Question: Sketch the vector function / and compute its
Sketch the vector function
and compute its divergence. The answer may surprise you. . . can you explain it?
> 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 system?
> A spherical conductor, of radius a, carries a charge Q (Fig. 4.29). It is surrounded by linear dielectric material of susceptibility χe, out to radius b. Find the energy of this configuration (Eq. 4.58).
> Suppose the region above the xy plane in Ex. 4.8 is also filled with linear dielectric but of a different susceptibility / . Find the potential everywhere.
> 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 the electric field in the insulator.
> Construct a vector function that has zero divergence and zero curl everywhere. (A constant will do the job, of course, but make it something a little more interesting than that!)
> Is the cross product associative? (A × B) × C =? A × (B × C). If so, prove it; if not, provide a counterexample (the simpler the better).
> 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 is just E0, and use Eq. 4.30 to write down the resulting po
> A very long cylinder of linear dielectric material is placed in an otherwise uniform electric field E0. Find the resulting field within the cylinder. (The radius is a, the susceptibility χe, and the axis is perpendicular to E0.)
> 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 material of dielectric constant εr
> A sphere of linear dielectric material has embedded in it a uniform free charge density ρ. Find the potential at the center of the sphere (relative to infinity), if its radius is R and the dielectric constant is εr .
> 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 polarizability of such an atom. [Hint: First calculate the
> 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 the material as in Fig. 4.25(a)? How about Fig. 4.25(b)?
> 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 2a. Slab 1 has a dielectric constant of 2
> 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. [Hint: E lines terminate on charges; D lines terminate on free charges.]
> Suppose the field inside a large piece of dielectric is E0, so that the electric displacement is / (a) Now a small spherical cavity (Fig. 4.19a) is hollowed out of the material. Find the field at the center of the cavity in terms of E0 and P. Also find th
> A thick spherical shell (inner radius a, outer radius b) is made of dielectric material with a “frozen-in” polarization where k is a constant and r is the distance from the center (Fig. 4.18). (There is no free charge
> Draw a circle in the xy plane. At a few representative points draw the vector v tangent to the circle, pointing in the clockwise direction. By comparing adjacent vectors, determine the sign of ∂vx /∂ y and ∂vy /∂ x . According to Eq. 1.41, then, what is
> When you polarize a neutral dielectric, the charge moves a bit, but the total remains zero. This fact should be reflected in the bound charges σb and ρb. Prove from Eqs. 4.11 and 4.12 that the total bound charge vanishes.
> 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 in the form / [Careful: I said “uniform,” not “radial”!]
> Calculate the potential of a uniformly polarized sphere (Ex. 4.2) directly from Eq. 4.9.
> 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
> 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 ρb. (b) Find the field inside and outside the sphere.
> 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 radius does the separation distance d amount to, roughly
> 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 V0 (relative, of course, to infinity). What charge should you use,
> (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 obvious that V = 0 on the sphere, r = R. (b) F
> Find the force on the charge +q in Fig. 3.14. (The xy plane is a grounded conductor.)
> A more elegant proof of the second uniqueness theorem uses Green’s identity (Prob. 1.61c), with T = U = V3. Supply the details.
> Calculate the curls of the vector functions in Prob. 1.15.
> Find the charge density σ(θ) on the surface of a sphere (radius R) that produces the same electric field, for points exterior to the sphere, as a charge q at the point a < R on the z axis. /
> A stationary electric dipole / is situated at the origin. A positive point charge q (mass m) executes circular motion (radius s) at constant speed in the field of the dipole. Characterize the plane of the orbit. Find the speed, angular momentum and total
> 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 forth in a semi-circular arc, as though it were a pendul
> (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 unit length on the side opposite to V0. [Hint: Use 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 V0. Find the potential inside the pipe. [No
> In Ex. 3.8 we determined the electric field outside a spherical conductor (radius R) placed in a uniform external field E0. Solve the problem now using the method of images, and check that your answer agrees with Eq. 3.76. [
> (a) Show that the quadrupole term in the multipole expansion can be written / (in the notation of Eq. 1.31), where / Here / is the Kronecker delta, and Qij is the quadrupole moment of the charge distribution. Notice the hierarchy: / The monopole mom
> Use Green’s reciprocity theorem (Prob. 3.50) to solve the following two problems. [Hint: for distribution 1, use the actual situation; for distribution 2, remove q, and set one of the conductors at potential V0.] (a) Both plates of a parallel-plate capac
> (a) Suppose a charge distribution ρ1(r) produces a potential V1(r), and some other charge distribution ρ2(r) produces a potential V2(r). [The two situations may have nothing in common, for all I care—perhaps number
> Prove that the field is uniquely determined when the charge density ρ is given and either V or the normal derivative ∂V /∂n is specified on each boundary surface. Do not assume the boundaries are conductors, or that V is constant over any given surface.
