Question

A box containing a resistance $R$, inductance $L$ and a capacitor of capacitance $C$ is connected in series to an alternating source of angular frequency $4rad/s$ and a capacitor of capacitance $C$ as shown in the circuit. Box has power factor of $\frac{1}{\sqrt{3}}$ and the circuit has overall power factor of $1$. The impedance of the box is

• A. $\frac{\sqrt{3}}{4\sqrt{2}C}$
• B. $\frac{1}{4C}$
• C. $\frac{1}{4\sqrt{2}C}$
• D. $\frac{3}{4\sqrt{2}C}$

Answer 1

The correct option is A $\frac{\sqrt{3}}{4\sqrt{2}C}$

Power factor = $\frac{R}{Z}$
For the box,
$\frac{1}{\sqrt{3}}=\frac{R}{Z_{box}}$....$\left(1\right)$
$\frac{1}{\sqrt{3}}=\frac{R}{\sqrt{R^2+(X_L-X_C{)}^2}}$....$\left(1\right)$

For the whole circuit since the power factor is $1$,$\Rightarrow Z=R$
reactance of capacitance outside the box is $=\frac{1}{\omega C}=\frac{1}{4C}$ $(\because \omega =4rad/s)$
and $X_L=X_C+\frac{1}{4C}$
$X_L-X_C=\frac{1}{4C}...\left(2\right)$

$\left(2\right)$ in $\left(1\right)$ gives,

$3R^2=R^2+{\left(\frac{1}{4c}\right)}^2$
$2R^2=\frac{1}{16C^2}$
$R=\frac{1}{4\sqrt{2}C}$
$\Rightarrow Z_{box}=\sqrt{3}R=\frac{\sqrt{3}}{4\sqrt{2}C}$