91Ó°ÊÓ

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.

Suppose

$\mathbf{E}(\mathbf{r}\mathbf{,}\mathbf{t})=\frac{1}{\mathbf{4}\mathrm{Ï€}{\mathbf{Î\mu }}_0}\frac{q}{{\mathbf{r}}^2}\mathrm{θ}(\mathbf{r}−\mathrm{Ï\dots }\mathbf{t})\widehat{\mathbf{r}}$; $\mathbf{B}(\mathbf{r}\mathbf{,}\mathbf{t})=\mathbf{0}$

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

Question: Assuming that "Coulomb's law" for magnetic charges ( q_m) reads

$\mathbf{F}\mathbf{=}\frac{{\mathbf{μ}}_{\mathbf{0}}}{\mathbf{4}\mathbf{π}}\frac{{\mathbf{qm}}_{\mathbf{1}}{\mathbf{qm}}_{\mathbf{2}}}{{\mathbf{r}}^{\mathbf{2}}}\hat{\mathbf{r}}$

Work out the force law for a monopole moving with velocity through electric and magnetic fields E and B.

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 $\mathit{σ}$. 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 $\mathbf{Â\pm }\raisebox{1ex}{${\mathbf{V}}_{\mathbf{0}}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.\mathbf{}\mathbf{}\mathit{a}\mathit{t}\mathbf{}\mathit{Ï•}\mathbf{=}\mathbf{Â\pm }\mathbf{Ï€}$, and a steady current flows over the surface, as indicated in Fig. 7.51. According to Ohm's law, then,

$\mathbf{V}\left(a,\mathrm{Ï•}\right)\mathbf{=}\frac{{\mathbf{V}}_{\mathbf{0}}\mathbf{Ï•}}{\mathbf{2}\mathbf{Ï€}}\mathbf{,}\left(-\mathrm{Ï€}<\mathrm{Ï•}<+\mathrm{Ï€}\right)$

Figure 7.51

(a) Use separation of variables in cylindrical coordinates to determine $\mathbf{V}\left(s,\mathrm{Ï•}\right)$ 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

$\mathit{B}\mathbf{=}\frac{{\mathbf{μ}}_{\mathbf{0}}}{\mathbf{2}\mathbf{π}}\frac{\mathbf{I}}{\mathbf{s}}\widehat{\mathbf{ϕ}}$

The electric field inside the wire is uniform:

$\mathit{E}\mathbf{=}\frac{\mathbf{I}\mathbf{ÒÏ}}{\mathbf{Ï€}{\mathbf{a}}^{\mathbf{2}}}\hat{\mathbf{z}}$,

Where $\mathit{ÒÏ}$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) $\mathit{V}\mathbf{(}\mathit{a}\mathbf{,}\mathit{z}\mathbf{)}\mathbf{=}\frac{\mathbf{I}\mathbf{ÒÏ}\mathbf{z}}{\mathbf{Ï€}{\mathbf{a}}^{\mathbf{2}}}$ ; (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 $\mathit{σ}\mathbf{\left(}\mathit{z}\mathbf{\right)}$on the wire.

[Answer: $\mathit{V}\mathbf{=}\left(-Iz\mathrm{ÒÏ}/\mathrm{Ï€}{a}^2\right)$ This is a peculiar result, since E_s and $\mathit{σ}\mathbf{\left(}\mathit{z}\mathbf{\right)}$are not independent of localid="1658816847863" $\mathit{z}\mathbf{→}\mathit{a}\mathit{s}$ one would certainly expect for a truly infinite wire.]

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?

Suppose the conductivity of the material separating the cylinders in Ex. 7.2 is not uniform; specifically, $\mathbf{σ}\left(s\right)\mathbf{=}\mathbf{k}\mathbf{/}\mathbf{s}$, 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.]

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 .$F=ma=qE$]

Question: An infinite wire carrying a constant current in the direction is moving in the direction at a constant speed . Find the electric field, in the quasistatic approximation, at the instant the wire coincides with the axis (Fig. 7.54).