Question

When $100\text{V}$ $DC$ is applied across a coil, a current of $1\text{A}$ flows through it. When $100\text{V}$ $AC$ of $50\text{Hz}$ is applied to the same coil, only $0.5\text{A}$ current flows. Calculate the inductance of the coil.

• A. $0.2\text{H}$
• B. $0.55\text{H}$
• C. $1\text{H}$
• D. $1.5\text{H}$

Answer 1

The correct option is B $0.55\text{H}$

In case of a coil, i.e. $LR$ circuit.

$i=\frac{V}{Z}$

And, $Z=\sqrt{R^2+X_L^2}=\sqrt{R^2+(\omega L{)}^2}$
Let $i_1$ be the current flowing in the circuit when $DC$ source is connected
$i_2$ be the current flowing in the circuit when $AC$ source is connected

So when $DC$ is applied, $\omega =0\Rightarrow Z=R$

and hence, $i_1=\frac{V}{R}$

$\Rightarrow R=\frac{V}{i_1}=\frac{100}{1}=100\Omega$

When $AC$ is applied

Current, $i_2=0.5\text{A}$

As, $i_2=\frac{V}{Z}$

$\therefore Z=\frac{V}{i_2}=\frac{100}{0.5}=200\Omega$

But, $Z=\sqrt{R^2+{\omega }^2{L}^2}$

$\Rightarrow {\omega }^2{L}^2=Z^2-R^2$

$(2\pi fL{)}^2={200}^2-{100}^2=3\times {10}^4$

So, $\text{L}=\frac{\sqrt{3}\times {10}^2}{2\pi \times 50}=\frac{\sqrt{3}}{\pi }=0.55\text{H}$

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