Electrodynamics

A long cylindrical shell of radius $R$carries a uniform surface charge on ${\sigma }_0$ the upper half and an opposite charge $-{\sigma }_0$ on the lower half (Fig. 3.40). Find the electric potential inside and outside the cylinder.

Two concentric metal spherical shells, of radius a and b, respectively, are separated by weakly conducting material of conductivity$\mathit{\sigma }$(Fig. 7 .4a). 

(a) If they are maintained at a potential difference V, what current flows from one to the other? 

(b) What is the resistance between the shells? 

(c) Notice that if b>>a the outer radius (b) is irrelevant. How do you account for that? Exploit this observation to determine the current flowing between two metal spheres, each of radius a, immersed deep in the sea and held quite far apart (Fig. 7 .4b ), if the potential difference between them is V. (This arrangement can be used to measure the conductivity of sea water.)

Question: A capacitor C has been charged up to potential ${\mathit{V}}_{\mathbf{0}}$ at time $\mathbf{ }\mathit{t}\mathbf{=}\mathbf{0}$ , it is connected to a resistor R, and begins to discharge (Fig. 7.5a).

(a) Determine the charge on the capacitor as a function of time, $\mathit{Q}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ What is the current through the resistor, $\mathit{l}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ ? 

(b) What was the original energy stored in the capacitor (Eq. 2.55)? By integrating Eq. 7.7, confirm that the heat delivered to the resistor is equal to the energy lost by the capacitor. 

Now imagine charging up the capacitor, by connecting it (and the resistor) to a battery of voltage ${\mathit{V}}_{\mathbf{0}}$ , at time t = 0 (Fig. 7.5b).

(c) Again, determine $\mathit{Q}\mathbf{\left(}\mathit{t}\mathbf{\right)}$and $\mathit{l}\mathbf{\left(}\mathit{t}\mathbf{\right)}$.

(d) Find the total energy output of the battery $\left(\int Vldt\right)$. Determine the heat delivered to the resistor. What is the final energy stored in the capacitor? What fraction of the work done by the battery shows up as energy in the capacitor? [Notice that the answer is independent of R!]

Suppose 

 $\mathbf{E}(\mathbf{r}\mathbf{,}\mathbf{t})=\frac{1}{\mathbf{4}\pi {\mathbf{\epsilon }}_0}\frac{q}{{\mathbf{r}}^2}\theta (\mathbf{r}-\upsilon \mathbf{t})\widehat{\mathbf{r}}$;  $\mathbf{B}(\mathbf{r}\mathbf{,}\mathbf{t})=\mathbf{0}$

(The theta function is defined in Prob. 1.46b). Show that these fields satisfy all of Maxwell's equations, and determine  $\rho$ and $\mathbf{J}$. Describe the physical situation that gives rise to these fields.

(a) Two metal objects are embedded in weakly conducting material of conductivity $\mathit{\sigma }$(Fig. 7 .6). Show that the resistance between them is related to the capacitance of the arrangement by

                                                                                     $\mathbf{R}\mathbf{=}\frac{{\mathbf{\in }}_{\mathbf{0}}}{\mathbf{\sigma }\mathbf{C}}$

(b) Suppose you connected a battery between 1 and 2, and charged them up to a potential difference ${\mathit{V}}_{\mathbf{0}}$ . If you then disconnect the battery, the charge will gradually leak off. Show that $\mathbf{V}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{V}}_{\mathbf{0}}{\mathbf{e}}^{\mathbf{-}\mathbf{t}\mathbf{/}\mathbf{r}}$ , and find the time constant,$\mathit{\tau }$, in terms of  ${\mathbf{\in }}_{\mathbf{0}}$and .$\mathit{\sigma }$

Suppose the conductivity of the material separating the cylinders in Ex. 7.2 is not uniform; specifically, $\mathbf{\sigma }\left(s\right)\mathbf{=}\mathbf{k}\mathbf{/}\mathbf{s}$, for some constant . Find the resistance between the cylinders. [Hint: Because a is a function of position, Eq. 7.5 does not hold, the charge density is not zero in the resistive medium, and E does not go like 1/s. But we do know that for steady currents is the same across each cylindrical surface. Take it from there.]

