First slide

Alternating Current and voltage

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 maintain the same voltage), the current in the circuit is found to be halved. Then the ratio of reactance to resistance at the original frequency $ω$ is

Difficult

A

$\sqrt{3 / 5}$
B

$\sqrt{5 / 3}$
C

$\sqrt{2 / 3}$
D

$\sqrt{3 / 2}$

Solution

According to the given problem
$I = \frac{V}{Z} = \frac{V}{{[R^{2} + {(1 / C ω)}^{2}]}^{1 / 2}} ...................... (i)$

And $\frac{1}{2} = \frac{V}{{[R^{2} + {(3 / C ω)}^{2}]}^{1 / 2}} ........... (i i)$

Substituting the value of I from equation (i) in (ii), we get

$4 (R^{2} + \frac{1}{C^{2} ω^{2}}) = R^{2} + \frac{9}{C^{2} ω^{2}}$

$= \frac{1}{C^{2} ω^{2}} = \frac{3}{5} R^{2}$

So that $\frac{X}{R} = \frac{(1 / C ω)}{R} = \frac{{(\frac{3}{5} R^{2})}^{1 / 2}}{R} = \sqrt{\frac{3}{5}}$

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