Question

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

Answer 1

Correct Answer - A
`I_(rms) = V_(rms)/(sqrt(R^(2) + (1/(omegaC))^(2)))`...........(i)
`I_(rms)/2= V_(rms)/(sqrt(R^(2)+[(1/((omegaC)//3))]^(2)))= V_(rms)/sqrt(R^(2)+9/(omega^(2)C^(2)))`..........(ii)
From eqs. (i) and (ii), we get
`3R^(2)= 5/(omega^(2)C^(2))`
`1/(omegaC//R)=sqrt(3/5)`
or `X_(C)/R = sqrt(3/5)` or `X_(C)/R = sqrt(0.6)`