >> a.
If the electric field in some region is given (in spherical coordinates) by the expression where A and B are constants, what is the charge density?
Find the net force that the southern hemisphere of a uniformly charged sphere exerts on the northern hemisphere. Express your answer in terms of the radius R and the total charge Q.
An inverted hemispherical bowl of radius R carries a uniform surface charge density σ. Find the potential difference between the "noah pole" and the center.
A sphere of radius R carries a charge density p (r) = kr (where k is a constant). Find the energy of the configuration. Check your answer by calculating it in at least two different ways.
The electric potential of some configuration is given by the expression where A and Î» are constants. Find the electric field E(r), the charge density p(r), and the total charge Q.
Two infinitely long wires running parallel to the x axis carry uniform charge densities + ? and ? ? (Fig. 2.54).? (a) Find the potential at any point (x, y, z), using the origin as your
In a vacuum diode, electrons are "boiled" off a hot cathode, at potential zero, and accelerated across a gap to the anode, which is held at positive potential V0. The cloud of moving electrons within
Problem 2.49 Imagine that new and extraordinarily precise measurements have revealed an error in Coulomb's law. The actual force of interaction between two point charges is found to be where Î» is
Suppose an electric field E(x, y, z) has the form Ex = ax, Ey = 0, Ez = 0 where a is a constant. What is the charge density? How do you account for the fact that the field points in a particular
All of electrostatics follows from the 1/r2 character of Coulomb's law, together with the principle of superposition. An analogous theory can therefore be constructed for Newton's law of universal
We know that the charge on a conductor goes to the surface, but just how it distributes itself there is not easy to determine. One famous example in which the surface charge density can be calculated
Find the average potential over a spherical surface of radius R due to a point charge q located inside (same as above, in other words, only with z
In one sentence, justify Earn Shaw’s Theorem: A charged particle cannot be held in a stable equilibrium by electrostatic forces alone. As an example, consider the cubical arrangement of fixed
Find the general solution to Laplace's equation in spherical coordinates, for the case where V depends only on r. Do the same for cylindrical coordinates, assuming V depends only on s.
Prove that the field is uniquely determined when the charge density p is given and either V or the normal derivative ∂V/∂n is specified on each boundary surface. Do not assume the boundaries are
A more elegant proof of the second uniqueness theorem uses Green's identity (Prob. 1.60c), with T = U = V3. Supply the details.
Find the force on the charge +q in Fig. 3.14. (The xy plane is a grounded conductor)
(a) Using the law of cosines, show that Eq. 3.17 can be written as follows where r and θ are the usual spherical polar coordinates, with the z axis along the line through q. In this form it is
In Ex. 3.2 we assumed that the conducting sphere was grounded (V = 0). But with the addition of a second image charge, the same basic model will handle the case of a sphere at any potential V 0
A uniform line charge λ is placed on an infinite straight wire, a distance d above a grounded conducting plane. (Let's say the wire runs parallel to the x-axis and directly above it, and the
Two semi-infinite grounded conducting planes meet at right angles. In the region between them, there is a point charge q, situated as shown in Fig. 3.15. Set up the image configuration, and calculate
Two long, straight coppek pipes, each of radius R, are held a distance 2d apart. One is at potential V0, the other at â€“ V0 (Fig. 3.16). Find the potential everywhere.
Find the potential in the infinite slot of Ex. 3.3 if the boundary at x = 0 consists of two metal strips: one, from y = 0 to y = a/2, is held at a constant potential V0, and the other, from y = a/2
For the infinite slot (Ex. 3.3) determine the charge density σ(y) on the strip at x = 0, assuming it is a conductor at constant potential V0.
A rectangular pipe, running parallel to the z-axis (from – ∞ to + ∞), has three grounded metal sides, at y = 0, y = a, and x = 0. The fourth side, at x = b, is maintained at a specified
A cubical box (sides of length a) consists of five metal plates, which are welded together and grounded (Fig. 3.23). The top is made of a separate sheet of metal, insulated from the others, and held
Derive P3 (x) from the Rodrigues formula, and check that P3 (cos θ) satisfies the angular equation (3.60) for l = 3. Check that P3 and P1 are orthogonal by explicit integration.
