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Chapter 7: Q7.38P (page 348)
Question: Assuming that "Coulomb's law" for magnetic charges ( qm) reads
Work out the force law for a monopole moving with velocity through electric and magnetic fields E and B.
Answer
The value of the magnetic charge is
.
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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 is Planck's constant. What, then, is the electron's magnetic dipole moment, in localid="1657713870556" ? [This semi classical 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(localid="1657713972487" , where e is the charge of the electron and m is its mass, is called the Bohr magneton.]
A perfectly conducting spherical shell of radius rotates about the z axis with angular velocity , in a uniform magnetic field . Calculate the emf developed between the 鈥渘orth pole鈥� and the equator. Answer:localid="1658295408106" .
A long solenoid with radius a and n turns per unit length carries a time-dependent current in the 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.
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 . 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 , and a steady current flows over the surface, as indicated in Fig. 7.51. According to Ohm's law, then,
Figure 7.51
(a) Use separation of variables in cylindrical coordinates to determine inside and outside the cylinder.
(b) Find the surface charge density on the cylinder.
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