Question: Find the vector potential above and below

> An iron rod of length L and square cross section (side a) is given a uniform longitudinal magnetization M, and then bent around into a circle with a narrow gap (width w), as shown in Fig. 6.14. Find the magnetic field at the center of the

> Calculate the line integral of the function / from the origin to the point (1,1,1) by three different routes: (a) (0, 0, 0) → (1, 0, 0) → (1, 1, 0) → (1, 1, 1). (b) (0, 0, 0) → (0, 0, 1) → (0, 1, 1) → (1, 1, 1). (c) The direct straight line. (d) What is

> Calculate the torque exerted on the square loop shown in Fig. 6.6, due to the circular loop (assume r is much larger than a or b). If the square loop is free to rotate, what will its equilibrium orientation be?

> Find the magnetic field at point P for each of the steady current configurations shown in Fig. 5.23.

> (a) Find the magnetic field at the center of a square loop, which carries a steady current I . Let R be the distance from center to side (Fig. 5.22). (b) Find the field at the center of a regular n-sided polygon, carrying a

> For a configuration of charges and currents confined within a volume V, show that where p is the total dipole moment. [Hint: evaluate /

> A thin glass rod of radius R and length L carries a uniform surface charge σ . It is set spinning about its axis, at an angular velocity ω. Find the magnetic field at a distance s((R from the axis, in the xy plane (Fi

> Using Eq. 5.88, calculate the average magnetic field of a dipole over a sphere of radius R centered at the origin. Do the angular integrals first. Compare your answer with the general theorem in Prob. 5.59. Explain the discr

> A uniformly charged solid sphere of radius R carries a total charge Q, and is set spinning with angular velocity ω about the z axis. (a) What is the magnetic dipole moment of the sphere? (b) Find the average magnetic field within the sphere (see Prob. 5.5

> (a) A phonograph record carries a uniform density of “static electricity” σ . If it rotates at angular velocity ω, what is the surface current density K at a distance r from the center? (b) A uniformly charged solid sphere, of radius R and total charge Q

> (a) Prove that the average magnetic field, over a sphere of radius R, due to steady currents inside / where m is the total dipole moment of the sphere. Contrast the electrostatic result, Eq. 3.105. [This is tough, so I’l

> A thin uniform donut, carrying charge Q and mass M , rotates about its axis as shown in Fig. 5.64. (a) Find the ratio of its magnetic dipole moment to its angular momentum. This is called the gyromagnetic ratio (or magnetomechanical ratio). (b) What is t

> Prove that the curl of a gradient is always zero. Check it for function (b) in Prob. 1.11.

> A magnetic dipole / is situated at the origin, in an otherwise uniform magnetic field / Show that there exists a spherical surface, centered at the origin, through which no magnetic field lines pass. Find the radius of this sphere, and sketch the field li

> Prove the following uniqueness theorem: If the current density J is specified throughout a volume V, and either the potential A or the magnetic field B is specified on the surface S bounding V, then the magnetic field itself is uniquely determined throughout

> Just as ∇ · B = 0 allows us to express B as the curl of a vector potential (B = ∇ × A), so ∇ · A = 0 permits us to write A itself as the curl of a “higher” potential: A = ∇ × W. (And this hierarchy can be extended ad infinitum.) (a) Find the general formu

> (a) Construct the scalar potential U(r) for a “pure” magnetic dipole m. (b) Construct a scalar potential for the spinning spherical shell (Ex. 5.11). [Hint: for r > R this is a pure dipole field, as you can see by comparing Eqs. 5.69 and 5.87.] (c) Try do

> Another way to fill in the “missing link” in Fig. 5.48 is to look for a magnetostatic analog to Eq. 2.21. The obvious candidate would be / (a) Test this formula for the simplest possible caseâ&#12

> (a) One way to fill in the “missing link” in Fig. 5.48 is to exploit the analogy between the defining equations for and Maxwell’s equations for / Evidently A depends on

> Consider a plane loop of wire that carries a steady current I ; we want to calculate the magnetic field at a point in the plane. We might as well take that point to be the origin (it could be inside or outside the loop). The shape of the w

> Magnetostatics treats the “source current” (the one that sets up the field) and the “recipient current” (the one that experiences the force) so asymmetrically that it

> (a) If it is uniformly distributed over the surface, what is the surface current density K ? (b) If it is distributed in such a way that the volume current density is inversely proportional to the distance from the axis, what is J(s)?

> Suppose you wanted to find the field of a circular loop (Ex. 5.6) at a point r that is not directly above the center (Fig. 5.60). You might as well choose your axes so that r lies in the yz plane at (0, y, z). The source poi

> Prove that the divergence of a curl is always zero. Check it for function va in Prob. 1.15.

> Use Eq. 5.41 to obtain the magnetic field on the axis of the rotating disk in Prob. 5.37(a). Show that the dipole field (Eq. 5.88), with the dipole moment you found in Prob. 5.37, is a good approximation if z ((R.

