First slide

AC Circuits

Question

An ac source of angular frequency $ω$ is fed across a resistor r and a capacitor C in series. The current registered is I. If now the frequency of source is changed to $ω$ /3 (but maintaining the same voltage), the current in then circuit is found to be halved. Calculate the ratio of reactance to resistance at the original frequency $ω$

Moderate

A

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

$\sqrt{\frac{2}{5}}$
C

$\sqrt{\frac{1}{5}}$
D

$\sqrt{\frac{4}{5}}$

Solution

At angular frequency $ω$ , the current in RC circuit is given by
$i_{rms} = \frac{V_{rms}}{\sqrt{R^{2} + {(\frac{1}{ωC})}^{2}}}$ ......(i)
Also $\frac{i_{rms}}{2} = \frac{V_{rms}}{\sqrt{R^{2} + {(\frac{1}{\frac{ω}{3} C})}^{2}}} = \frac{V_{rms}}{\sqrt{R^{2} + \frac{9}{ω^{2} C^{2}}}}$ ......(ii)
From equation (i) and (ii) we get
$3 R^{2} = \frac{5}{ω^{2} C^{2}} \Rightarrow \frac{\frac{1}{ωC}}{R} = \sqrt{\frac{3}{5}} \Rightarrow \frac{X_{C}}{R} = \sqrt{\frac{3}{5}}$

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