Can anyone help me derive an expression for the e.m.f induced in a coil rotating in a uniform magnetic field.
let
w= angular velocity etc..........
When the coil is rotated at a uniform rate in a uniform horizontal magnetic field of intensity $\displaystyle \vec{B}$, then the induced e.m.f produced is given by:
$\displaystyle e = -\frac{d\phi}{dt}$
$\displaystyle = -\frac{d}{dt}[N\vec{B}\cdot \hat{n} A]$
Where N = number of turns in coil
and A = Area of coil
$\displaystyle = -\frac{d}{dt}[NBcos{\theta}A]$
$\displaystyle = -NAB\frac{d}{dt}[cos{\theta}]$
$\displaystyle = -NAB\left[-sin{\theta}\frac{d\theta}{dt}\right]$
$\displaystyle = NAB\omega sin{\omega t}$
Where $\displaystyle \omega = \frac{d\theta}{dt} = \frac{\theta}{t}, \text{as it is uniform}$
$\displaystyle \therefore \ e = (NAB\omega)sin(\omega t)$
$\displaystyle e = e_0 \ sin{(\omega t)}$
And there you go.