Question

Find the average of the squares of emf induced over a long period.

Answer 1

The frequency will be

$f=\frac{300}{60}$

So the angular velocity will be

$\omega =2\pi f$

$\omega =10\pi rad/\sec$

The flux is given as

$\varphi =B\times A$

$\varphi =BA\cos \omega t$

The emf will be given as

$\epsilon =-N\left(\frac{d\varphi }{dt}\right)$

$\epsilon =NBA\omega \sin \omega t$

By substituting the values in the above equation we get instantaneous emf

$\epsilon =50\times 2\times {10}^{-4}\times 0.1\times 10\pi \times \sin 10\pi tV$

The time period will be

$T=\frac{1}{f}$

$T=\frac{1}{5}=0.2\sec$

Foe the half cycle the emf induced will be

$\epsilon =\underset{0}{\overset{0.1}{\int }}50\times 2\times {10}^{-4}\times 0.1\times 10\pi \times \sin 10\pi tV$

The average of the square of the emf for the full cycle will be

$\epsilon ={10}^{-2}\times \frac{1}{0.1}\times \frac{1}{10\pi }\left(-\cos \left(10\pi \right)0.1+\cos 0\right)$

$\epsilon ={\left(0.02\right)}^2+{\left(0.02\right)}^2$

$\epsilon =8mV$