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Chapter 7: Q10P (page 311)
A square loop (side a) is mounted on a vertical shaft and rotated at angular velocity (Fig. 7.19). A uniform magnetic field B points to the right. Find thefor this alternating current generator.
The induced emf in the square loop is .
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Prove Alfven's theorem: In a perfectly conducting fluid (say, a gas of free electrons), the magnetic flux through any closed loop moving with the fluid is constant in time. (The magnetic field lines are, as it were, "frozen" into the fluid.)
(a) Use Ohm's law, in the form of Eq. 7.2, together with Faraday's law, to prove that if and is J finite, then
(b) Let S be the surface bounded by the loop at time t , and a surface bounded by the loop in its new position at time t+dt (see Fig. 7.58). The change in flux is
Use to show that
(Where R is the "ribbon" joining P and P' ), and hence that
(For infinitesimal dt ). Use the method of Sect. 7.1.3 to rewrite the second integral as
And invoke Stokes' theorem to conclude that
Together with the result in (a), this proves the theorem.
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.)
Suppose the circuit in Fig. 7.41 has been connected for a long time when suddenly, at time , switch S is thrown from A to B, bypassing the battery.
Notice the similarity to Eq. 7.28-in a sense, the rectangular toroid is a short coaxial cable, turned on its side.
(a) What is the current at any subsequent time t?
(b) What is the total energy delivered to the resistor?
(c) Show that this is equal to the energy originally stored in the inductor.
A circular wire loop (radius, resistance) encloses a region of uniform magnetic field,, perpendicular to its plane. The field (occupying the shaded region in Fig. 7.56) increases linearly with time. An ideal voltmeter (infinite internal resistance) is connected between pointsand.
(a) What is the current in the loop?
(b) What does the voltmeter read? [Answer: ]
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.
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