ac generator

[logistic_guy]

Show that the $\displaystyle \text{rms}$ output of an ac generator is $\displaystyle V_{\text{rms}} = NAB\omega/\sqrt{2}$.

 

[khansaheb]

Show that the $\displaystyle \text{rms}$ output of an ac generator is $\displaystyle V_{\text{rms}} = NAB\omega/\sqrt{2}$.

Please show us what you have tried and exactly where you are stuck.

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[logistic_guy]

The induced emf is given by:

$\displaystyle \mathscr{E} = NBA\omega\sin\omega t$

The relation between $\displaystyle V_{\text{rms}}$ and $\displaystyle \mathscr{E}$ is:

$\displaystyle V^2_{\text{rms}} = \mathscr{E}^2_{\text{average}} = N^2B^2A^2\omega^2\sin^2_{\text{average}}\omega t$

$\displaystyle 0 \leq \sin^2\omega t \leq 1$

Then, it has an average value of $\displaystyle \frac{1}{2}$. This gives:

$\displaystyle V^2_{\text{rms}} = \frac{1}{2}N^2B^2A^2\omega^2$

Then,

$\displaystyle V_{\text{rms}} = \sqrt{\frac{N^2B^2A^2\omega^2}{2}} = \textcolor{blue}{\frac{NBA\omega}{\sqrt{2}}}$