The magnetic field of a strong electromagnet is 1.4 T when the current in its 230 coils is 150 A. The coils are 25 cm in diameter and have a resistance of 120 ohms. If a square wave is fed to the coils to produce this maximum field, what must be its frequency?

Solution:

Some approximations are necessary to do this problem. A pure square wave would instantaneously change from its maximum voltage to zero, thus producing an infinite induced current.

Assuming that the magnetic field is perpendicular to the coils and is independent of area, B can be taken outside the integral. For one of the loops,

Pure square waves are unattainable. Let's assume we have a "very poor" square wave, so that the decrease from its maximum voltage to zero requires the entire period. Let's further assume that the time - the period - is small so that we can approximate the derivative with "deltas". Then

But the emf is given by Ohm's Law. Dropping the sign, since it only indicates a direction for the current, and including the 230 turns,

The reciprocal of this period is the frequency: 1.14 kHz.