Question

A series $LCR$ circuit containing a resistor of resistance $120\Omega$ has resonant frequency $4\times {10}^5\text{rad/s}$. At resonance, the voltage across resistor and inductor are $60\text{V}$ and $40\text{V}$ respectively. Find the value of $C$.

• A. $\frac{1}{2}\mu \text{F}$
• B. $2\mu \text{F}$
• C. $\frac{1}{9}\mu \text{F}$
• D. $\frac{1}{32}\mu \text{F}$

Answer 1

The correct option is D $\frac{1}{32}\mu \text{F}$

In a series $LCR$ circuit, at resonance,

$X_L=X_C$ and $Z=R$

Also, current,

$i=\frac{\Delta V_R}{R}=\frac{60}{120}=0.5\text{A}$

Voltage across the capacitor is the same as voltage across the inductor at resonance.

$\Delta V_L=\Delta V_C=i{X}_C=\frac{i}{\omega C}$

$\Rightarrow 40=\frac{0.5}{4\times {10}^5\times C}$

$\Rightarrow C=\frac{1}{32\times {10}^6}\text{F}$

$\Rightarrow C=\frac{1}{32}\mu \text{F}$

Hence, option $\left(D\right)$ is the correct answer.