1. When an AC source is connected to an ideal capacitor, show tha… - Vidyadip

Question

When an AC source is connected to an ideal capacitor, show that the average power supplied by the source over a complete cycle is zero.
A bulb is connected in series with a variable capacitor and an A.C. source as shown. What happens to the brightness of the bulb when the key is the capacitor is gradually reduced?

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Answer

Let the applied voltage be

$\text{V}= \text{V}_{o}\sin\omega\text{t}$
The current through an ideal capacitor, would then be
$\text{I} = \text{I}_{o}\sin(\omega\text{t} + \frac{\pi}{2}) = \text{I}_{0}\cos\omega\text{t}$
$\therefore\text{P}_{inst} = \text{VI}$
$\therefore\text{P}_{AV} = \frac{1}{\text{T}}\int^{T}_{0}\text{VIdt}$
$\therefore\text{P}_{AV} = \frac{\text{V}_{0}\text{I}_{0}}{2}\langle\sin2\omega\text{t}\rangle$
$=0.$

$\text{X}_{c} = \frac{1}{\omega\text{C}}$

$\therefore\text{X}_{c}$ increases as C decreases. Hence, with decreasing C, the brightness of the bulb would decrease.

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