Question

A resistor of resistance $R$, capacitor of capacitance $C$ and inductor of inductance $L$ are connected in parallel to $AC$ power source of voltage ${ϵ}_0\sin {\omega }_t$. The maximum current through the resistance is half of the maximum current through the power source. Then value of $R$ is

• A. $\frac{\sqrt{3}}{|\omega C-\frac{1}{\omega L}|}$
• B. $\sqrt{3}\left|\frac{1}{\omega C}-\omega L\right|$
• C. $\sqrt{5}\left|\frac{1}{\omega C}-\omega L\right|$
• D. None of these

Answer 1

Answer: (A)

$\frac{i_{R_0}}{\sqrt{{\left(i_{R_0}\right)}^2+{\left(i_{c_0}-i_{L_0}\right)}^2}}=\frac{1}{2}$
$\Rightarrow \frac{{ϵ}_0/R}{\sqrt{{\left({ϵ}_0/R\right)}^2{\left({ϵ}_0\omega C-\frac{{ϵ}_0}{{\omega }_L}\right)}^2}}=\frac{1}{2}$
$\Rightarrow R=\frac{\sqrt{3}}{\left(\omega C-\frac{1}{\omega L}\right)}$