Question

For the given AC RLC circuit, at a particular frequency $\left(f\right)$ of the AC source, the current leads the voltage by ${45}^{\circ }$. The relation between $R,L$ and $C$ is -

• A. $C=\frac{1}{2\pi f(2\pi fL-R)}$
• B. $C=\frac{1}{2\pi f(2\pi fL+R)}$
• C. $C=\frac{1}{\pi f(2\pi fL-R)}$
• D. $C=\frac{1}{\pi f(2\pi fL+R)}$

Answer 1

The correct option is B $C=\frac{1}{2\pi f(2\pi fL+R)}$


Suppose, the AC source voltage is given by,

$V=V_m\sin \left(\omega t\right)$

Then, the AC in the circuit is,

$I=I_m\sin (\omega t+\varphi )$

So,

$\tan \varphi =\frac{X_C-X_L}{R}$

$\Rightarrow \tan {45}^{\circ }=\frac{X_C-X_L}{R}$

$\Rightarrow R=X_C-X_L$

$\Rightarrow R=\frac{1}{\omega C}-\omega L$

$\Rightarrow R=\frac{1}{2\pi fC}-2\pi fL$

​​​​​​​$\Rightarrow C=\frac{1}{2\pi f(2\pi fL+R)}$

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