A series combination of resistor (R), capacitor (C) is connected to an AC source of angular frequency `omega`. Keeping the voltage same, If the frequency is changed to `Omega/3`, the current becomes half of the original current. Then, the ratio of the capacitance reactance and resistance at the former frequency is

Options

A. `sqrt(0.6)`
B. `sqrt(3)`
C. `sqrt(2)`
D. `sqrt(6)`

Answer

A

Explanation

(a) `I_("rms")=(V)/(sqrt(R^(2)+X_(C)^(2)))"and"" " I_("rms")=(V)/(sqrt(R^(2)(X_(C))^(2)))`
`because I_("rms") = (1)/(2)I_("rms")`
`implies (V)/(sqrt(R^(2)+(X_(C))^(2)))=(V)/(2sqrt(R^(2)+X_(C)^(2)))`
`implies R^(2)+(X_(C))^(2)=4(R^(2)+X_(C)^(2))`
`implies X_(C)=(3)/(omegaC),X_(C)=(1)/(omegaC)`
`3R^(2)=5X_(C)`
`(X_(C))/(R) = sqrt(0.6)`