(a) Suppose the potential is a constant V0 over the surface of the sphere. Use the results of Ex. 3.6 and Ex. 3.7 to find the potential inside and outside the sphere. (Of course, you know the answers
The potential at the surface of a sphere (radius R) is given by V0 = k cos 3θ, where k is a constant. Find the potential inside and outside the sphere, as well as the surface charge density σ (θ)
Suppose the potential V0 (θ) at the surface of a sphere is specified, and there is no charge inside or outside the sphere. Show that the charge density on the sphere is given by where
Find the potential outside a charged metal sphere (charge Q, radius R) placed in an otherwise uniform electric field E 0. Explain clearly where you are setting the zero of potential.
In Prob. 2.25 you found the potential on the axis of a uniformly charged disk: (a) Use this, together with the fact that Pl(1) = 1, to evaluate the first three terms in the expansion (3.72) for the
A spherical shell of radius R carries a uniform surface charge σ0 on the "northern" hemisphere and a uniform surface charge – σ0 on the "southern" hemisphere. Find the potential inside and
Solve Laplace's equation by separation of variables in cylindrical coordinates, assuming there is no dependence on z (cylindrical symmetry). [Make sure you find all solutions to the radial equation;
Find the potential outside an infinitely long metal pipe, of radius R, placed at right angles to an otherwise uniform electric field E0. Find the surface charge induced on the pipe.
Charge density ?? (Ф) = a sin 5Ф (where a is a constant) is glued over the surface of an infinite cylinder of radius R (Fig. 3.25). Find the potential inside and outside the cylinder.
A sphere of radius R, centered at the origin, carries charge density where k is a constant, and r, ? are the usual spherical coordinates. Find the approximate potential for points on the z axis, far
Four particles (one of charge q, one of charge 3q, and two of charge – 2q) are placed as shown in Fig. 3.31, each a distance a from the origin. Find a simple approximate formula for the potential,
In Ex. 3.9 we derived the exact potential for a spherical shell of radius R, which carries a surface charge σ = k cos θ.(a) Calculate the dipole moment of this charge distribution.(b) Find the
For the dipole in Ex. 3.10, expand 1/π ± to order (d/r)3, and use this to determine the quadruple and octopole terms in the potential
Two point charges, 3q and -q, are separated by a distance a. For each of the arrangements in Fig. 3.35, find (i) the monopole moment, (ii) the dipole moment, and (iii) the approximate potential (in
A "pure" dipole p is situated at the origin, pointing in the z direction.(a) What is the force on a point charge q at (a, 0, 0) (Cartesian coordinates)?(b) What is the force on q at (0, 0, a)?(c) How
Three point charges are located as shown in Fig. 3.38, each a distance a from the origin. Find the approximate electric field at points far from the origin. Express your answer in spherical
Show that the electric field of a ("pure") dipole (Eq. 3.103) can be written in the coordinate-free form
A point charge q of mass m is released from rest at a distance d from an infinite grounded conducting plane. How long will it take for the charge to hit the plane?
Two infinite parallel grounded conducting planes are held a distance a apart. A point charge q is placed in the region between them, a distance x from one plate. Find the force on q. Check that your
Two long straight wires, carrying opposite uniform line charges ± λ, are situated on either side of a long conducting cylinder (Fig. 3.39). The cylinder (which carries no net charge) has radius R,
A conducting sphere of radius a, at potential V0, is surrounded by a thin concentric spherical shell of radius b, over which someone has glued a surface charge σ(θ) = k cosθ, where k is a
A charge + Q is distributed uniformly along the z axis from z = ? a to z = + a. Show that the electric potential at a point r is given by for r > a.
