A resistor of resistance $\mathrm{R}$ and an inductor of inductive reactance $\mathrm{R}$ are connected in…
A resistor of resistance $\mathrm{R}$ and an inductor of inductive reactance $\mathrm{R}$ are connected in series to an ac source. A capacitor of capacitive reactance $2 \mathrm{R}$ is then connected in series with $L$ and $R$. The ratio of the power fators of $L R$ and LCR circuits is
$1: 1$
$1: 2$
$1: 3$
$2: 3$
Solution
Power factor of LCR circuit is
$\begin{aligned}
\mathrm{P}_1=\cos \phi & =\frac{\mathrm{R}}{\sqrt{\mathrm{R}^2+\left(\mathrm{X}_{\mathrm{L}}-\mathrm{X}_{\mathrm{C}}\right)^2}}=\frac{\mathrm{R}}{\sqrt{\mathrm{R}^2+(2 \mathrm{R}-\mathrm{R})^2}} \\
& =\frac{1}{\sqrt{2}}
\end{aligned}$
Power factor of L R circuit is
$\begin{aligned} & \mathrm{P}_2=\frac{\mathrm{R}}{\sqrt{\mathrm{R}^2+\mathrm{X}_{\mathrm{L}}^2}}=\frac{\mathrm{R}}{\sqrt{\mathrm{R}^2+\mathrm{R}^2}}=\frac{1}{\sqrt{2}} \\ & \text { Ratio }=\frac{\mathrm{P}_2}{\mathrm{P}_1}=\frac{1}{1}\end{aligned}$