Question

A $10\Omega$ resistance is connected across $220\text{V}–50\text{Hz}AC$ supply. The time taken by the current to change from its maximum value to the $RMS$ value is

• A. $2.5\text{ms}$
• B. $1.5\text{ms}$
• C. $4.5\text{ms}$
• D. $3.0\text{ms}$

Answer 1

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

As the given $AC$ circuit is a pure $R$ circuit,

$\Rightarrow i=i_0\sin \omega t$

Let $t_1$ be the time taken by the current from zero to maximum value,

When $i=i_0$

$\Rightarrow i_0=i_0\sin \omega t_1\Rightarrow \omega t_1=\frac{\pi }{2}...\left(1\right)$

Let $t_2$ be the time taken by the current from zero to $RMS$ value,

When $i=\frac{i_0}{\sqrt{2}}$

$\Rightarrow \frac{i_0}{\sqrt{2}}=i_0\sin \omega t_2\Rightarrow \omega t_2=\frac{\pi }{4}...\left(2\right)$

Time taken by current from maximum value to $RMS$ value

$\Rightarrow (t_1-t_2)=\frac{\pi }{2\omega }-\frac{\pi }{4\omega }=\frac{\pi }{4\omega }$

$\Rightarrow (t_1-t_2)=\frac{\pi }{4\times 2\pi f}$

$\Rightarrow (t_1-t_2)=\frac{1}{8\times 50}$

$\Rightarrow (t_1-t_2)=\frac{1}{400}\text{s}$

$\Rightarrow (t_1-t_2)=2.5\text{ms}$

Hence, option $\left(A\right)$ is correct.