A battery of emf $\epsilon$and internal resistance r is hooked up to a variable "load" resistance,R . If you want to deliver the maximum possible power to the load, what resistance R should you choose? (You can't change e and R , of course.)

A rectangular loop of wire is situated so that one end (height h) is between the plates of a parallel-plate capacitor (Fig. 7.9), oriented parallel to the field E. The other end is way outside, where the field is essentially zero. What is the emf in this loop? If the total resistance is R, what current flows? Explain. [Warning: This is a trick question, so be careful; if you have invented a perpetual motion machine, there's probably something wrong with it.]

A metal bar of mass m slides frictionlessly on two parallel conducting rails a distance l apart (Fig. 7 .17). A resistor R is connected across the rails, and a uniform magnetic field B, pointing into the page, fills the entire region.

(a) If the bar moves to the right at speed V, what is the current in the resistor? In what direction does it flow? 

(b) What is the magnetic force on the bar? In what direction? 

(c) If the bar starts out with speed${\mathbf{V}}_{\mathbf{0}}$at time t=0, and is left to slide, what is its speed at a later time t?

(d) The initial kinetic energy of the bar was, of course,$\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{mv}}^{\mathbf{2}}$ Check that the energy delivered to the resistor is exactly $\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{mv}}^{\mathbf{2}}$.

A square loop of wire (side a) lies on a table, a distance s from a very long straight wire, which carries a current I, as shown in Fig. 7.18.

(a) Find the flux of B through the loop. 

(b) If someone now pulls the loop directly away from the wire, at speed , V what emf is generated? In what direction (clockwise or counter clockwise) does the current flow? 

(c) What if the loop is pulled to the right at speed V ?

An infinite number of different surfaces can be fit to a given boundary line, and yet, in defining the magnetic flux through a loop, $\varphi =\int B.da$ da, I never specified the particular surface to be used. Justify this apparent oversight.

A square loop (side a) is mounted on a vertical shaft and rotated at angular velocity $\omega$  (Fig. 7.19). A uniform magnetic field  B points to the right. Find the$\epsilon \left(t\right)$ for this alternating current generator.

A square loop is cut out of a thick sheet of aluminum. It is then placed so that the top portion is in a uniform magnetic field , B and is allowed to fall under gravity (Fig. 7 .20). (In the diagram, shading indicates the field region; points into the page.) If the magnetic field is 1 T (a pretty standard laboratory field), find the terminal velocity of the loop (in m/s ). Find the velocity of the loop as a function of time. How long does it take (in seconds) to reach, say, $90\%$  of the terminal velocity? What would happen if you cut a tiny slit in the ring, breaking the circuit? [Note: The dimensions of the loop cancel out; determine the actual numbers, in the units indicated.]

A long solenoid, of radius a, is driven by an alternating current, so that the field inside is sinusoidal:$B\left(t\right)=B_0\cos \left(\omega t\right)\hat{z}$ . A circular loop of wire, of radius $a/2$ and resistance R , is placed inside the solenoid, and coaxial with it. Find the current induced in the loop, as a function of time.

A square loop of wire, with sides of length a , lies in the first quadrant of the xy plane, with one comer at the origin. In this region, there is a nonuniform time-dependent magnetic field $\mathbf{B}\mathbf{(}\mathbf{y}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}{\mathbf{ky}}^{\mathbf{3}}{\mathbf{t}}^{\mathbf{2}}\hat{\mathbf{z}}$  (where k is a constant). Find the emf induced in the loop.

As a lecture demonstration a short cylindrical bar magnet is dropped down a vertical aluminum pipe of slightly larger diameter, about 2 meters long. It takes several seconds to emerge at the bottom, whereas an otherwise identical piece of unmagnetized iron makes the trip in a fraction of a second. Explain why the magnet falls more slowly.

A long solenoid with radius a and n turns per unit length carries a time-dependent current $\mathbf{l}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$ in the $\hat{\mathbf{\varphi }}$ direction. Find the electric field (magnitude and direction) at a distance s from the axis (both inside and outside the solenoid), in the quasistatic approximation.

An alternating current $\mathbf{l}\mathbf{=}{\mathbf{l}}_{\mathbf{0}}\mathbf{cos}\left(\mathrm{wt}\right)$ flows down a long straight wire, and returns along a coaxial conducting tube of radius a.

