A stationary electric dipole is situated at the origin. A

Problem 57P 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 energy of the charge.29

Problem 2P 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, off hand, as though a positive charge at the center would be suspended in midair, since it is repelled away from each corner. Where is the leak in this “electrostatic bottle”? [To harness nuclear fusion as a practical energy source it is necessary to heat a plasma (soup of charged particles) to fantastic temperatures—so hot that contact would vaporize any ordinary pot. Earnshaw’s theorem says that electrostatic containment is also out of the question. Fortunately, it is possible to confine a hot plasma magnetically.]

Problem 1P 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, where Vcenter is the potential at the center due to all the external charges, and Qenc is the total enclosed charge.

Problem 3P 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.

Problem 5P Prove that the field is uniquely determined when the charge density ? 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.

Problem 4P (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?

Problem 6P A more elegant proof of the second uniqueness theorem uses Green’s identity (Prob. 1.61c), with T = U = V3. Supply the details. Reference prob 1.61c Although the gradient, divergence, and curl theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that:

Problem 7P Find the force on the charge +q in Fig. 3.14. (The xy plane is a grounded conductor.)

Problem 8P (a) Using the law of cosines, show that Eq. 3.17 can be written as follows: where r and ? 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) Find the induced surface charge on the sphere, as a function of ?. Integrate this to get the total induced charge. (What should it be?) Reference equation 3.17

Problem 9P 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, and where should you put it? Find the force of attraction between a point charge q and a neutral conducting sphere. Reference example 3.2 A point charge q is situated a distance a from the center of a grounded conducting sphere of radius R (Fig. 3.12). Find the potential outside the sphere. Reference figure 3.12

Problem 11P 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 region. What charges do you need, and where should they be located? What is the force on q? How much work did it take to bring q in from infinity? Suppose the planes met at some angle other than 90?; would you still be able to solve the problem by the method of images? If not, for what particular angles does the method work? Reference figure 3.15

Problem 10P uniform line charge ? 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 the region above the plane. [Hint: Refer to Prob. 2.52.] (b) Find the charge density ? induced on the conducting plane. Reference prob 2.52 For Theorem 2, show that (d)?(a), (a)?(c), (c)?(b), (b)?(c), and (c)?(a). Reference Theorem 2, Divergence-less (or “solenoidal”) fields. The following conditions are equivalent: (a) ? ? F = 0 everywhere. (b) ? F ? da is independent of surface, for any given boundary line. (c) for any closed surface. (d) F is the curl of some vector function: F = ? × A.

Problem 12P 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.] Reference figure 3.16 Reference Prob. 2.52 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 reference. (b) Show that the equipotential surfaces are circular cylinders, and locate the axis and radius of the cylinder corresponding to a given potential V0.

Problem 13P 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. Reference example 3.3 Two infinite grounded metal plates lie parallel to the xz plane, one at y = 0, the other at y = a (Fig. 3.17). The left end, at x = 0, is closed off with an infinite strip insulated from the two plates, and maintained at a specific potential V0(y). Find the potential inside this “slot.”

Problem 14P For the infinite slot (Ex. 3.3), determine the charge density ?(y) on the strip at x = 0, assuming it is a conductor at constant potential V0. Reference example 3.3 Two infinite grounded metal plates lie parallel to the xz plane, one at y = 0, the other at y = a (Fig. 3.17). The left end, at x = 0, is closed off with an infinite strip insulated from the two plates, and maintained at a specific potential V0(y). Find the potential inside this “slot.”

Problem 16P 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 potential inside the box. [What should the potential at the center (a/2, a/2, a/2) be? Check numerically that your formula is consistent with this value.]11

Problem 17P Derive P3(x) from the Rodrigues formula, and check that P3(cos ?) satisfies the angular equation (3.60) for l = 3. Check that P3 and P1 are orthogonal by explicit integration. Reference equation 3.60

Problem 18P (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 method.) (b) Find the potential inside and outside a spherical shell that carries a uniform surface charge ?0, using the results of Ex. 3.9. Reference 3.6 The potential V0(?) is specified on the surface of a hollow sphere, of radius R. Find the potential inside the sphere. Reference 3.7 The potential V0(?) is again specified on the surface of a sphere of radius R, but this time we are asked to find the potential outside, assuming there is no charge there. Reference 3.9 A specified charge density ?0(?) is glued over the surface of a spherical shell of radius R. Find the resulting potential inside and outside the sphere.

Problem 15P 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 the pipe. (b) Find the potential explicitly, for the case V0(y) = V0 (a constant).

Problem 19P The potential at the surface of a sphere (radius R) is given by where k is a constant. Find the potential inside and outside the sphere, as well as the surface charge density ?(?) on the sphere. (Assume there’s no charge inside or outside the sphere.)

Problem 20P Suppose the potential V0(?) 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

Problem 21P 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.

