Question

Show mathematically that the rotation of a coil in a magnetic field over one rotation induces an alternating emf of one cycle.

Answer 1

Consider a rectangular coil of N turns kept in a uniform magnetic field `vec"B"`. The coil rotates in an anti-clockwise direction with an angular velocity ω about an axis, perpendicular to the field. At time = 0, the plane of the coil is perpendicular to the field and the flux linked with the coil has its maximum value Φ_m = BA (where A is the area of the coil).

The coil has rotated through an angle θ ω = t

Variation of induced emf as a function of ωt

ε = `"d"/"dt"` (NΦ_B ) = `"d"/"dt"` (NΦ_m cos ωt)

= -NΦ_m (-sin ωt)ω = NΦ_m ω sin ωt

When the coil is rotated through 90° from initial position, sin ωt = 1, Then the maximum value of induced emf is

ε_m = NΦ_mω = NBAω since Φ_m = BA

Therefore, the value of induced emf at that instant is then given by

ε = ε_m sin ωt

It is seen that the induced emf varies as sine function of the time angle ωt. The graph between – induced emf and time angle for one rotation of coil will be a sine curve and the emf varying in this manner is called sinusoidal emf or alternating emf.