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

Answer

B

Explanation

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