Question

The rms value of alternating EMF is $ 8\sin \omega t + 6\sin 2\omega t $
(A) 7.07 V
(B) 14.14 V
(C) 10 V
(D) 20 V

Answer 1

Hint
The rms value of any quantity is also known as its root mean square value. Now, to find the mean square value, we will first square the quantity whose value is to be found. Then we will integrate this value from the initial to the final point. After integration, we need to divide it by the difference in final and initial value.

Complete step by step answer
The rms value of any quantity is given by
$\Rightarrow {E_{rms\,}}\, = \,\sqrt {\dfrac{1}{T}\int_0^T {{E^2}} } $
In our case E = $ 8\sin \omega t + 6\sin 2\omega t $ , substituting this value we get:
$\Rightarrow {E_{rms\,}} = \sqrt {\dfrac{1}{T}\int_0^T {{{(8\sin \omega t + 6\sin 2\omega t)}^2}dt} }$
$\Rightarrow {E_{rms\,}}\, = \,\sqrt {\dfrac{1}{T}\int_0^T {(64{{\sin }^2}\omega t + 36{{\sin }^2}2\omega t + 96\sin \omega t\sin 2\omega t)dt} }$
As we already know that:
$\Rightarrow \dfrac{{\int_0^T {{{\sin }^2}\omega t} \;}}{T} = \dfrac{1}{2} $
$\Rightarrow \dfrac{{\int_0^T {{{\sin }^2}2\omega t} }}{T}\; = \dfrac{1}{2} $
$\Rightarrow \dfrac{{\int_0^T {\sin \omega t\sin 2\omega t} }}{T} = 0 $
$\Rightarrow {E_{rms\,}}\, = \,\sqrt {64x\dfrac{1}{2} + 36x\dfrac{1}{2} + 96x0} $
$\Rightarrow {E_{rms\,}}\, = \,\sqrt {50} $
$\Rightarrow {E_{rms\,}}\, = \,7.07 $
Therefore the correct answer is option (A).

Note
This formula for finding the root mean square value or the mean value of a quantity is universal. This means that if we have to find the root mean square velocity of the gas particles, we have to follow this procedure.