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Physics-

General

Easy

Question

An ac source of angular frequency $omega$ 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 $omega$ /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 $omega$

$\sqrt{\frac{3}{5}}$
$\sqrt{\frac{2}{5}}$
$\sqrt{\frac{1}{5}}$
$\sqrt{\frac{4}{5}}$

The correct answer is: $square root of fraction numerator 3 over denominator 5 end fraction end root$

At angular frequency w, the current in RC circuit is given by
$i subscript r m s end subscript equals fraction numerator V subscript r m s end subscript over denominator square root of R to the power of 2 end exponent plus open parentheses fraction numerator 1 over denominator omega C end fraction close parentheses to the power of 2 end exponent end root end fraction$ ......(i)
Also $fraction numerator i subscript r m s end subscript over denominator 2 end fraction equals fraction numerator V subscript r m s end subscript over denominator square root of R to the power of 2 end exponent plus open parentheses fraction numerator 1 over denominator fraction numerator omega over denominator 3 end fraction C end fraction close parentheses to the power of 2 end exponent end root end fraction equals fraction numerator V subscript r m s end subscript over denominator square root of R to the power of 2 end exponent plus fraction numerator 9 over denominator omega to the power of 2 end exponent C to the power of 2 end exponent end fraction end root end fraction$ ......(ii)
From equation (i) and (ii) we get
$3 R to the power of 2 end exponent equals fraction numerator 5 over denominator omega to the power of 2 end exponent C to the power of 2 end exponent end fraction rightwards double arrow fraction numerator fraction numerator 1 over denominator omega C end fraction over denominator R end fraction equals square root of fraction numerator 3 over denominator 5 end fraction end root$ Þ $fraction numerator X subscript C end subscript over denominator R end fraction equals square root of fraction numerator 3 over denominator 5 end fraction end root$

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