> In two dimensions, show that the divergence transforms as a scalar under rotations. [Hint: Use Eq. 1.29 to determine and the method of Prob. 1.14 to calculate the derivatives. Your aim is to show that / /
> In Ex. 3.9, we obtained the potential of a spherical shell with surface charge σ(θ) k cos θ . In Prob. 3.30, you found that the field is pure dipole out- side; it’s uniform inside (Eq. 3.86). Show that the limit R 0 reproduces the delta function term
> (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 integrals first. [Note: You must express / (see back cover) before integrati
> 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 result. Here’s one meth
> 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 cos(π z/2a), (b) λ k sin(πz/a), (c) λ k cos(π z/a), where k is a constant.
> 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 the cylinder.
> 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 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 cos θ, where k is a constant and θ
> You can use the superposition principle to combine solutions obtained by separation of variables. For example, in Prob. 3.16 you found the potential inside a cubical box, if five faces are grounded and the sixth is at a constant potential V0; by a six-fol
> Buckminsterfullerine is a molecule of 60 carbon atoms arranged like the stitching on a soccer-ball. It may be approximated as a conducting spherical shell of radius R=3.5 Å. A nearby electron would be attracted, according to Prob. 3.9, so it is not surpr
> 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, and the wires are a distance a from the axis.
> (a) Show that the average electric field over a spherical surface, due to charges outside the sphere, is the same as the field at the center. (b) What is the average due to charges inside the sphere?
> 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.20 Check that your answer is correct for the special cases a →∞ and x = a
> Here’s an alternative derivation of Eq. 3.10 (the surface charge density induced on a grounded conducted plane by a point charge q a distance d above the plane). This approach19 (which generalizes to many other problems) does not rely on the method of im
> In Section 3.1.4, I proved that the electrostatic potential at any point P in a charge-free region is equal to its average value over any spherical surface (radius R) centered at P. Here’s an alternative argument that does not rely on Coulomb’s law, only
> Show that the electric field of a (perfect) dipole (Eq. 3.103) can be written in the coordinate-free form
> A solid sphere, radius R, is centered at the origin. The “northern” hemisphere carries a uniform charge density ρ0, and the “southern” hemisphere a uniform charge density −ρ0. Find the approximate field E(r,θ) for points far from the sphere (r((R).
> 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 coordinates, and include the two lowest orders 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 much work does it take to move q from (a, 0, 0) to
> 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 spherical coordinates) at large r (inc
> For the dipole in Ex. 3.10, expand 1/r± to order (d/r)3, and use this to determine the quadrupole and octopole terms in the potential.
> Calculate the divergence of the following vector functions: (a) va = x 2 xˆ + 3xz2 yˆ − 2xz zˆ. (b) vb = xy xˆ + 2yz yˆ + 3zx zˆ. (c) vc = y2 xˆ + (2xy + z2) yˆ + 2yz zˆ.
> 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 approximate potential, at points far from the sphere, and
> 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.
> 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  from the origin. Find a simple approximate formula for the potential, valid at points far from the origin. (Ex
> A circular ring in the xy plane (radius R, centered at the origin) carries a uniform line charge λ. Find the first three terms (n =0, 1, 2) in the multipole expansion for V (r,θ).
