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Chapter 7: Q9P (page 311)
An infinite number of different surfaces can be fit to a given boundary line, and yet, in defining the magnetic flux through a loop, da, I never specified the particular surface to be used. Justify this apparent oversight.
It is necessary to specify particular surface oversight.
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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 , what current flows in the loop, and which way (left or right) does it pass through the resistor?
(b) If the currentin 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?
In a perfect conductor, the conductivity is infinite, so (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 , 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 , 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 while cooling it below . Find the induced surface current density K, as a function of the polar angle.
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 .]
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.
Where is nonzero in Figure 7.21(b)? Exploit the analogy between Faraday's law and Ampere's law to sketch (qualitatively) the electric field.
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