An a.c. 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 the 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 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

`I_(rms) = (V_(rms))/(sqrt(R^(2) + ((1)/(omegaC))^(2)))" "` . . . (i)
and` (I_(rms))/(2) = (V_(rms))/(sqrt(R^(2)((1)/((omegaC)/3))^(2))) = (V_(rms))/( sqrt(R^(2) + (9)/(omega^(2) C^(2))))" "` . . . (ii)
From eq. (i) and (ii)
`3 R^(2) = (5)/( omega^(2) C^(2)) rArr sqrt((3)/(5)) xx R = (1)/(omegaC)`
`(X_(C))/(R) = sqrt((3)/(5))`