Two electric circuits $A$ and $B$ are shown in the figure. The ratio of power factor of circuit $B$ to that…
Two electric circuits $A$ and $B$ are shown in the figure. The ratio of power factor of circuit $B$ to that of circuit $A$ is
$\sqrt{3}: 2$
$\sqrt{2}: 1$
$2: 3$
$4: 3$
Solution
The power factor of the circuit $A$ is
$\begin{aligned}
& \cos \phi_A=\frac{R}{\sqrt{R^2+X_L^2}} \\
& =\frac{R}{\sqrt{R^2+(3 R)^2}}=\frac{1}{\sqrt{10}} .
\end{aligned}$
Similarly, the power factor of the circuit $B$ is
$\begin{aligned}
\cos \phi_B=\frac{R}{\sqrt{R^2+(X_L^2-X_C)^2}} \\
=\frac{R}{\sqrt{R^2+(3 R-R)^2}}=\frac{1}{\sqrt{5}} .
\end{aligned}$
$\therefore$ ratio of power factor of the circuit $B$ to the circuit $A$ is
$\frac{\cos \phi_B}{\cos \phi_A}=\frac{1 / \sqrt{5}}{1 / \sqrt{10}}=\sqrt{2}$
