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Chapter 7: Q7.21P (page 321)
Imagine a uniform magnetic field, pointing in the direction and filling all space . A positive charge is at rest, at the origin. Now somebody turns off the magnetic field, thereby inducing an electric field. In what direction does the charge move?
The direction of the charge is indetermined.
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A square loop is cut out of a thick sheet of aluminum. It is then placed so that the top portion is in a uniform magnetic field , B and is allowed to fall under gravity (Fig. 7 .20). (In the diagram, shading indicates the field region; points into the page.) If the magnetic field is 1 T (a pretty standard laboratory field), find the terminal velocity of the loop (in m/s ). Find the velocity of the loop as a function of time. How long does it take (in seconds) to reach, say, of the terminal velocity? What would happen if you cut a tiny slit in the ring, breaking the circuit? [Note: The dimensions of the loop cancel out; determine the actual numbers, in the units indicated.]
Refer to Prob. 7.16, to which the correct answer was
(a) Find the displacement current density ·
(b) Integrate it to get the total displacement current,
Compare and I. (What's their ratio?) If the outer cylinder were, say, 2 mm in diameter, how high would the frequency have to be, for to be 1% of I ? [This problem is designed to indicate why Faraday never discovered displacement currents, and why it is ordinarily safe to ignore them unless the frequency is extremely high.]
An alternating current flows down a long straight wire, and returns along a coaxial conducting tube of radius a.
(a) In what direction does the induced electric field point (radial, circumferential, or longitudinal)?
(b) Assuming that the field goes to zero as , find
The current in a long solenoid is increasing linearly with time, so the flux is proportional :.Two voltmeters are connected to diametrically opposite points (A and B), together with resistors ( and ), as shown in Fig. 7.55. What is the reading on each voltmeter? Assume that these are ideal voltmeters that draw negligible current (they have huge internal resistance), and that a voltmeter register -between the terminals and through the meter. [Answer: . Notice that , even though they are connected to the same points]
A familiar demonstration of superconductivity (Prob. 7.44) is the levitation of a magnet over a piece of superconducting material. This phenomenon can be analyzed using the method of images. Treat the magnet as a perfect dipole , m a height z above the origin (and constrained to point in the z direction), and pretend that the superconductor occupies the entire half-space below the xy plane. Because of the Meissner effect, B = 0 for , and since B is divergenceless, the normal ( z) component is continuous, so just above the surface. This boundary condition is met by the image configuration in which an identical dipole is placed at - z , as a stand-in for the superconductor; the two arrangements therefore produce the same magnetic field in the region z>0.
(a) Which way should the image dipole point (+ z or -z)?
(b) Find the force on the magnet due to the induced currents in the superconductor (which is to say, the force due to the image dipole). Set it equal to Mg (where M is the mass of the magnet) to determine the height h at which the magnet will "float." [Hint: Refer to Prob. 6.3.]
(c) The induced current on the surface of the superconductor ( xy the plane) can be determined from the boundary condition on the tangential component of B (Eq. 5.76): . Using the field you get from the image configuration, show that
where r is the distance from the origin.
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