Question

In the circuit shown in the figure, the power factor of the box is $0.5$ and the power factor of the circuit is $\sqrt{3}/2$. Current leads the voltage. Find the effective resistance of the box.

• A. $1\Omega$
• B. $3\Omega$
• C. $5\Omega$
• D. $7\Omega$

Answer 1

The correct option is C $5\Omega$

The power factor of the box is $0.5$.

So, $\cos {\varphi }_1=0.5\Rightarrow {\varphi }_1={60}^{\circ }$

Also, the power factor of the circuit is $\sqrt{3}/2$.

So, $\cos {\varphi }_2=\sqrt{3}/2\Rightarrow {\varphi }_2={30}^{\circ }$

Let $R$ be the effective resistance of the box. Then,

$\tan {\varphi }_1=\frac{X_C}{R}\Rightarrow \sqrt{3}=\frac{X_C}{R}...\left(1\right)$

Also,

$\tan {\varphi }_2=\frac{X_C}{R+10}\Rightarrow \frac{1}{\sqrt{3}}=\frac{X_C}{R+10}...\left(2\right)$

On solving equations $\left(1\right)$ and $\left(2\right)$, we get,

$R=5\Omega$

Hence, option $\left(C\right)$ is the correct answer.