Problem 22P 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, assuming r > R. (b) Find the potential for r < R by the same method, using Eq. 3.66. [Note: You must break the interior region up into two hemispheres, above and below the disk. Do not assume the coefficients Al are the same in both hemispheres.] Reference 3.72 Reference 3.66 Reference prob 25 Using Eqs. 2.27 and 2.30, find the potential at a distance z above the center of the charge distributions in Fig. 2.34. In each case, compute E = ??V, and compare your answers with Ex. 2.1, Ex. 2.2, and Prob. 2.6, respectively. Suppose that we changed the right-hand charge in Fig. 2.34a to ?q; what then is the potential at P? What field does that suggest? Compare your answer to Prob. 2.2, and explain carefully any discrepancy. Eqs. 2.27 Eqs. 2.30 Reference example 2.1 Find the electric field a distance z above the midpoint between two equal charges (q), a distance d apart (Fig. 2.4a). Reference example 2.1 Find the electric field a distance z above the midpoint of a straight line segment of length 2L that carries a uniform line charge ? (Fig. 2.6). Reference prob.2.6 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 ?.What does your formula give in the limit R??? Also check the case z ?R. Reference figure 2.10 Reference prob.2.2 Find the electric field (magnitude and direction) a distance z above the midpoint between equal and opposite charges (±q), a distance d apart (same as Example 2.1, except that the charge at x = +d/2 is ?q). Reference example 2.1 Find the electric field a distance z above the midpoint between two equal charges (q), a distance d apart (Fig. 2.4a).

Problem 23P spherical shell of radius R carries a uniform surface charge ?0 on the “northern” hemisphere and a uniform surface charge ??0 on the “southern” hemisphere. Find the potential inside and outside the sphere, calculating the coefficients explicitly up to A6 and B6.

Problem 24P 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 an infinite line charge, for which (of course) we already know the answer.]

Problem 25P 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.] reference 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 an infinite line charge, for which (of course) we already know the answer.]

Problem 26P Charge density (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 Prob. 3.24.] Reference prob 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 an infinite line charge, for which (of course) we already know the answer.]

Problem 27P 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 from the sphere.

Problem 30P In Ex. 3.9, we derived the exact potential for a spherical shell of radius R, which carries a surface charge ? = k cos ?. (a) Calculate the dipole moment of this charge distribution. (b) Find the approximate potential, at points far from the sphere, and compare the exact answer (Eq. 3.87). What can you conclude about the higher multipoles? Reference equation 3.87 Reference example 3.9 A specified charge density ?0(?) is glued over the surface of a spherical shell of radius R. Find the resulting potential inside and outside the sphere.

Problem 31P For the dipole in Ex. 3.10, expand and use this to determine the quadrupole and octopole terms in the potential. Reference 3.10 A (physical) electric dipole consists of two equal and opposite charges (±q) separated by a distance d. Find the approximate potential at points far from the dipole.

Problem 29P 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, valid at points far from the origin. (Express your answer in spherical coordinates.)

Problem 32P 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 (include both the monopole and dipole contributions).

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 \((0,0, a)\) ?

Problem 34P 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 the multipole expansion.

A solid sphere, radius R, is centered at the origin. The “northern” hemisphere carries a uniform charge density \(\rho_{0}\), and the “southern” hemisphere a uniform charge density ?\(\rho_{0}\). Find the approximate field E(r,?) for points far from the sphere (r ? R). . .

Problem 36P Show that the electric field of a (perfect) dipole (Eq. 3.103) can be written in the coordinate-free form Reference equation 3.103

Problem 40P Two long straight wires, carrying opposite uniform line charges ±?, 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. Find the potential.

Problem 39P 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/2.

Problem 43P 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 where k is a constant and ? is the usual spherical coordinate. (a) Find the potential in each region: (i) r > b, and (ii) a < r < b. (b) Find the induced surface charge ?i (?) on the conductor. (c) What is the total charge of this system? Check that your answer is consistent with the behavior of V at large r .

Problem 44P 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.

Problem 45P A long cylindrical shell of radius R carries a uniform surface charge ?0 on the upper half and an opposite charge ??0 on the lower half (Fig. 3.40). Find the electric potential inside and outside the cylinder.

Problem 46P 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) ? = k cos(?z/2a), (b) ? = k sin(?z/a), (c) ? = k cos(?z/a), where k is a constant.

Problem 47P 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 method:22 (a) Show that the average field due to a single charge q at point r inside the sphere is the same as the field at r due to a uniformly charged sphere with namely Where is the vector from r to d? ? (b) The latter can be found from Gauss’s law (see Prob. 2.12). Express the answer in terms of the dipole moment of q. (c) Use the superposition principle to generalize to an arbitrary charge distribution. (d) While you’re at it, show that the average field over the volume of a sphere, due to all the charges outside, is the same as the field they produce at the center. Reference prob 2.12

Problem 50P (a) Suppose a charge distribution ?1(r) produces a potential V1(r), and some other charge distribution ?2(r) produces a potential V2(r). [The two situations may have nothing in common, for all I care—perhaps number 1 is a uniformly charged sphere and number 2 is a parallel-plate capacitor. Please understand that ?1 and ?2 are not present at the same time; we are talking about two different problems, one in which only ?1 is present, and another in which only ?2 is present.] Prove Green’s reciprocity theorem:25 [Hint: Evaluate ? E1 . E2 d? two ways, first writing E1 = ??V1 and using integration by parts to transfer the derivative to E2, then writing E2 = ??V2 and transferring the derivative to E1.] (b) Suppose now that you have two separated conductors (Fig. 3.41). If you charge up conductor a by amount Q (leaving b uncharged), the resulting potential of b is, say, Vab. On the other hand, if you put that same charge Q on conductor b (leaving a uncharged), the potential of a would be Vba . Use Green’s reciprocity theorem to show that Vab = Vba (an astonishing result, since we assumed nothing about the shapes or placement of the conductors).