(a) In what direction does the induced electric field point (radial, circumferential, or longitudinal)? 

(b) Assuming that the field goes to zero as $\mathbf{s}\mathbf{\to }\mathbf{\infty }$, find $\mathbf{E}\mathbf{=}\mathbf{(}\mathbf{s}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{.}$

A long solenoid of radius a, carrying n turns per unit length, is looped by a wire with resistance R, as shown in Fig. 7.28.

(a) If the current in the solenoid is increasing at a constant rate $\mathbf{(}\mathbf{dl}\mathbf{/}\mathbf{dt}\mathbf{=}\mathbf{k}\mathbf{)}\mathbf{,}$, what current flows in the loop, and which way (left or right) does it pass through the resistor?

(b) If the current $\mathit{l}$ in the solenoid is constant but the solenoid is pulled out of the loop (toward the left, to a place far from the loop), what total charge passes through the resistor?

 Imagine a uniform magnetic field, pointing in the $\mathbf{z}$ direction and filling all space $\left(B=B_{0}z\right)$. A positive charge is at rest, at the origin. Now somebody turns off the magnetic field, thereby inducing an electric field. In what direction does the charge move?

Where is $\frac{\mathbf{\partial }\mathbf{B}}{\mathbf{\partial }\mathbf{t}}$  nonzero in Figure 7.21(b)? Exploit the analogy between Faraday's law and Ampere's law to sketch (qualitatively) the electric field.

A square loop, side a , resistance R , lies a distance from an infinite straight wire that carries current l (Fig. 7.29). Now someone cuts the wire, so l drops to zero. In what direction does the induced current in the square loop flow, and what total charge passes a given point in the loop during the time this current flows? If you don't like the scissors model, turn the current down gradually:

$\mathbf{I}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{\{}\begin{array}{l}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{t}\mathbf{)}\mathbf{I}\\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{0}\end{array}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\begin{array}{c}\mathbf{ }\mathbf{for}\mathbf{ }\mathbf{0}\mathbf{\le }\mathbf{t}\mathbf{\le }\mathbf{1}\mathbf{/}\mathbf{a}\\ \mathbf{for}\mathbf{ }\mathbf{t}\mathbf{>}\mathbf{/}\mathbf{a}\end{array}$

A toroidal coil has a rectangular cross section, with inner radius a , outer radius $\mathbf{a}\mathbf{+}\mathbf{w}$ , and height h . It carries a total of N tightly wound turns, and the current is increasing at a constant rate $\mathbf{(}\mathbf{dl}\mathbf{/}\mathbf{dt}\mathbf{=}\mathbf{k}\mathbf{)}$. If w and h are both much less than a , find the electric field at a point z above the center of the toroid. [Hint: Exploit the analogy between Faraday fields and magnetostatic fields, and refer to Ex. 5.6.]

A square loop of wire, of side $\mathbf{a}$, lies midway between two long wires,$\mathbf{3}\mathbf{a}$ apart, and in the same plane. (Actually, the long wires are sides of a large rectangular loop, but the short ends are so far away that they can be neglected.) A clockwise current  $\mathbf{I}$in the square loop is gradually increasing: $\raisebox{1ex}{$\mathbf{d}\mathbf{l}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{d}\mathbf{t}$}\right.\mathbf{=}\mathbf{k}$ (a constant). Find the emf induced in the big loop. Which way will the induced current flow?

A capacitor $\mathbf{C}$   is charged up to a voltage  $\mathbf{V}$ and connected to an inductor $\mathbf{L}$, as shown schematically in Fig. 7.39. At time $t=\mathbf{0}$, the switch  $\mathbf{S}$ is closed. Find the current in the circuit as a function of time. How does your answer change if a resistor R is included in series with  $\mathbf{C}$  and $\mathbf{L}$ ?

A small loop of wire (radius a) is held a distance z above the center of a large loop (radius b ), as shown in Fig. 7.37. The planes of the two loops are parallel, and perpendicular to the common axis. 

(a) Suppose current I flows in the big loop. Find the flux through the little loop. (The little loop is so small that you may consider the field of the big loop to be essentially constant.) 

