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Unformatted text preview: Physics 21 Fall, 2009 Solution to HW-17 29-8 A at, circular, steel loop of radius 75 cm is at rest in a uniform magnetic field, as shown in an edge-on view in the figure. The field is changing with time, according to B ( t ) = 1 . 4 exp( . 057 t ) ( B in Tesla; t in seconds.) (a) Find the emf induced in the loop as a function of time. (b) When is the induced emf equal to 1 / 10 of its initial value? (c) Find the direction of the current induced in the loop, as viewed from above the loop. (a) The magnetic ux B through the loop is B = Z B d A . Since B is constant, the surface integral is just B = BA cos = BA cos 30 = B ( r 2 ) 3 2 . The factor cos 30 comes from the dot product of B and d A . From the figure, the angle between B and d A is the complement of 60 , or 30 . The magnitude of the induced emf is proportional to the time derivative of the magnetic ux through the loop: |E| = d B dt = dB dt r 2 3 2 = 1 . 4(0 . 057) exp( . 057 t ) (0 . 75 m) 2 3 2 = 0 . 122 exp( . 057 t ) V (b) When the induced emf is one-tenth of its initial value, the exponential factor, initially equal to unity and the only factor changing in time, is one-tenth: exp( . 057 t ) = 0 . 1 . 057 t = ln 0 . 1 = ln 10 Therefore t = ln 10 . 057 = 40 . 4 s (c) According to Lenzs Law, the induced current will be a direction such that the magnetic field it causes opposes the change in magnetic ux through the loop. Here, the induced magnetic field will attempt to replace the decaying field. To an observer looking down at the loop, this current will be counterclockwise, and the magnetic field it causes will be toward the observer....
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This note was uploaded on 11/06/2009 for the course PHYS 21 taught by Professor Hickman during the Spring '07 term at Lehigh University .
- Spring '07