Question

An ac source of voltage $V_0$ and variable frequency is connected to a R-C circuit. At frequency $\omega_0$ current in the circuit is $I_0$. When frequency is changed to $\frac{\omega_0}{3}$ current in the circuit becomes $\frac{I_0}{2}$. Find value of $\omega_0RC$.

Answer 1

$\sqrt{\frac{5}{3}}$

Answer 2

The impedance of a series R-C circuit is $Z = \sqrt{R^2 + (1/\omega C)^2}$. The current $I = V_0/Z$. For $\omega_0$, $I_0 = V_0 / \sqrt{R^2 + (1/\omega_0 C)^2}$. For $\omega_0/3$, $I_0/2 = V_0 / \sqrt{R^2 + (3/\omega_0 C)^2}$. The ratio $I_0 / (I_0/2) = 2$ leads to $2 = \sqrt{(R^2 + 9/(\omega_0 C)^2) / (R^2 + 1/(\omega_0 C)^2)}$. Squaring and simplifying gives $3R^2 = 5/(\omega_0 C)^2$, which rearranges to $3(\omega_0 RC)^2 = 5$. Thus, $\omega_0 RC = \sqrt{5/3}$.