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

Answer

A

Explanation

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