(b) Suppose current I flows in the little loop. Find the flux through the big loop. (The little loop is so small that you may treat it as a magnetic dipole.) 

(c) Find the mutual inductances, and confirm that ${\mathbf{M}}_{\mathbf{12}}\mathbf{=}{\mathbf{M}}_{\mathbf{21}}$ ·

Find the self-inductance per unit length of a long solenoid, of radius R , carrying n turns per unit length.

Try to compute the self-inductance of the "hairpin" loop shown in Fig. 7.38. (Neglect the contribution from the ends; most of the flux comes from the long straight section.) You'll run into a snag that is characteristic of many self-inductance calculations. To get a definite answer, assume the wire has a tiny radius$\mathbf{\in }$, and ignore any flux through the wire itself.

An alternating current I(t)$\mathbf{=}{\mathit{I}}_{\mathbf{0}}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathit{\omega }\mathit{t}\mathbf{\right)}$ (amplitude 0.5 A, frequency  ) flows down a straight wire, which runs along the axis of a toroidal coil with rectangular cross section (inner radius 1cm , outer radius 2 cm , height 1 cm, 1000 turns). The coil is connected to a 500 $\mathbf{\Omega }$ resistor. 

(a) In the quasistatic approximation, what emf is induced in the toroid? Find the current, ${\mathit{I}}_{\mathbf{R}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ , in the resistor. 

(b) Calculate the back emf in the coil, due to the current ${\mathit{I}}_{\mathbf{R}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ . What is the ratio of the amplitudes of this back emf and the "direct" emf in (a)?

Find the energy stored in a section of length $\mathit{l}$ of a long solenoid (radius $\mathit{R}$ , current $\mathit{I}$ ,  $\mathit{n}$  turns per unit length), 

(a) using Eq. 7.30 (you found $\mathit{L}$  in Prob. 7.24); 

(b) using Eq. 7.31 (we worked out $\mathit{A}$  in Ex. 5.12); 

(c) using Eq. 7.35; 

(d)  using Eq. 7.34 (take as your volume the cylindrical tube from radius $\mathbf{a}\mathbf{<}\mathbf{R}$  out to radius $\mathbf{b}\mathbf{>}\mathbf{R}$ ). 

Question: A long cable carries current in one direction uniformly distributed over its (circular) cross section. The current returns along the surface (there is a very thin insulating sheath separating the currents). Find the self-inductance per unit length.

Question: Calculate the energy stored in the toroidal coil of Ex. 7.11, by applying Eq. 7.35. Use the answer to check Eq. 7.28.

Suppose the circuit in Fig. 7.41 has been connected for a long time when suddenly, at time $\mathbf{t}\mathbf{=}\mathbf{0}$, switch S is thrown from A to B, bypassing the battery.

Notice the similarity to Eq. 7.28-in a sense, the rectangular toroid is a short coaxial cable, turned on its side.

(a) What is the current at any subsequent time t? 

(b) What is the total energy delivered to the resistor? 

(c) Show that this is equal to the energy originally stored in the inductor.

Two tiny wire loops, with areas and , are situated a displacement apart (Fig. 7 .42). FIGURE7.42 

(a) Find their mutual inductance. [Hint: Treat them as magnetic dipoles, and use Eq. 5.88.] Is your formula consistent with Eq. 7.24? 

(b) Suppose a current is flowing in loop 1, and we propose to turn on a current in loop 2. How much work must be done, against the mutually induced emf, to keep the current flowing in loop 1? In light of this result, comment on Eq. 6.35.

An infinite cylinder of radius R carries a uniform surface charge $\mathbf{\sigma }$. We propose to set it spinning about its axis, at a final angular velocity $\mathbf{\omega }$. How much work will this take, per unit length? Do it two ways, and compare your answers:

(a) Find the magnetic field and the induced electric field (in the quasistatic approximation), inside and outside the cylinder, in terms of $\mathbf{\omega }\mathbf{,}\mathbf{\omega }\mathbf{,}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{s}$(the distance from the axis). Calculate the torque you must exert, and from that obtain the work done per unit length $\left(W=\int \mathrm{Nd\varphi }\right)$ .

(b) Use Eq. 7.35 to determine the energy stored in the resulting magnetic field.