Problem 51P 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 capacitor are grounded, and a point charge q is placed between them at a distance x from plate 1. The plate separation is d. Find the induced charge on each plate. [Answer: Q1 = q(x/d ? 1); Q2 = ?qx/d] (b) Two concentric spherical conducting shells (radii a and b) are grounded, and a point charge q is placed between them (at radius r ). Find the induced charge on each sphere. Reference prob 3.50 (a) Suppose a charge distribution ?1(r) produces a potential V1(r), and some other charge distribution ?2(r) produces a potential V2(r). [The two situations may have nothing in common, for all I care—perhaps number 1 is a uniformly charged sphere and number 2 is a parallel-plate capacitor. Please understand that ?1 and ?2 are not present at the same time; we are talking about two different problems, one in which only ?1 is present, and another in which only ?2 is present.] Prove Green’s reciprocity theorem:25 [Hint: Evaluate ? E1 . E2 d? two ways, first writing E1 = ??V1 and using integration by parts to transfer the derivative to E2, then writing E2 = ??V2 and transferring the derivative to E1.] (b) Suppose now that you have two separated conductors (Fig. 3.41). If you charge up conductor a by amount Q (leaving b uncharged), the resulting potential of b is, say, Vab. On the other hand, if you put that same charge Q on conductor b (leaving a uncharged), the potential of a would be Vba . Use Green’s reciprocity theorem to show that Vab = Vba (an astonishing result, since we assumed nothing about the shapes or placement of the conductors).

Problem 52P (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 Qi j is the quadrupole moment of the charge distribution. Notice the hierarchy: The monopole moment (Q) is a scalar, the dipole moment (p) is a vector, the quadrupole moment (Qi j ) is a second-rank tensor, and so on. (b) Find all nine components of Qi j for the configuration in Fig. 3.30 (assume the square has side a and lies in the xy plane, centered at the origin). (c) Show that the quadrupole moment is independent of origin if the monopole and dipole moments both vanish. (This works all the way up the hierarchy—the lowest nonzero multipole moment is always independent of origin.) (d) How would you define the octopole moment? Express the octopole term in the multipole expansion in terms of the octopole moment. Reference figure 3.30

Problem 54P 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. [Note: This is a rotated version of Prob. 3.15(b), but set it up as in Ex. 3.4, using sinusoidal functions in y and hyperbolics in x. It is an unusual case in which k = 0 must be included. Begin by finding the general solution to Eq. 3.26 when k = 0.]26 Reference prob 3.15 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 the pipe. (b) Find the potential explicitly, for the case V0(y) = V0 (a constant). Reference equation 3.26

Problem 53P 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. [Hint: Use Ex. 3.2, but put another charge, ?q, diametrically opposite q. Let a??, with Reference equation 3.76 Reference example 3. 2 A point charge q is situated a distance a from the center of a grounded conducting sphere of radius R (Fig. 3.12). Find the potential outside the sphere.

Problem 48P (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 in terms of (see back cover) before integrating. If you don’t understand why, reread the discussion in Sect. 1.4.1.] Compare your answer with the general theorem (Eq. 3.105). The discrepancy here is related to the fact that the field of a dipole blows up at r = 0. The angular integral is zero, but the radial integral is infinite, so we really don’t know what to make of the answer. To resolve this dilemma, let’s say that Eq. 3.103 applies outside a tiny sphere of radius ?—its contribution to Eave is then unambiguously zero, and the whole answer has to come from the field inside the -sphere. (b) What must the field inside the ? - sphere be, in order for the general theorem (Eq. 3.105) to hold? [Hint: since ? is arbitrarily small, we’re talking about something that is infinite at r = 0 and whose integral over an infinitesimal volume is finite.] Evidently, the true field of a dipole is You may wonder how we missed the delta-function term23 when we calculated the field back in Sect. 3.4.4. The answer is that the differentiation leading to Eq. 3.103 is valid except at r = 0, but we should have known (from our experience in Sect. 1.5.1) that the point r = 0 would be problematic.24 Reference equation 3.105 Reference equation 3.103

Problem 55P (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 answer to Prob. 3.15 or Prob. 3.54.] (b) A long metal pipe of circular cross-section (radius R) is divided (lengthwise) into four equal sections, three of them grounded and the fourth maintained at constant potential V0. Find the net charge per unit length on the section opposite to V0. [Answer to both (a) and (b): Reference prob 3.15

Problem 56P 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 pendulum supported at the origin.28 Reference figure 3.36