Question

An ac circuit has an inductor and a resistor of resistance $R$ in series, such that $X_L=3R$. Now, a capacitor is added in series such that $X_C=2R$. The ratio of new power factor with the old power factor of the circuit is $\sqrt{5}:x$. The value of $x$ is _____

Answer 1

Answer: 1

$\cos \varphi =\frac{R}{\sqrt{R^2+3R^2}}$
$\cos {\varphi }^{\prime }=\frac{R}{\sqrt{R^2+R^2}}$
$=\frac{1}{\sqrt{10}}$
$=\frac{1}{\sqrt{2}}$
$\frac{\cos {\varphi }^{\prime }}{\cos \varphi }=\frac{\sqrt{10}}{\sqrt{2}}=\frac{\sqrt{5}}{1}$
$\therefore x=1$