Question

When $100\text{V}\text{dc}$ is applied across a coil, a current of one ampere flows through it, but when $100\text{V}\text{ac}$ of $50\text{Hz}$ is applied to the same coil, only $0.5\text{amp}$ flows. Calculate the resistance and inductance of the coil.

• A. $100\Omega ,\frac{\sqrt{3}}{\pi }H$
• B. $10\Omega ,\frac{\sqrt{3}}{\pi }H$
• C. $100\Omega ,\frac{2\sqrt{3}}{\pi }H$
• D. $100\Omega ,\frac{3\sqrt{3}}{\pi }H$

Answer 1

The correct option is A $100\Omega ,\frac{\sqrt{3}}{\pi }H$

For dc supply:
Given $V=100\text{V}$, $I=1A$
$i=\frac{V}{R_{\text{coil}}}=\frac{100}{R_{\text{coil}}}=1$
$\Rightarrow R_{\text{coil}}=100\Omega$

For ac supply:
when $100\text{V A.C.}$ of $50\text{Hz}$ then $i=\frac{1}{2}A$
Impedance of the circuit
$Z=\frac{V}{i}=\frac{100}{\frac{1}{2}}=200\Omega$
$\Rightarrow 200=\sqrt{R_{\text{coil}}^2+{\omega }^2{L}^2}$
$\omega =2\pi \times 50=100\pi$
$\Rightarrow {200}^2={100}^2+{100}^2{\pi }^2\times L^2$
On solving we get
$4-1={\pi }^2{L}^2$
i.e $L=\frac{\sqrt{3}}{\pi }H$