Ratio of Resistance to Reactance in Series RC Circuit

To find the ratio of resistance $R$ to capacitive reactance $X_C = \frac{1}{\omega C}$ at angular frequency $\omega$, we set up the impedance equations for both frequencies.

For initial frequency $\omega$, the impedance is $Z_1 = \sqrt{R^2 + X_C^2}$ and current $I = \frac{V}{Z_1}$.

When the frequency is reduced to $\omega/4$, the capacitive reactance increases to $4X_C$, giving an impedance of $Z_2 = \sqrt{R^2 + 16X_C^2}$. Since the current reduces to $I/3$, the impedance triples ($Z_2 = 3Z_1$).

Equating the square of the impedances gives: $R^2 + 16X_C^2 = 9(R^2 + X_C^2)$

Solving for the ratio $\frac{R}{X_C}$ yields: $\frac{R}{X_C} = \sqrt{\frac{7}{8}}$

Thus, the correct option is C.