Question: A fat wire, radius  a, carries a constant current I , uniformly distributed over its cross section. A narrow gap in the wire, of width w << a, forms a parallel-plate capacitor, as shown in Fig. 7.45. Find the magnetic field in the gap, at a distance s < a  from the axis.

Question: The preceding problem was an artificial model for the charging capacitor, designed to avoid complications associated with the current spreading out over the surface of the plates. For a more realistic model, imagine thin wires that connect to the centers of the plates (Fig. 7.46a). Again, the current  I is constant, the radius of the capacitor is a, and the separation of the plates is  w << a. Assume that the current flows out over the plates in such a way that the surface charge is uniform, at any given time, and is zero at t = 0. 

(a) Find the electric field between the plates, as a function of t. 

(b) Find the displacement current through a circle of radius   in the plane mid-way between the plates. Using this circle as your "Amperian loop," and the flat surface that spans it, find the magnetic field at a distance s from the axis.

                                                            Figure 7.46

(c) Repeat part (b), but this time uses the cylindrical surface in Fig. 7.46(b), which is open at the right end and extends to the left through the plate and terminates outside the capacitor. Notice that the displacement current through this surface is zero, and there are two contributions to I_enc. 

Refer to Prob. 7.16, to which the correct answer was

$\mathit{E}\mathit{(}\mathit{s}\mathit{,}\mathit{t}\mathit{)}\mathit{=}\frac{{\mathit{\mu }}_{\mathit{0}}{\mathit{I}}_{\mathit{0}}\mathit{\omega }}{\mathit{2}\mathit{\tau }\mathit{\tau }}\mathit{s}\mathit{i}\mathit{n}\mathit{\left(}\mathit{\omega }\mathit{t}\mathit{\right)}\mathit{I}\mathit{n}\mathit{\left(}\frac{\mathit{a}}{\mathit{s}}\mathit{\right)}\hat{\mathit{z}}$

(a) Find the displacement current density ${\mathit{J}}_{\mathbf{d}}$·

(b) Integrate it to get the total displacement current,

${\mathit{I}}_{\mathbf{d}}\mathbf{=}\mathbf{\int }{\mathit{J}}_{\mathbf{d}}\mathbf{.}\mathit{d}\mathit{a}$

Compare ${\mathit{I}}_{\mathbf{d}}$ and I. (What's their ratio?) If the outer cylinder were, say, 2 mm in diameter, how high would the frequency have to be, for ${\mathit{I}}_{\mathbf{d}}$ to be 1% of I ? [This problem is designed to indicate why Faraday never discovered displacement currents, and why it is ordinarily safe to ignore them unless the frequency is extremely high.]

Suppose a magnetic monopole ${\mathbf{q}}_{\mathbf{m}}$ passes through a resistanceless loop of wire with self-inductance L . What current is induced in the loop?

Sea water at frequency $\mathbf{v}\mathbf{=}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{Hz}$ has permittivity $\mathbf{\in }\mathbf{=}\mathbf{81}\mathbf{ }{\mathbf{\in }}_{\mathbf{0}}$ , permeability $\mathbf{\mu }\mathbf{=}{\mathbf{\mu }}_{\mathbf{0}}$ , and resistivity $\mathbf{\rho }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{23}\mathbf{\Omega }\mathbf{.}\mathbf{m}$ . What is the ratio of conduction current to displacement current? [Hint: Consider a parallel-plate capacitor immersed in sea water and driven by a voltage ${\mathbf{V}}_{\mathbf{0}}\mathbf{cos}\mathbf{\left(}\mathbf{2}\mathbf{\pi vt}\mathbf{\right)}$ .]

Two long, straight copper pipes, each of radius a, are held a distance 2d apart (see Fig. 7.50). One is at potential V₀, the other at -V₀. The space surrounding the pipes is filled with weakly conducting material of conductivity $\mathit{\sigma }$. Find the current per unit length that flows from one pipe to the other. [Hint: Refer to Prob. 3.12.]