A long cylindrical shell of radius R carries a uniform surface charge σ0 on the upper half and an opposite charge σ0 on the lower half (Fig. 3.40). Find the electric potential inside and outside
A thin insulating rod, running from z = – a to z = + a, carries the indicated line charges. In each case, find the leading term in the multipole expansion of the potential:(a) λ = k
Show that the average field inside a sphere of radius R, due to all the charge within the sphere, is where p is the total dipole moment. There are several ways to prove this delightfully simple
(a) Using Eq. 3.103, calculate the average electric field of a dipole, over a spherical volume of radius R, centered at the origin. Do the angular intervals first. [Note: You must express r and ? in
(a) Suppose a charge distribution P1 (r) produces a potential V1 (r), and some other charge distribution p2(r) produces a potential V2(r). [The two situations may have nothing in common, for all I
Use Green's reciprocity theorem (Prob. 3.43) to solve the following two problems.(a) Both plates of a parallel-plate capacitor are grounded, and a point charge q is placed between them at a distance
(a) Show that the quadrupole term in the multi pole expansion can be written where here is the Kronecker delta, and Qij is the quadrupole moment of the charge distribution. Notice the hierarchy: The
In Ex. 3.8 we determined the electric field outside a spherical conductor (radius R) placed in a uniform external field E 0. Solve the problem now using the method of images, and check that your
For the infinite rectangular pipe in Ex. 3.4, suppose the potential on the bottom (y = 0) and the two sides (x = ± b) is zero, but the potential on the top (y = a) is a nonzero constant V 0. Find
(a) A long metal pipe of square cross-section (side a) is grounded on three sides, while the fourth (which is insulated from the rest) is maintained at constant potential V0. Find the net charge per
An ideal electric dipole is situated at the origin, and points in the z direction, as in Fig. 3.36. An electric charge is released from rest at a point in the xy plane. Show that it swings back and
A hydrogen atom (with the Bohr radius of half an angstrom) is situated between two metal plates 1 mm apart, which are connected to opposite terminals of a 500 V battery. What fraction of the atomic
According to quantum mechanics, the electron cloud for a hydrogen atom in the ground state has a charge density. where q is the charge of the electron and a is the Bohr radius. Find the atomic
According to Eq. 4.1, the induced dipole moment of an atom is proportional to the external field. This is a "rule of thumb," not a fundamental law, and it is easy to concoct exceptions––in
A point charge q is situated a large distance r from a neutral atom of polarizability a Find the force of attraction between them.
In Figure 4.6, P1 and P2 are (perfect) dipoles a distance r apart. What is the torque on P1 due to p2? What is the torque on P2 due to P1? [In each case I want the torque on the dipole about its own
(Perfect) dipole p is situated a distance z above an infinite grounded conducting plane (Fig. 4.7). The dipole makes an angle 0 with the perpendicular to the plane. Find the torque on p. If the
Show that the energy of an ideal dipole p in an electric field E is given by U = – p ∙ E. (4.6)
Show that the interaction energy of two dipoles separated by a displacement r is
A dipole p is a distance r from a point charge q, and oriented so that p makes an angle θ with the vector r from q to p.(a) What is the force on p?(b) What is the force on q?
A sphere of radius R carries a polarization P(r) = kr, where k is a constant and r is the vector from the center.(a) Calculate the bound charges σb and pb.(b) Find the field inside and outside the
A short cylinder, of radius a and length L, carries a "frozen-in" uniform polarization P, parallel to its axis. Find the bound charge, and sketch the electric field (i) for L >> a. (ii) for L
Calculate the potential of a uniformly polarized sphere (Ex. 4.2) directly from Eq. 4.9.
A very long cylinder, of radius a, carries a uniform polarization P perpendicular to its axis. Find the electric field inside the cylinder. Show that the field outside the cylinder can be expressed
When you polarize a neutral dielectric, charge moves a bit, but the total remains zero. This fact should be reflected in the bound charges crb and Pb. Prove from Eqs. 4.11 and 4.12 that the total
A thick spherical shell (inner radius a, outer radius b) is made of dielectric material with a "frozen-in" polarization P(r) = k/r r. where k is a constant and r is the distance from the center (Fig.
