Question

Interactive LearningWare 22.2 reviews the fundamental approach in problems such as this. A constant magnetic field passes through a single rectangular loop whose dimensions are 0.29 m x 0.61 m. The magnetic field has a magnitude of 1.7 T and is inclined at an angle of 72o with respect to the normal to the plane of the loop. (a) If the magnetic field decreases to zero in a time of 0.61 s, what is the magnitude of the average emf induced in the loop? (b) If the magnetic field remains constant at its initial value of 1.7 T, what is the magnitude of the rate A/ t at which the area should change so that the average emf has the same magnitude?

Answer 1

| emf | = d(flux)/dt

Since flux = (magnetic field)*(area), differentiate to get:

d(flux)/dt = (magnetic field)*d(area)/dt + (area)*d(magnetic field)/dt

In part A, the area was constant, so d(area)/dt = 0, and the equation for d(flux)/dt becomes:
d(flux)/dt = (area)*d(magnetic field)/dt

Plug in the following:
area = (0.29m)(0.61 m)cos(72 deg)
d(magnetic field)/dt = (1.7T)/(0.61 s)

d(magnetic field)/dt = (1.7 T)/(0.61 s)

This gives:
| emf | = d(flux)/dt = (area)*d(magnetic field)/dt = [(0.29m)(0.61 m)cos(72 deg) ][(1.7 T)/(0.61 s)] = 0.15234 V

In part B, the magnetic field is constant. So, d(magnetic field)/dt = 0, and the equation for d(flux)/dt becomes:
d(flux)/dt = (magnetic field)*d(area)/dt
Plug in the following:
magnetic field = 1.7 T
d(flux)/dt = 0.15234 V
This gives:
d(flux)/dt = (magnetic field)*d(area)/dt
d(area)/dt = [d(flux)/dt]/[magnetic field] = [0.15234 V]/[1.7 T] = 0.0896 m^2 / sec