Electromotive Force in Spiral

A flat metal spiral (with a constant distance h between coils) and a total number of coils N is placed in a uniform magnetic field B = B₀ cos ωt perpendicular to the plane of the spiral (see Figure 1.1). Evaluate the total electromotive force induced in the spiral (between points A and C). Assume N >> 1.

Figure 1.1

SOLUTION

The electromotive force ε_e is given by

(1)

where Φ is the magnetic flux, and we used r(φ) = hφ/2π. As another solution, in the approximation of a large number of coils, we can consider the nth coil to be a circle of radius r_n = nh. Then,

where S_n = πr²_n = πh²n² is the area of the nth coil. So we find for the electromotive force between the ends of a single coil

(2)

Summing up over all the coils from (2), we obtain the total electromotive force (1)

(3)

where we have used the approximation

for N >> 1.