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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) Show that Maxwell's equations with magnetic charge (Eq. 7.44) are invariant under the duality transformation
Where and is an arbitrary rotation angle in 鈥淓/B-space.鈥� Charge and current densities transform in the same way as and . [This means, in particular, that if you know the fields produced by a configuration of electric charge, you can immediately (using ) write down the fields produced by the corresponding arrangement of magnetic charge.]
(b) Show that the force law (Prob. 7.38)
is also invariant under the duality transformation.
A long solenoid, of radius a, is driven by an alternating current, so that the field inside is sinusoidal:. A circular loop of wire, of radius 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, of side , lies midway between two long wires,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 in the square loop is gradually increasing: role="math" localid="1658127306545" (a constant). Find the emf induced in the big loop. Which way will the induced current flow?
(a) Referring to Prob. 5.52(a) and Eq. 7.18, show that
(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 carries a uniform surface charge . It spins about a fixed axis at an angular velocity 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 . Refer to Ex. 5.11.]
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.
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