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`.

Explanation

`X_(C) = (1)/(omegaC)`
`Z = sqrt(R^(2) + X_(C)^(2)`
`I = (V)/(Z) = (V)/sqrt(R^(2) + X_(C)^(2))`
`X_(C) = (1)/(omega/(3)C) = (3)/(omegaC) = 3X_(C)`
`Z' = sqrt(R^(2) + X_(C)^(2)) = sqrt(R^(2) + (3X_(C))^(2)`
`(1)/(2) = (V)/(sqrtR^(2) + 9X_(C)^(2))`
`(2)/(X) (V)/(sqrt(R^(2) + X_(C)^(2)))= (V)/(sqrt(R^(2) + 9X_(C)^(2))`
`R^(2) + 9X_(C)^(2) =4R^(2) + 4X_(C)^(2)`
`5X_(C)^(2) = 3R^(2)`
`(X_(C))/(R) = sqrt(3/(5)`
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