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((5)/(3))`
C. `sqrt((5)/(4))`
D. `sqrt((3)/(4))`

Answer

A

Explanation

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

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