Introduction to Electrodynamics 4th Edition solutions

Introduction to Electrodynamics 4
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, Laplaces equation does not hold within the sphere.) Show that, in general, where Vcenter is the potential a

Introduction to Electrodynamics 4
Problem 2P In one sentence, justify Earnshaws 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 suspende

Introduction to Electrodynamics 4
Problem 3P Find the general solution to Laplaces 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.

Introduction to Electrodynamics 4
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?

Introduction to Electrodynamics 4
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.

Introduction to Electrodynamics 4
Problem 6P A more elegant proof of the second uniqueness theorem uses Greens 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 numb

Introduction to Electrodynamics 4
Problem 7P Find the force on the charge +q in Fig. 3.14. (The xy plane is a grounded conductor.)

Introduction to Electrodynamics 4
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,

Introduction to Electrodynamics 4
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

Introduction to Electrodynamics 4
Problem 10P uniform line charge ? is placed on an infinite straight wire, a distance d above a grounded conducting plane. (Lets 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

Introduction to Electrodynamics 4
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?

Introduction to Electrodynamics 4
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

Introduction to Electrodynamics 4
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

Introduction to Electrodynamics 4
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,

Introduction to Electrodynamics 4
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 e

Introduction to Electrodynamics 4
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

Introduction to Electrodynamics 4
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

Introduction to Electrodynamics 4
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 advancethis is just a consistency check on the method.) (b) Find the potential inside

Introduction to Electrodynamics 4
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 theres no charge inside or outside the sphere.)

Introduction to Electrodynamics 4
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

Introduction to Electrodynamics 4
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.

Introduction to Electrodynamics 4
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 potent

Introduction to Electrodynamics 4
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.

Introduction to Electrodynamics 4
Problem 24P Solve Laplaces 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

Introduction to Electrodynamics 4
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 Laplaces equation by separation of variables i

Introduction to Electrodynamics 4
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 Laplaces equation by separation of variables in cylindri

Introduction to Electrodynamics 4
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.

Introduction to Electrodynamics 4
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.)

Introduction to Electrodynamics 4
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)

Introduction to Electrodynamics 4
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 d

Introduction to Electrodynamics 4
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).

Introduction to Electrodynamics 4
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 \((0,0, a)\)

Introduction to Electrodynamics 4
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.

Introduction to Electrodynamics 4
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). . .

Introduction to Electrodynamics 4
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

Introduction to Electrodynamics 4
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.

Introduction to Electrodynamics 4
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.

Introduction to Electrodynamics 4
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 <

Introduction to Electrodynamics 4
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.

Introduction to Electrodynamics 4
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.

Introduction to Electrodynamics 4
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.

Introduction to Electrodynamics 4
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. Heres one method:22 (a) Show that the average field due to a single charge q at poi

Introduction to Electrodynamics 4
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 dont understand why, reread the discussio

Introduction to Electrodynamics 4
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 careperhaps number 1 is a uniformly charged sphere and number 2 is a parallel-plate capacitor.

Introduction to Electrodynamics 4
Problem 51P Use Greens 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 charg

Introduction to Electrodynamics 4
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

Introduction to Electrodynamics 4
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 oppo

Introduction to Electrodynamics 4
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

Introduction to Electrodynamics 4
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

Introduction to Electrodynamics 4
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 Refer

Introduction to Electrodynamics 4
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