# 6.6 Moving Conductor in a Time-Varying Magnetic Field

For the general case of a single-turn conducting loop moving in a time-varying magnetic field, the induced emf is the sum of a transformer component and a motional component. Thus, the sum of Eqs. (6.8) and (6.26) gives

V e m f=V e m f t r+V e m f m=−∫S∂B∂t⋅d s+∮C(u×B)⋅d l.(6.39)

V e m f is also given by the general expression of Faraday’s law:

In fact, it can be shown mathematically that the right-hand side of Eq. (6.39) is equivalent to the right-hand side of Eq. (6.40). For a particular problem, the choice between using Eq. (6.39) or Eq. (6.40) is usually made on the basis of which is the easier to apply. In either case, for an N-turn loop, the right-hand sides of Eqs. (6.39) and (6.40) should be multiplied by N.

# Example 6-6: Electromagnetic Generator

Find the induced voltage when the rotating loop of the electromagnetic generator of Section 6-5 is in a magnetic field B=z ˆ B 0 cos ω t. Assume that α=0 at t=0.

Solution: The flux Φ is given by Eq. (6.37) with B 0 replaced with B 0 cos ω t. Thus,

Φ=B 0 A cos 2 ω t,

and

V emf=−d Φ d t=−d d t(B 0 A cos 2 ω t)=2 B 0 A ω cos ω t sin ω t=B 0 A ω sin 2 ω t.