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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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An alternating current I(t) (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 resistor.
(a) In the quasistatic approximation, what emf is induced in the toroid? Find the current, , in the resistor.
(b) Calculate the back emf in the coil, due to the current . What is the ratio of the amplitudes of this back emf and the "direct" emf in (a)?
An infinite wire runs along the z axis; it carries a current I (z) that is a function ofz(but not of t ), and a charge density that is a function of t (but not of z ).
(a) By examining the charge flowing into a segment dz in a time dt, show that . If we stipulate that and , show that , , where k is a constant.
(b) Assume for a moment that the process is quasistatic, so the fields are given by Eqs. 2.9 and 5.38. Show that these are in fact the exact fields, by confirming that all four of Maxwell's equations are satisfied. (First do it in differential form, for the region s > 0, then in integral form for the appropriate Gaussian cylinder/Amperian loop straddling the axis.)
A toroidal coil has a rectangular cross section, with inner radius a , outer radius , and height h . It carries a total of N tightly wound turns, and the current is increasing at a constant rate . 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.]
Suppose
;
(The theta function is defined in Prob. 1.46b). Show that these fields satisfy all of Maxwell's equations, and determine and . Describe the physical situation that gives rise to these fields.
Suppose a magnetic monopole passes through a resistanceless loop of wire with self-inductance L . What current is induced in the loop?
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