Effect of Angular Frequency on Bulb Brightness in RC Circuit

To determine the effect of increasing the angular frequency $\omega$ on the brightness of the bulb, let us analyze the circuit parameters.

1. Impedance of the Circuit:
   The circuit is a series $RC$ circuit connected to an AC voltage source of fixed amplitude $V_0$ and angular frequency $\omega$.
   The capacitive reactance $X_C$ is given by:
   $X_C = \frac{1}{\omega C}$
   The total impedance $Z$ of the series circuit is:
   $Z = \sqrt{R^2 + X_C^2} = \sqrt{R^2 + \left(\frac{1}{\omega C}\right)^2}$

2. Effect of Increasing $\omega$:
   When the angular frequency $\omega$ increases:

   • The capacitive reactance $X_C = \frac{1}{\omega C}$ decreases.
   • Consequently, the total impedance $Z = \sqrt{R^2 + X_C^2}$ decreases.

3. Current and Power Dissipated in the Bulb:
   The root-mean-square (rms) current in the circuit is:
   $I_{\text{rms}} = \frac{V_{\text{rms}}}{Z}$
   Since the source amplitude $V_0$ is fixed, $V_{\text{rms}} = \frac{V_0}{\sqrt{2}}$ is also constant.
   As $Z$ decreases, the current $I_{\text{rms}}$ flowing through the circuit increases.

   The brightness of the bulb depends on the average electrical power dissipated as heat and light across its resistance $R$:
   $P = I_{\text{rms}}^2 R$
   Since $I_{\text{rms}}$ increases and the resistance $R$ remains constant, the power dissipated $P$ increases.

Therefore, when $\omega$ is increased, the bulb glows brighter.

Correct Option: B