Question

An electric lamp is connected in series with a capacitor and an AC source is glowing with a certain brightness. How does the brightness of the lamp change on increasing the capacitance ?

Answer 1

Impedance, Z = $\sqrt{R^2+\frac{1}{{\omega }^2{C}^2}}$, where R is the resistance of the lamp, w is the angular frequency of AC and C is the capacitance of the capacitor connected in series with the AC source and the lamp. When C is increased, $$ decreases. Hence, Z increases.
Power factor, cos Φ = $\frac{R}{Z}$
As Z increases, the power factor decreases.
Now, the average power over one cycle,
P_av = v_rms i_rms cos Φ
= V_rms $\left(\frac{V_{\mathrm{rms}}}{Z}\right)$ cos Φ
= $\frac{V_{\mathrm{rms}}^2}{\mathrm{Z}}\cos \varphi$
∴ P_av decreases as Z increases and cos Φ decreases.
As the current through the lamp $\left(\frac{V_{\mathrm{rms}}}{Z}\right)$ decreases, the brightness of the lamp will decrease when C is increased