Suppose the field inside a large piece of dielectric is E0, so that the electric displacement is D0 = ?0E0 + P.? (a) Now a small spherical cavity (Fig. 4.19a) is hollowed out of the material. Find
For the bar electret of Prob. 4.11, make three careful sketches: one of P, one of E, and one of D. Assume L is about 2a.
The space between the plates of a parallel-plate capacitor (Fig. 4.24) is filled with two slabs of linear dielectric material. Each slab has thickness a, so the total distance between the plates is
Suppose you have enough linear dielectric material, of dielectric constant εr, to half-fill a parallel-plate capacitor (Fig. 4.25). By what fraction is the capacitance increased when you distribute
A sphere of linear dielectric material has embedded in it a uniform free charge density p. Find the potential at the center of the sphere (relative to infinity), if its radius is R and its dielectric
A certain coaxial cable consists of a copper wire, radius a, surrounded by a concentric copper tube of inner radius c (Fig. 4.26). The space between is partially filled (from b out to c) with
A very long cylinder of linear dielectric material is placed in an otherwise uniform electric field E 0. Find the resulting field within the cylinder. (The radius is a, the susceptibility Xe, and the
Find the field inside a sphere of linear dielectric material in an otherwise uniform electric field E0 (Ex. 4.7) by the following method of successive approximations: First pretend the field inside
An uncharged conducting sphere of radius a is coated with a thick insulating shell (dielectric constant εr) out to radius b. This object is now placed in an otherwise uniform electric field E0. Find
Suppose the region above the xy plane in Ex. 4.8 is also filled with linear dielectric but of a different susceptibility Xe'. Find the potential everywhere.
A spherical conductor, of radius a, carries a charge Q (Fig. 4.29). It is surrounded by linear dielectric material of susceptibility Xe, out to radius b. Find the energy of this configuration
Calculate W, using both Eq. 4.55 and Eq. 4.58, for a sphere of radius R with frozen-in uniform polarization P (Ex. 4.2). Comment on the discrepancy. Which (if either) is the "true" energy of the
Two long coaxial cylindrical metal tubes (inner radius a, outer radius b) stand vertically in a tank of dielectric oil (susceptibility Xe, mass density p). The inner one is maintained at potential V,
(a) For the configuration in Problem 4.5, calculate the force on P2 due to P1, and the force on P1 due to P2. Are the answers consistent with Newton's third law?(b) Find the total torque on P2 with
An electric dipole p, pointing in the y direction, is placed midway between two large conducting plates, as shown in Fig. 4.33. Each plate makes a small angle ? with respect to the x axis, and they
A dielectric cube of side a, centered at the origin, carries a "frozen-in" polarization P = kr, where k is a constant. Find all the bound charges, and check that they add up to zero.
A point charge q is imbedded at the center of a sphere of linear dielectric material (with susceptibility Xe and radius R). Find the electric field, the polarization, and the bound charge densities,
At the interface between one linear dielectric and another the electric field lines bend (see Fig. 4.34). Show that tan ?2/tan ?1 = ?2/ ?1, (4.68) assuming there is no free charge at the boundary.
A point dipole p is imbedded at the center of a sphere of linear dielectric material (with radius R and dielectric constant εr). Find the electric potential inside and outside the sphere.
Prove the following uniqueness theorem: A volume 1; contains a specified free charge distribution, and various pieces of linear dielectric material, with the susceptibility of each one given. If the
A conducting sphere at potential V0 is half embedded in linear dielectric material of susceptibility Xe, which occupies the region z (a) Write down the formula for the proposed potential V(r), in
According to Eq. 4.5, the force on a single dipole is (p ?? ??)E, so the net force on a dielectric object is [Here Eext is the field of everything except the dielectric. You might assume that it
In a linear dielectric, the polarization is proportional to the field: P = ε0XeE. If the material consists of atoms (or nonpolar molecules), the induced dipole moment of each one is likewise
Check the Clausius-Mossotti relation (Eq. 4.72) for the gases listed in Table 4.1. Dielectric constants are given in Table 4.2.) (The densities here are so small that Eqs. 4.70 and 4.72 are

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