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 the circuit is found to be halved. Calculate the ratio of the reactance to resistance at the original frequency `omega`.

Options

A. `sqrt((3)/(5))`
B. `sqrt((2)/(5))`
C. `sqrt((1)/(5))`
D. `sqrt((4)/(5))`

Answer

A

Explanation

(a) At angular frequency `omega`, the current in `RC` circuit is given by
`I_(rms) = (V_(rms))/(sqrt(R^(2) + ((1)/(omega C))^(2)))`
Also, `(I_(rms))/(2) = (V_(rms))/(sqrt(R^(2) + [(1)/((omega C)/(3))]^(2))) = (V_(rms))/(sqrt(R^(2) + (9)/(omega^(2) C^(2)))`
From Eqs. (i) and (ii), we get, `3 R^(2) = (5)/(omega^(2) C^(2))`
`implies (1)/((omega C)/(R )) = sqrt((3)/(5))` `implies (X_(C ))/(R ) = sqrt((3)/(5))`