> A sphere of radius R, centered at the origin, carries charge density where k is a constant, and r , θ are the usual spherical coordinates. Find the approximate potential for points on the z axis, far from the sphere.
> Charge density σ(φ) = a sin 5φ (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. [Use your result from Pr
> 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. [Use your result from Prob. 3.24.]
> 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; in particular, your result must accommodate the case of
> 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 outside the sphere, calculating the coefficients explicitly up to A6
> 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 (Eq. 3.72) for the potential of the disk at points off the axis, assu
> Suppose that f is a function of two variables (y and z) only. Show that the gradient / transforms as a vector under rotations, Eq. 1.29. and the analogous formula for /. We know that / And / “solve” these equat
> Find the potential outside a charged metal sphere (charge Q, radius R) placed in an otherwise uniform electric field E0. Explain clearly where you are setting the zero of potential.
> 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 /
> In one sentence, justify Earnshaw’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 charges in Fig. 3.4. It looks,
> 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) on the sphere. (Assume there’s no charge inside or outside
> (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 in advance—this is just a consistency check on the meth
> 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 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 at a constant potential V0. Find the pot
> 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 potential V0(y). (a) Develop a general formula for the potential inside
> 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.
> 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 to y = a, is at potential −V0.
> Let r be the separation vector from a fixed point (x r, yr, zr) to the point (x, y, z), and let r be its length. Show that (a) ∇(r2) = 2r. (b) ∇(1/r) = −rˆ/r2. (c) What is the general formula for ∇(rn)?
> Two long, straight copper pipes, each of radius R, are held a distance 2d apart. One is at potential V0, the other at V0 (Fig. 3.16). Find the potential everywhere. [Hint: Exploit the result of Prob. 2.52.]
> 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 the potential in this re
> 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 conducting plane is the xy plane.) (a) Find the potential in th
> 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 < R). (In this case, of course, Laplace’s equation does not hold within the sphere.) Show that, in general,
> Suppose the electric field in some region is found to be / in spherical coordinates (k is some constant). (a) Find the charge density ρ. (b) Find the total charge contained in a sphere of radius R, centered at the origin. (Do it two different ways.)
> Use your result in Prob. 2.7 to find the field inside and outside a solid sphere of radius R that carries a uniform volume charge density ρ. Express your answers in terms of the total charge of the sphere, q. Draw a graph of (E( as a function of the distan
> Find the electric field a distance z from the center of a spherical surface of radius R (Fig. 2.11) that carries a uniform charge density σ. Treat the case z R (outside). Express your answers in terms of the total charge q on t
> What is the minimum-energy configuration for a system of N equal point charges placed on or inside a circle of radius R?17 Because the charge on a conductor goes to the surface, you might think the N charges would arrange themselves (uniformly) around th
> A point charge q is at the center of an uncharged spherical conducting shell, of inner radius a and outer radius b. Question: How much work would it take to move the charge out to infinity (through a tiny hole drilled in the shell)? [Answer: /
> Find the electric field a distance z above the center of a flat circular disk of radius R (Fig. 2.10) that carries a uniform surface charge σ. What does your formula give in the limit R → â
> The height of a certain hill (in feet) is given by h(x, y) = 10(2xy − 3x 2 − 4y2 − 18x + 28y + 12), where y is the distance (in miles) north, x the distance east of South Hadley. (a) Where is the top of the hill located? (b) How high is the hill? (c) How
> Prove or disprove (with a counterexample) the following Theorem: Suppose a conductor carrying a net charge Q, when placed in an external electric field Ee, experiences a force F; if the external field is now reversed (Ee → −Ee), the force also reverses (
> (a) Consider an equilateral triangle, inscribed in a circle of radius a, with a point charge q at each vertex. The electric field is zero (obviously) at the center, but (surprisingly) there are three other points inside the triangle where the field is zero
> 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 explicitly is the ellipsoid: In this case15 / where Q
> 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 gravitation. What is the gravitational energy of a sphere
> 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 direction, when the charge density is uniform? [This is a mor
> 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 λ is a new constant of nature (it has dimensi
> 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 the gap (called sp
