Question

A resistance of $20\Omega$ is connected to a source of alternating current rated $110\text{V},50\text{Hz}$. Find the time taken by the current to change from its first maximum value to the consecutive rms value.

• A. $5.5\text{ms}$
• B. $4.5\text{ms}$
• C. $2.5\text{ms}$
• D. $3.5\text{ms}$

Answer 1

The correct option is C $2.5\text{ms}$

Let the current be, $i=i_0\sin \omega t$.

Also, let $t_1$ and $t_2$ be the time instants for consecutive appearances of the maximum value and the rms value of the current, respectively.

$i_0=i_0\sin \omega t_1$

$\Rightarrow \omega t_1=\frac{\pi }{2}\Rightarrow t_1=\frac{\pi }{2\omega }$

$\frac{i_0}{\sqrt{2}}=i_0\sin \omega t_2$

$\Rightarrow \omega t_2=\frac{3\pi }{4}\Rightarrow t_2=\frac{3\pi }{4\omega }\left[\text{As we are taking the rms value after the first peak value}\right]$

So, the time taken by the current to change from its first maximum value to the consecutive rms value is,

$t_2-t_1=\frac{3\pi }{4\omega }-\frac{\pi }{2\omega }=\frac{\pi }{4\omega }$

$\Rightarrow t_2-t_1=\frac{\pi }{4\left(2\pi f\right)}=\frac{1}{8\times 50}=2.5\times {10}^{-3}\text{s}=2.5\text{ms}$