Show that the magnetic field of an infinite solenoid runs

Problem 1P 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. Is the charge positive or negative? In terms of a, d, B and q, find the momentum of the particle reference figure 5.8

Problem 3P 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 fields E and B (mutually perpendicular, and both of them perpendicular to the beam), and adjusted the electric field until he got zero deflection. What, then, was the speed of the particles (in terms of E and B)? (b) Then he turned off the electric field, and measured the radius of curvature, R, of the beam, as deflected by the magnetic field alone. In terms of E, B, and R, what is the charge-to-mass ratio (q/m) of the particles?

Problem 4P Suppose that the magnetic field in some region has the form (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 down the x axis.

Problem 62P thin glass rod of radius R and length L carries a uniform surface charge ?. It is set spinning about its axis, at an angular velocity ?. Find the magnetic field at a distance s ?R from the axis, in the xy plane (Fig. 5.66). [Hint: treat it as a stack of magnetic dipoles.]

Problem 2P Find and sketch the trajectory of the particle in Ex. 5.2, if it starts at the origin with velocity Reference example 5.2 Cycloid Motion. A more exotic trajectory occurs if we include a uniform electric field, at right angles to the magnetic one. Suppose, for instance, that B points in the x-direction, and E in the z-direction, as shown in Fig. 5.7. A positive charge is released from the origin; what path will it follow?

Problem 6P (a) A phonograph record carries a uniform density of “static electricity” ?. If it rotates at angular velocity ?, 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, is centered at the origin and spinning at a constant angular velocity ? about the z axis. Find the current density J at any point (r , ?, ?) within the sphere.

Problem 5P A current I flows down a wire of radius a. (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)?

Problem 7P For a configuration of charges and currents confined within a volume V, show that Reference equation 5.31 where p is the total dipole moment. [Hint: evaluate

Problem 9P Find the magnetic field at point P for each of the steady current configurations shown in Fig. 5.23. Reference figure 5.23

Problem 8P (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 steady current I. Again, let R be the distance from the center to any side. (c) Check that your formula reduces to the field at the center of a circular loop, in the limit n??. Reference figure 5.22

Problem 10P (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). Reference figure 5.24

Problem 11P 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 of ?1 and ?2 (it’s easiest that way). Consider the turns to be essentially circular, and use the result of Ex. 5.6. What is the field on the axis of an infinite solenoid (infinite in both directions)? Figure 5.25 Reference example 5.6 Find the magnetic field a distance z above the center of a circular loop of radius R, which carries a steady current I (Fig. 5.21).

Problem 13P Suppose you have two infinite straight line charges ?, 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? Work out the actual number. Is this a reasonable sort of speed?12 Figure 5.26

Problem 15P 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. Figure 5.41

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 solenoid, (ii) between them, and (iii) outside both. Fig. 5.42

Problem 17P A large parallel-plate capacitor with uniform surface charge ? on the upper plate and ?? 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 above and below them. (b) Find the magnetic force per unit area on the upper plate, including its direction. (c) At what speed v would the magnetic force balance the electrical force?15 Fig. 5.43.

Problem 18P 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 of such a coil? Show that the toroid field (Eq. 5.60) reduces to the solenoid field, when the radius of the donut is so large that a segment can be considered essentially straight. Equation 5.60

Problem 14P 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) The current is distributed in such a way that J is proportional to s, the distance from the axis. Figure 5.40

Problem 19P 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?

Problem 20P (a) Find the density ? 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 of 1 A. [Note: This is literally a snail’s pace. How, then, can you carry on a long distance telephone conversation?] (c) What is the force of attraction between two such wires, 1 cm apart? (d) If you could somehow remove the stationary positive charges, what would the electrical repulsion force be? How many times greater than the magnetic force is it?

Problem 21P 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?

Problem 22P 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.

What current density would produce the vector potential, \(\mathbf{A}=k \hat{\boldsymbol{\phi}}\) (where k is a constant), in cylindrical coordinates?

Problem 25P If B is uniform, show that A(r) = (r x B) works. That is, check that A = 0 and x A = B. Is this result unique, or are there other functions with the same divergence and curl?

Problem 26P (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 radius R and the current is uniformly distributed.

Problem 23P 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. Equation 5.66 Equation 5.37

Problem 27P Find the vector potential above and below the plane surface current in Ex.5.8. Reference: Ex.5.8. Find the magnetic field of an infinite uniform surface current K = K ˆx, flowing over the xy plane (Fig.5.33) (one Bl comes from the top segment and the other from the bottom), so B =(?0/2)K, or, more precisely, Notice that the field is independent of the distance from the plane, just like the electric field of a uniform surface charge

Problem 28P (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 Equation 5.65 Equation 5.63 Equation 5.47 Equation 5.64

Problem 29P 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 (there ? × B ? 0); but that’s not enough. Show, by applying Ampère’s law to a path that starts at a and circles the wire, returning to b (Fig. 5.47), that the scalar potential cannot be single-valued (that is, U(a)?U(b), even if they represent the same physical point). As an example, find the scalar potential for an infinite Figure 5.47 straight wire. (To avoid a multivalued potential, you must restrict yourself to simply connected regions that remain on one side or the other of every wire, never allowing you to go all the way around.) equation 5.67 B = ??U,

Problem 30P Use the results of Ex. 5.11 to find the magnetic field inside a solid sphere, of uniform charge density ? and radius R, that is rotating at a constant angular velocity ?.

