Question

An ac circuit has an inductor and a resistor of resistance $R$ in series, such that $X _{ L }=3 R$. Now, a capacitor is added in series such that $X_{C}=2 R$. 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

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