Question: A rare case in which the electrostatic field  E for a circuit can actually be calculated is the following:  Imagine an infinitely long cylindrical sheet, of uniform resistivity and radius a . A slot (corresponding to the battery) is maintained at  $\mathbf{\pm }\raisebox{1ex}{${\mathbf{V}}_{\mathbf{0}}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.\mathbf{ }\mathbf{ }\mathit{a}\mathit{t}\mathbf{ }\mathit{\varphi }\mathbf{=}\mathbf{\pm }\mathbf{\pi }$, and a steady current flows over the surface, as indicated in Fig. 7.51. According to Ohm's law, then,

 $\mathbf{V}\left(a,\varphi \right)\mathbf{=}\frac{{\mathbf{V}}_{\mathbf{0}}\mathbf{\varphi }}{\mathbf{2}\mathbf{\pi }}\mathbf{,}\left(-\pi <\varphi <+\pi \right)$

                                             Figure 7.51

(a) Use separation of variables in cylindrical coordinates to determine  $\mathbf{V}\left(s,\varphi \right)$  inside and outside the cylinder. 

(b) Find the surface charge density on the cylinder. 

The magnetic field outside a long straight wire carrying a steady current I is

 $\mathit{B}\mathbf{=}\frac{{\mathbf{\mu }}_{\mathbf{0}}}{\mathbf{2}\mathbf{\pi }}\frac{\mathbf{I}}{\mathbf{s}}\hat{\mathbf{\varphi }}$

The electric field inside the wire is uniform:

$\mathit{E}\mathbf{=}\frac{\mathbf{I}\mathbf{\rho }}{\mathbf{\pi }{\mathbf{a}}^{\mathbf{2}}}\hat{\mathbf{z}}$ ,

Where $\mathit{\rho }$ is the resistivity and a is the radius (see Exs. 7.1 and 7 .3). Question: What is the electric field outside the wire? 29 The answer depends on how you complete the circuit. Suppose the current returns along a perfectly conducting grounded coaxial cylinder of radius b (Fig. 7.52). In the region a < s < b, the potential V (s, z) satisfies Laplace's equation, with the boundary conditions

(i) $\mathit{V}\mathbf{(}\mathit{a}\mathbf{,}\mathit{z}\mathbf{)}\mathbf{=}\frac{\mathbf{I}\mathbf{\rho }\mathbf{z}}{\mathbf{\pi }{\mathbf{a}}^{\mathbf{2}}}$  ; (ii) $\mathit{V}\mathbf{(}\mathit{b}\mathbf{,}\mathit{z}\mathbf{)}\mathbf{=}\mathbf{0}$ 

Figure 7.52

This does not suffice to determine the answer-we still need to specify boundary conditions at the two ends (though for a long wire it shouldn't matter much). In the literature, it is customary to sweep this ambiguity under the rug by simply stipulating that V (s,z) is proportional to V (s,z) = zf (s) . On this assumption: 

(a) Determine (s). 

(b) E (s,z). 

(c) Calculate the surface charge density $\mathit{\sigma }\mathbf{\left(}\mathit{z}\mathbf{\right)}$ on the wire.

[Answer: $\mathit{V}\mathbf{=}\left(-Iz\rho /\pi a^2\right)$ This is a peculiar result, since E_s and $\mathit{\sigma }\mathbf{\left(}\mathit{z}\mathbf{\right)}$ are not independent of $\mathit{z}\mathbf{\to }\mathit{a}\mathit{s}$ one would certainly expect for a truly infinite wire.]

In a perfect conductor, the conductivity is infinite, so $\mathbf{E}\mathbf{=}\mathbf{0}$ (Eq. 7.3), and any net charge resides on the surface (just as it does for an imperfect conductor, in electrostatics).

(a) Show that the magnetic field is constant $\left(\frac{\partial B}{\partial t}=0\right)$, inside a perfect conductor.

(b) Show that the magnetic flux through a perfectly conducting loop is constant.

A superconductor is a perfect conductor with the additional property that the (constant) B inside is in fact zero. (This "flux exclusion" is known as the Meissner effect.)

(c) Show that the current in a superconductor is confined to the surface.

(d) Superconductivity is lost above a certain critical temperature $\mathbf{\left(}{\mathbf{T}}_{\mathbf{c}}\mathbf{\right)}$, which varies from one material to another. Suppose you had a sphere (radius ) above its critical temperature, and you held it in a uniform magnetic field ${\mathbf{B}}_{\mathbf{0}}\hat{\mathbf{z}}$ while cooling it below ${\mathbf{T}}_{\mathbf{c}}$ . Find the induced surface current density K, as a function of the polar angle $\mathit{\theta }$.