Problem 31P (a) Complete the proof of Theorem 2, Sect. 1.6.2. That is, show that any divergenceless 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 ? ? Az/?x = Fy ; and (iii) ? Ay/?x ? ? Ax/?y = Fz . Here’s one way to do it: Pick Ax = 0, and solve (ii) and (iii) for Ay and Az . Note that the “constants of integration” are themselves functions of y and z—they’re constant only with respect to x. Now plug these expressions into (i), and use the fact that ? · F = 0 to obtain (b) By direct differentiation, check that the A you obtained in part (a) satisfies ? × A = F. Is A divergenceless? [This was a very asymmetrical construction, and it would be surprising if it were—although we know that there exists a vector whose curl is F and whose divergence is zero.] (c) As an example, let Calculate A, and confirm that ? × A = F. (For further discussion, see Prob. 5.53.) Reference prob 5.53 Reference Theorem 2 Divergence-less (or “solenoidal”) fields. The following conditions are equivalent:

Problem 33P 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.] Reference equation 5.78, Reference equation 5.63, Reference equation 5.76, Reference equation 5.77 A above = Abelow,

Problem 34P Show that the magnetic field of a dipole can be written in coordinate free form: Equation 5.89

Problem 35P 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 at points far from the origin? (c) Show that, for points on the z axis, your answer is consistent with the exact field (Ex. 5.6), when z ? R.

Problem 36P 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.

Problem 32P (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. Eq. 5.76 Eqs. 5.77 A above = Abelow, Eqs 5.78 Example 5.9 Find the magnetic field of a very long solenoid, consisting of n closely wound turns per unit length on a cylinder of radius R, each carrying a steady current I (Fig. 5.34). [The point of making the windings so close is that one can then pretend each turn is circular. If this troubles you (after all, there is a net current I in the direction of the solenoid’s axis, no matter how tight the winding), picture instead a sheet of aluminum foil wrapped around the cylinder, carrying the equivalent uniform surface current K = nI (Fig. 5.35). Or make a double winding, going up to one end and then—always in the same sense— going back down again, thereby eliminating the net longitudinal current. But, in truth, this is all unnecessary fastidiousness, for the field inside a solenoid is huge (relatively speaking), and the field of the longitudinal current is at most a tiny refinement.] Figure 5.34 and 5.35 Reference example 5.11 A spherical shell of radius R, carrying a uniform surface charge ?, is set spinning at angular velocity ?. Find the vector potential it produces at point r (Fig. 5.45).

Problem 37P (a) A phonograph record of radius R, carrying a uniform surface charge ?, is rotating at constant angular velocity ?. Find its magnetic dipole moment. (b) Find the magnetic dipole moment of the spinning spherical shell in Ex. 5.11. Show that for points r > R the potential is that of a perfect dipole.

Problem 38P 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 to Eq. 5.80. (b) Write down the monopole potential, and prove that it vanishes. (c) Using Eqs. 1.107 and 5.86, show that the dipole moment can be written Equation 1.107 Equation 5.86

Problem 40P 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 do you account for this? If the positive charges (density ?+) are “nailed down,” and the negative charges (density ??) move at speed v (and none of these depends on the distance from the axis), show that ?? = ??+? 2, where If the wire as a whole is neutral, where is the compensating charge located?22 [Notice that for typical velocities (see Prob. 5.20), the two charge densities are essentially unchanged by the current (since ? ? 1). In plasmas, however, where the positive charges are also free to move, this so-called pinch effect can be very significant.] Reference prob 5.20 (a) Find the density ? 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 of 1 A. [Note: This is literally a snail’s pace. How, then, can you carry on a long distance telephone conversation?] (c) What is the force of attraction between two such wires, 1 cm apart? (d) If you could somehow remove the stationary positive charges, what would the electrical repulsion force be? How many times greater than the magnetic force is it?