> The magnetic field on the axis of a circular current loop (Eq. 5.41) is far from uniform (it falls off sharply with increasing z). You can produce a more nearly uniform field by using two such loops a distance d apart (Fig. 5.59). (a) Find the field (B) as

> Use the Biot-Savart law (most conveniently in the form of Eq. 5.42 appropriate to surface currents) to find the field inside and outside an infinitely long solenoid of radius R, with n turns per unit length, c

> Consider the motion of a particle with mass m and electric charge qe in the field of a (hypothetical) stationary magnetic monopole qm at the origin: / (a) Find the acceleration of qe, expressing your answer in terms of q, qm , m, r (the p

> Calculate the magnetic force of attraction between the northern and southern hemispheres of a spinning charged spherical shell (Ex. 5.11). [Answer: (π/4)μ0σ 2ω2 R4.]

> A circularly symmetrical magnetic field (B depends only on the distance from the axis), pointing perpendicular to the page, occupies the shaded region in Fig. 5.58. If the total flux / is zero, show that a charged particle

> A plane wire loop of irregular shape is situated so that part of it is in a uniform magnetic field B (in Fig. 5.57 the field occupies the shaded region, and points perpendicular to the plane of the loop). The loop carries a

> A current I flows to the right through a rectangular bar of conducting material, in the presence of a uniform magnetic field B pointing out of the page (Fig. 5.56). (a) If the moving charges are positive, in which direction

> It may have occurred to you that since parallel currents attract, the current within a single wire should contract into a tiny concentrated stream along the axis. Yet in practice the current typically distributes itself quite uniformly over the wire. How

> Suppose that the magnetic field in some region has the form B = kz xˆ (where k is a constant). Find the force on a square loop (side a), lying in the yz plane and centered at the origin, if it carries a current I , flowing counterclockwise, when you look d

> Calculate the Laplacian of the following functions: (a) Ta = x 2 + 2xy + 3z + 4. (b) Tb = sin x sin y sin z. (c) Tc = e−5x sin 4y cos 3z. (d) v = x 2 xˆ + 3xz2 yˆ− 2xz zˆ.

> Analyze the motion of a particle (charge q, mass m) in the magnetic field of a long straight wire carrying a steady current I . (a) Is its kinetic energy conserved? (b) Find the force on the particle, in cylindrical coordinates, with I along the z axis. (

> I worked out the multipole expansion for the vector potential of a line current because that’s the most common type, and in some respects the easiest to handle. For a volume current J: (a) Write down the multipole expansion, analogous t

> (a) A phonograph record of radius R, carrying a uniform surface charge σ , is rotating at constant angular velocity ω. Find its magnetic dipole moment. (b) Find the magnetic dipole moment of the spinning spherical shell in Ex. 5.11. Show that for points

> Find the exact magnetic field a distance z above the center of a square loop of side w, carrying a current I . Verify that it reduces to the field of a dipole, with the appropriate dipole moment, when z ((w.

> A circular loop of wire, with radius R, lies in the xy plane (centered at the origin) and carries a current I running counterclockwise as viewed from the positive z axis. (a) What is its magnetic dipole moment? (b) What is the (approximate) magnetic field

> Show that the magnetic field of a dipole can be written in coordinate- free form:

> Prove Eq. 5.78, using Eqs. 5.63, 5.76, and 5.77. [Suggestion: I’d set up Cartesian coordinates at the surface, with z perpendicular to the surface and x parallel to the current.]

> (a) Check Eq. 5.76 for the configuration in Ex. 5.9. (b) Check Eqs. 5.77 and 5.78 for the configuration in Ex. 5.11.

> (a) Complete the proof of Theorem 2, Sect. 1.6.2. That is, show that any diver genceless vector field F can be written as the curl of a vector potential A. What you have to do is find Ax , Ay , and Az such that (i) ∂ Az/∂ y ∂ Ay /∂ z Fx ; (ii) ∂ Ax /∂ z

> Use the results of Ex. 5.11 to find the magnetic field inside a solid sphere, of uniform charge density ρ and radius R, that is rotating at a constant angular velocity ω.

> (a) Check product rule (iv) (by calculating each term separately) for the functions A = x xˆ + 2y yˆ + 3z zˆ; B = 3y xˆ − 2x yˆ. (b) Do the same for product rule (ii). (c) Do the same for rule (vi).

> In 1897, J. J. Thomson “discovered” the electron by measuring the charge-to-mass ratio of “cathode rays” (actually, streams of electrons, with charge q and mass m) as follows: (a) First he passed the beam through uniform crossed electric and magnetic fiel

> Suppose you want to define a magnetic scalar potential U (Eq. 5.67) in the vicinity of a current-carrying wire. First of all, you must stay away from the wire itself / but that’s not enough. Show, by applying AmpÃ

> (a) Check that Eq. 5.65 is consistent with Eq. 5.63, by applying the divergence. (b) Check that Eq. 5.65 is consistent with Eq. 5.47, by applying the curl. (c) Check that Eq. 5.65 is consistent with Eq. 5.64, by applying the Laplacian.

> (a) By whatever means you can think of (short of looking it up), find the vector potential a distance s from an infinite straight wire carrying a current I . Check that ∇ · A = 0 and ∇ × A = B. (b) Find the magnetic potential inside the wire, if it has rad

> If B is uniform, show that / works. That is, check that / Is this result unique, or are there other functions with the same divergence and curl?

> 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!)