Question

The power factor of an L-R series circuit is $0.5$ and that of a C-R series circuit is $0.2$ . If the elements $\left(\right.L, \, C \, and \, R\left.\right)$ of the two circuits are joined in series and connected to the same ac source, the power factor of this circuit is found to be $1$ . The ratio of the resistance in the L-R circuit to the resistance in the C-R circuit is

• A. $2$
• B. $\sqrt{2}$
• C. $2\sqrt{2}$
• D. $4$

Answer 1

Answer: (C)

$\frac{R_{1}}{\sqrt{X_{L}^{2} + R_{1}^{2}}}=0.5$
$\left(2 R_{1}\right)^{2}=R_{1}^{2}+X_{L}^{2}$
$3R_{1}^{2}=X_{L}^{2}$ .....(i)
$\frac{R_{2}}{\sqrt{X_{C}^{2} + R_{2}^{2}}}=0.2$
$25R_{2}^{2}=X_{C}^{2}+R_{2}^{2}$
$24R_{2}^{2}=X_{C}^{2}=X_{L}^{2}$ (Since the LCR circuit is in resonance, the capacitive reactance and the inductive reactance are equal)
$3R_{1}^{2}=24R_{L}^{2}$
$\frac{R_{1}}{R_{2}}=2\sqrt{2}$