Is the cross product associative? (A × B) × C =? A × (B × C). If so, prove it; if not, provide a counterexample (the simpler the better).
> What current density would produce the vector potential, A=k φˆ (where k is a constant), in cylindrical coordinates?
> Find the magnetic vector potential of a finite segment of straight wire carrying a current I . [Put the wire on the z axis, from z1 to z2, and use Eq. 5.66.] Check that your answer is consistent with Eq. 5.37.
> Suppose there did exist magnetic monopoles. How would you modify Maxwell’s equations and the force law to accommodate them? If you think there are several plausible options, list them, and suggest how you might decide experimentally which one is right.
> Is Ampère’s law consistent with the general rule (Eq. 1.46) that divergence-of-curl is always zero? Show that Ampère’s law cannot be valid, in general, outside magnetostatics. Is there any such “defect” in the other three Maxwell equations?
> Derive the three quotient rules.
> (a) Find the density ρ of mobile charges in a piece of copper, assuming each atom contributes one free electron. [Look up the necessary physical constants.] (b) Calculate the average electron velocity in a copper wire 1 mm in diameter, carrying a current
> Find and sketch the trajectory of the particle in Ex. 5.2, if it starts at the origin with velocity (a) v(0) = (E/B)yˆ, (b) v(0) = (E/2B)yˆ, (c) v(0) = (E/B)(yˆ + zˆ).
> In calculating the current enclosed by an Amperian loop, one must, in general, evaluate an integral of the form The trouble is, there are infinitely many surfaces that share the same boundary line. Which one are we supposed to use?
> Show that the magnetic field of an infinite solenoid runs parallel to the axis, regardless of the cross-sectional shape of the coil, as long as that shape is constant along the length of the solenoid. What is the magnitude of the field, inside and outside o
> A large parallel-plate capacitor with uniform surface charge σ on the upper plate and σ on the lower is moving with a constant speed v, as shown in Fig. 5.43. (a) Find the magnetic field between the plates and also ab
> Two long coaxial solenoids each carry current I, but in opposite directions, as shown in Fig. 5.42. The inner solenoid (radius a) has n1 turns per unit length, and the outer one (radius b) has n2. Find B in each of the three regions: (i) inside the inner
> A thick slab extending from z=-a to z=+a (and infinite in the x and y directions) carries a uniform volume current / (Fig. 5.41). Find the magnetic field, as a function of z, both inside and outside the slab.
> A steady current I flows down a long cylindrical wire of radius a (Fig. 5.40). Find the magnetic field, both inside and outside the wire, if (a) The current is uniformly distributed over the outside surface of the wire. (b)
> Suppose you have two infinite straight line charges λ, a distance d apart, moving along at a constant speed v (Fig. 5.26). How great would v have to be in order for the magnetic attraction to balance the electrical repulsion?
> Use the result of Ex. 5.6 to calculate the magnetic field at the center of a uniformly charged spherical shell, of radius R and total charge Q, spinning at constant angular velocity ω.
> Prove product rules (ii) and (vi). Refer to Prob. 1.22 for the definition of (A · ∇)B.
> Find the magnetic field at point P on the axis of a tightly wound solenoid (helical coil) consisting of n turns per unit length wrapped around a cylindrical tube of radius a and carrying current I (Fig. 5.25). Express your answer in terms
> (a) Find the force on a square loop placed as shown in Fig. 5.24(a), near an infinite straight wire. Both the loop and the wire carry a steady current I . (b) Find the force on the triangular loop in Fig. 5.24(b).
> A particle of charge q enters a region of uniform magnetic field B (pointing into the page). The field deflects the particle a distance d above the original line of flight, as shown in Fig. 5.8.
> 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?
> Show that the interaction energy of two dipoles separated by a displacement r is [Hint: Use Prob. 4.7 and Eq. 3.104.]
> Show that the energy of an ideal dipole p in an electric field E is given by
> A (perfect) dipole p is situated a distance z above an infinite grounded conducting plane (Fig. 4.7). The dipole makes an angle θ with the perpendicular to the plane. Find the torque on p. If the dipole is free to rotate, in w
> In Fig. 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 center. If it bothers you that the answers are not equal
> The Clausius-Mossotti equation (Prob. 4.41) tells you how to calculate the susceptibility of a nonpolar substance, in terms of the atomic polarizability α. The Langevin equation tells you how to calculate the susceptibility of a polar subst
> 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 indistinguishable. For experimental data that confirm the Clausi
> (a) If A and B are two vector functions, what does the expression (A. ∇)B mean? (That is, what are its x , y, and z components, in terms of the Cartesian components of A, B, and ∇?) (b) Compute (rˆ · ∇)rˆ, where rˆ is the unit vector defined in Eq. 1.21.
> In a linear dielectric, the polarization is proportional to the field: / If the material consists of atoms (or nonpolar molecules), the induced dipole moment of each one is likewise proportional to the field p=Î&plusm
> 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 wouldn’t matter if you used the total field; after all, the d
> A point charge q is situated a large distance r from a neutral atom of polarizability α. Find the force of attraction between them.
> A conducting sphere at potential V0 is half embedded in linear dielectric material of susceptibility χe, which occupies the region z (a) Write down the formula for the proposed potential V (r), in terms of V0, R, and r . Use it to determine th
> Prove the following uniqueness theorem: A volume contains a specified free charge distribution, and various pieces of linear dielectric material, with the susceptibility of each one given. If the potential is specified on th
> 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.
> At the interface between one linear dielectric and another, the electric field lines bend (see Fig. 4.34). Show that assuming there is no free charge at the boundary. [Comment: Eq. 4.68 is reminiscent of Snell’s law in op
> A point charge q is imbedded at the center of a sphere of linear dielectric material (with susceptibility χe and radius R). Find the electric field, the polarization, and the bound charge densities, ρb and σb. What is the total bound charge on the surface
> The space between the plates of a parallel-plate capacitor is filled with dielectric material whose dielectric constant varies linearly from 1 at the bottom plate (x=0) to 2 at the top plate (x=d). The capacitor is connected to a battery of voltage V . Fi
> 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.
> Prove product rules (i), (iv), and (v).
> Earnshaw’s theorem (Prob. 3.2) says that you cannot trap a charged particle in an electrostatic field. Question: Could you trap a neutral (but polarizable) atom in an electrostatic field? (a) Show that the
> A point charge Q is “nailed down” on a table. Around it, at radius R, is a frictionless circular track on which a dipole p rides, constrained always to point tangent to the circle. Use Eq. 4.5 to show that the electric
> 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 θ with respect to the x axis, and they are maintained at potentials / . What is th
> 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 theory. Suppose, for example, the charge density of the electron cl
> (a) For the configuration in Prob. 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 respect to the center of p1, and compare it with the torqu
> Two long coaxial cylindrical metal tubes (inner radius a, outer radius b) stand vertically in a tank of dielectric oil (susceptibility χe, mass density ρ). The inner one is maintained at potential V , and the outer one is grounded (
> 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!)
> 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.
> Sketch the vector function and compute its divergence. The answer may surprise you. . . can you explain it?
> (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