Problem 41P 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 are they deflected by the magnetic field? This deflection results in an accumulation of charge on the upper and lower surfaces of the bar, which in turn produces an electric force to counteract the magnetic one. Equilibrium occurs when the two exactly cancel. (This phenomenon is known as the Hall effect.) (b) Find the resulting potential difference (the Hall voltage) between the top and bottom of the bar, in terms of B, v (the speed of the charges), and the relevant dimensions of the bar.23 (c) How would your analysis change if the moving charges were negative? [The Hall effect is the classic way of determining the sign of the mobile charge carriers in a material.] Figure 5.56

Problem 43P 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(? B · da) is zero, show that a charged particle that starts out at the center will emerge from the field region on a radial path (provided it escapes at all). On the reverse trajectory, a particle fired at the center from outside will hit its target (if it has sufficient energy), though it may follow a weird route getting there. [Hint: Calculate the total angular momentum acquired by the particle, using the Lorentz force law.]

Problem 44P ce of attraction between the northern and southern hemispheres of a spinning charged spherical shell (Ex. 5.11). Calculate the magnetic force of attraction between the northern and southern hemispheres of a spinning charged spherical shell (Ex. 5.11).

Problem 45P 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 position of the particle), and v (its velocity). (b) Show that the speed v = |v| is a constant of the motion. (c) Show that the vector quantity is a constant of the motion. [Hint: differentiate it with respect to time, and prove—using the equation of motion from (a)—that the derivative is zero.] (d) Choosing spherical coordinates (r, ?, ?), with the polar (z) axis along Q, (i) calculate and show that ? is a constant of the motion (so qe moves on the surface of a cone—something Poincaré first discovered in 1896)24; (ii) calculate and show that the magnitude of Q is (iii) calculate show that (that is: determine the function f (r )). (f) Solve this equation for r(?).

Problem 46P 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, carrying a steady current I . Equation 5.42

Problem 47P 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 a function of z, and show that ? B/?z is zero at the point midway between them (z = 0). (b) If you pick d just right, the second derivative of B will also vanish at the midpoint. This arrangement is known as a Helmholtz coil; it’s a convenient way of producing relatively uniform fields in the laboratory. Determine d such that ?2B/?z2 = 0 at the midpoint, and find the resulting magnetic field at the center. Reference figure 5.59

Problem 42P 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 current I . Show that the net magnetic force on the loop is F = I Bw, where w is the chord subtended. Generalize this result to the case where the magnetic field region itself has an irregular shape. What is the direction of the force? Figure 5.57

Problem 48P 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. Eq. 5.41 Eq. 5.88, Reference Prob. 5.37. 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 a function of z, and show that ? B/?z is zero at the point midway between them (z = 0). (b) If you pick d just right, the second derivative of B will also vanish at the midpoint. This arrangement is known as a Helmholtz coil; it’s a convenient way of producing relatively uniform fields in the laboratory. Determine d such that ?2B/?z2 = 0 at the midpoint, and find the resulting magnetic field at the center. Reference figure 5.59

Problem 49P 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 point is (R cos ?? , R sin ? ?, 0), and ?? runs from 0 to 2??. Set up the integrals 25 from which you could calculate Bx , By, and Bz , and evaluate Bx explicitly Figure 5.60

Problem 50P 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 is by no means obvious that the magnetic force between two current loops is consistent with Newton’s third law. Show, starting with the Biot-Savart law (Eq. 5.34) and the Lorentz force law (Eq. 5.16), that the force on loop 2 due to loop 1 (Fig. 5.61) can be written as Figure 5.61 In this form, it is clear that F2 = ?F1, since

Problem 52P (a) One way to fill in the “missing link” in Fig. 5.48 is to exploit the analogy between the defining equations for A (viz. ? · A = 0, ? × A = B) and Maxwell’s equations for B (viz. ? · B = 0, ? × B = ?0J). Evidently A depends on B in exactly the same way that B depends on ?0J (to wit: the Biot-Savart law). Use this observation to write down the formula for A in terms of B. (b) The electrical analog to your result in (a) is Derive it, by exploiting the appropriate analogy. Figure 5.48

Problem 53P 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 case—uniform B (use the origin as your reference point). Is the result consistent with Prob. 5.25? You could cure this problem by throwing in a factor of 1 2 , but the flaw in this equation runs deeper. (b) Show that ?(B × dl) is not independent of path, by calculating As far as I know,28 the best one can do along these lines is the pair of equations [Equation (i) amounts to selecting a radial path for the integral in Eq. 2.21; equation (ii) constitutes a more “symmetrical” solution to Prob. 5.31.] (c) Use (ii) to find the vector potential for uniform B. (d) Use (ii) to find the vector potential of an infinite straight wire carrying a steady current I . Does (ii) automatically satisfy ? · A = 0? Prob. 5.25 If B is uniform, show that works. That is, check that ? · A = 0 and ? × A = B. Is this result unique, or are there other functions with the same divergence and curl? Prob. 5.31 a) Complete the proof of Theorem 2, Sect. 1.6.2. That is, show that any divergenceless 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 ? ? Az/?x = Fy ; and (iii) ? Ay/?x ? ? Ax/?y = Fz . Here’s one way to do it: Pick Ax = 0, and solve (ii) and (iii) for Ay and Az . Note that the “constants of integration” are themselves functions of y and z—they’re constant only with respect to x. Now plug these expressions into (i), and use the fact that ? · F = 0 to obtain (b) By direct differentiation, check that the A you obtained in part (a) satisfies ? × A = F. Is A divergenceless? [This was a very asymmetrical construction, and it would be surprising if it were—although we know that there exists a vector whose curl is F and whose divergence is zero.] (c) As an example, let Calculate A, and confirm that ? × A = F. (For further discussion, see Prob. 5.53.) Reference prob 5.53 Reference Theorem 2 Divergence-less (or “solenoidal”) fields. The following conditions are equivalent:

Problem 55P 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 formula for W (as an integral over B), which holds when B ? 0 at?. (b) Determine W for the case of a uniform magnetic field B. [Hint: see Prob. 5.25.] (c) Find W inside and outside an infinite solenoid. [Hint: see Ex. 5.12.] Prob. 5.25 Ex. 5.12 Find the vector potential of an infinite solenoid with n turns per unit length, radius R, and current I .

Problem 56P 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 V. [Hint: First use the divergence theorem to show that for arbitrary vector functions U and V.]

Problem 57P A magnetic dipole m 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 lines, inside and out.

Problem 58P 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 the gyromagnetic ratio for a uniform spinning sphere? [This requires no new calculation; simply decompose the sphere into infinitesimal rings, and apply the result of part (a).] (c) According to quantum mechanics, the angular momentum of a spinning electron is , where ¯h is Planck’s constant. What, then, is the electron’s magnetic dipole moment, in A · m2? [This semiclassical value is actually off by a factor of almost exactly 2. Dirac’s relativistic electron theory got the 2 right, and Feynman, Schwinger, and Tomonaga later calculated tiny further corrections. The determination of the electron’s magnetic dipole moment remains the finest achievement of quantum electrodynamics, and exhibits perhaps the most stunningly precise agreement between theory and experiment in all of physics. Incidentally, the quantity where e is the charge of the electron and m is its mass, is called the Bohr magneton.]

Problem 60P A uniformly charged solid sphere of radius R carries a total charge Q, and is set spinning with angular velocity ? 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. 5.59). (c) Find the approximate vector potential at a point (r, ?) where r ?R. (d) Find the exact potential at a point (r, ?) outside the sphere, and check that it is consistent with (c). [Hint: refer to Ex. 5.11.] (e) Find the magnetic field at a point (r, ?) inside the sphere (Prob. 5.30), and check that it is consistent with (b). Reference Ex. 5.11.] A spherical shell of radius R, carrying a uniform surface charge ?, is set spinning at angular velocity ?. Find the vector potential it produces at point r (Fig. 5.45).

Problem 54P (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 doing the same for the interior of a solid spinning sphere. [Hint: If you solved Prob. 5.30, you already know the field; set it equal to ??U, and solve for U. What’s the trouble?] Eqs. 5.69 Eqs. 5.87 Prob. 5.30, Use the results of Ex. 5.11 to find the magnetic field inside a solid sphere, of uniform charge density ? and radius R, that is rotating at a constant angular velocity ?.

Problem 59P (a) Prove that the average magnetic field, over a sphere of radius R, due to steady currents inside the sphere, is where m is the total dipole moment of the sphere. Contrast the electrostatic result, Eq. 3.105. [This is tough, so I’ll give you a start: Write B as (? × A), and apply Prob. 1.61(b). Now put in Eq. 5.65, and do the surface integral first, showing that (see Fig. 5.65). Use Eq. 5.90, if you like.] (b) Show that the average magnetic field due to steady currents outside the sphere is the same as the field they produce at the center. Equation 5.90 Fig. 5.65. Reference Prob. 1.61(b). 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 61P 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 discrepancy, and indicate how Eq. 5.89 can be corrected to resolve the ambiguity at r = 0. (If you get stuck, refer to Prob. 3.48.) Evidently the true field of a magnetic dipole is29 Compare the electrostatic analog, Eq. 3.106. Reference equation 5.88 Reference equation 5.89 Prob. 5.59 (a) Prove that the average magnetic field, over a sphere of radius R, due to steady currents inside the sphere, is where m is the total dipole moment of the sphere. Contrast the electrostatic result, Eq. 3.105. [This is tough, so I’ll give you a start: Write B as (? × A), and apply Prob. 1.61(b). Now put in Eq. 5.65, and do the surface integral first, showing that (see Fig. 5.65). Use Eq. 5.90, if you like.] (b) Show that the average magnetic field due to steady currents outside the sphere is the same as the field they produce at the center. Equation 5.90 Fig. 5.65. Reference Prob. 1.61(b). 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: Reference prob 3.48 (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