If a magnetic dipole levitating above an infinite superconducting plane (Pro b. 7 .45) is free to rotate, what orientation will it adopt, and how high above the surface will it float?

Refer to Prob. 7.11 (and use the result of Prob. 5.42): How long does is take a falling circular ring (radius $a$, mass $m$, resistance $R$) to cross the bottom of the magnetic field $B$, at its (changing) terminal velocity?

A familiar demonstration of superconductivity (Prob. 7.44) is the levitation of a magnet over a piece of superconducting material. This phenomenon can be analyzed using the method of images. Treat the magnet as a perfect dipole , m a height z above the origin (and constrained to point in the z direction), and pretend that the superconductor occupies the entire half-space below the xy plane. Because of the Meissner effect, B = 0 for $\mathit{Z}\mathbf{\le }\mathbf{0}$, and since B is divergenceless, the normal ( z) component is continuous, so ${\mathbf{B}}_{\mathbf{z}}\mathbf{=}\mathbf{0}$just above the surface. This boundary condition is met by the image configuration in which an identical dipole is placed at - z , as a stand-in for the superconductor; the two arrangements therefore produce the same magnetic field in the region z>0.

(a) Which way should the image dipole point (+ z or -z)? 

(b) Find the force on the magnet due to the induced currents in the superconductor (which is to say, the force due to the image dipole). Set it equal to Mg (where M is the mass of the magnet) to determine the height h at which the magnet will "float." [Hint: Refer to Prob. 6.3.] 

(c) The induced current on the surface of the superconductor ( xy the plane) can be determined from the boundary condition on the tangential component of B (Eq. 5.76): $\mathbf{B}\mathbf{=}{\mathbf{\mu }}_{\mathbf{0}}\mathbf{(}\mathbf{K}\mathbf{\times }\hat{\mathbf{z}}\mathbf{)}$. Using the field you get from the image configuration, show that

$\mathbf{K}\mathbf{=}\mathbf{-}\frac{\mathbf{3}\mathbf{mrh}}{\mathbf{2}\mathbf{\pi }{\mathbf{(}{\mathbf{r}}^{\mathbf{2}}\mathbf{+}{\mathbf{h}}^{\mathbf{2}}\mathbf{)}}^{\raisebox{1ex}{$\mathbf{5}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.}}\hat{\mathbf{\varphi }}$

where r is the distance from the origin.

A perfectly conducting spherical shell of radius rotates about the z axis with angular velocity $\mathbf{\omega }$, in a uniform magnetic field $\mathbf{B}\mathbf{=}{\mathbf{B}}_{\mathbf{0}}\hat{\mathbf{Z}}$ . Calculate the emf developed between the “north pole” and the equator. Answer:$\left[\frac{1}{2}{B}_0{\mathrm{\omega \alpha }}^2\right]$.

(a) Referring to Prob. 5.52(a) and Eq. 7.18, show that 

                                     $E=-\frac{\partial A}{\partial t}$                     (7.66) for Faraday-induced electric fields. Check this result by taking the divergence and curl of both sides. 

(b) A spherical shell of radius $R$ carries a uniform surface charge $\sigma$. It spins about a fixed axis at an angular velocity $\omega \left(t\right)$ that changes slowly with time. Find the electric field inside and outside the sphere. [Hint: There are two contributions here: the Coulomb field due to the charge, and the Faraday field due to the changing $\mathit{B}$. Refer to Ex. 5.11.]

Electrons undergoing cyclotron motion can be sped up by increasing the magnetic field; the accompanying electric field will impart tangential acceleration. This is the principle of the betatron. One would like to keep the radius of the orbit constant during the process. Show that this can be achieved by designing a magnet such that the average field over the area of the orbit is twice the field at the circumference (Fig. 7.53). Assume the electrons start from rest in zero field, and that the apparatus is symmetric about the center of the orbit. (Assume also that the electron velocity remains well below the speed of light, so that nonrelativistic mechanics applies.) [Hint: Differentiate Eq. 5.3 with respect to time, and use .$F=ma=qE$]