Question

A magnetic field is passing through a loop of wire whose area is 0.014 m². The direction of the magnetic field is parallel to the normal to the loop, and the magnitude of the field is increasing at the rate of 0.29 T/s. (a) Determine the magnitude of the emf induced in the loop. (b) Suppose the area of the loop can be enlarged or shrunk. If the magnetic field is increasing as in part (a), at what rate (in m^2/s) should the area be changed at the instant when B = 2.4 T if the induced emf is to be zero? Use a minus sign if the area is to be shrunk.

Answer 1

The magnitude of the induced emf is the time rate of change in flux, B dot A. For your case the dot product is just BA and the induced voltage is;

V = d(BA)/dt

(a) the area is constant so;

V = AdB/dt = (.014)(.29) = .004 volts

(b) Both B & A change so;

V = AdB/dt + BdA/dt

You want to find dA/dt such that V=0, when B=2.4 and,presumably, when A=.014:

dA/dt = - (A/B)dB/dt
= -(.014/2.4)(.29)
= -.0017 m^2/s

Since it is a negative rate the Area must be decreasing.