Question

Obtain an expression for the e.m.f. induced in a coil rotating in uniform magnetic field.

Answer 1

Consider a coil of cross section area $S$ rotating with an angular speed $\omega$ in a uniform magnetic field $B$ as shown in the figure.

Magnetic flux through the loop in the given position is given by:

$\varphi =\overrightarrow{B}\cdot \overrightarrow{S}$

$=BS\cos (\pi /2-\theta )$

$=BS\sin \left(\theta \right)$

$\theta ={\theta }_o+\omega t$

$⟹\varphi =BS\sin ({\theta }_o+\omega t)$

Induced emf is given by:

$\epsilon =\frac{d\varphi }{dt}$

$\epsilon =\frac{d\left(BS\sin \right({\theta }_o+\omega t\left)\right)}{dt}$

$\epsilon =BS\omega \cos ({\theta }_o+\omega t)$

Note that the induced emf is alternating in nature.

The rms value of induced emf is: ${\epsilon }_{rms}=\frac{BS\omega }{\sqrt{2}}$