Set 54 Problem number 11

Problem

A square loop with side 2.5 meters rotates at 7 Hz. A magnetic field of magnitude 5.8 Tesla passes through a loop, with the field perpendicular to the axis of rotation of the loop. What is the average magnitude of the voltage produced?

Solution

Since the field is perpendicular to the axis of the loop, at two instants in its cycle the field will be perpendicular to the loop.

7 Hz means 7 cycles per second. This implies a time of 1/ 7 seconds = .1428 sec for each cycle.

One cycle corresponds to 360 degrees; a 90 degree turn corresponds to a time of ( .1428 sec)/4 = .0357 sec.
The area of the loop is ( 2.5 m) ^ 2 = 6.25 m ^ 2, so the flux when the plane of the loop is prependicular to the field is ( 5.8 T)( 6.25 m ^ 2) = 36.25 T m ^ 2.
After being turned 90 degrees, the loop will be parallel to the field, and will intercept none of the field. The flux will then be zero.
The magnitude of the flux change from 36.25 T m ^ 2 to zero is 36.25 T m ^ 2.
This occurs in .0357 seconds, so the average rate is ( 36.25 T m ^ 2)/( .0357 sec) = 2030 T m ^ 2 / sec = 2030 Volts.
Since this is the average during each quarter of a cycle, it is equal to the overall average.

Generalized Solution

If a loop rotates at frequency f in a magnetic field, then the time required for a complete rotation is 1/f (think of 10 cycles per second; the time per cycle is obviously 1/10 sec). In a quarter of a rotation the loop will rotate through 90 degrees. This will require 1/4 of the time of a cycle. Since a complete rotation takes time 1/f, the time to complete 1/4 of a cycle is `dt = 1/4 (1/f) = 1 / (4f).

If the loop rotates 90 degrees, starting from a position where the field is perpendicular to the loop, the magnitude of the flux change will be | 3.8 `phi | = B * A. Since this takes place in time `dt = 1 / (4f), the rate of flux change will be

average rate of flux change = | `d`phi | / `dt = B * A / ( 1 / (4f) ) = 4f * B * A.

This is in fact the average rate, since the magnitude of the flux change will be B * A through every subsequent 90 degree rotation.