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

In `RC` circuit, `I_(upsilon) = (E_(upsilon))/(sqrt(R^(2) + ((1)/(omega C))^(2))`
and `(I_(upsilon))/(2) = (E_(upsilon))/(sqrt(R^(2) + ((1)/((omega C)/(3)))^(2)`
`:. 2 sqrt(R^(2) + ((1)/(omega C))^(2)) = sqrt(R^(2) + (9)/(omega^(2) C^(2))`
or `3 R^(2) = (5)/(omega^(2) C^(2) = 5 X_(C )^(2)`
`:. (X_(C ))/(R ) = sqrt((3)/(5))`