Question

When $100$ volt dc is appiled across a coil, a current of 1 amp flows through it. When $100V$ A.C of $50Hz$ is appiled to the same coil, only $0.5$ amp flows. What is the value of inductance of the coil ?

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

Answer 1

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

When connected to dc source, in steady state, current is constant. So inductor just acts like a conducting wire.
We can draw the circuit as follows



$\therefore R=\frac{100V}{1A}=100\Omega$

When we connect it to an a.c source, circuit is as follows



Given $V_{rms}=100V,i_{rms}=0.5A$
$V_{rms}=i_{rms}\times Z$
$\Rightarrow Z=\frac{100}{0.5}=200\Omega$

Impedance $Z$ for an $LR$ circuit is
$Z=\sqrt{R^2+X_L^2}$
$\Rightarrow Z^2=R^2+X_L^2$
$\Rightarrow {200}^2={100}^2+X_L^2$

$\Rightarrow X_L=100\sqrt{3}=2\pi fL=2\pi \left(50\right)L$
$\Rightarrow L=\frac{\sqrt{3}}{\pi }H$