For the `LCR` circuit, shown here, the current is observed to lead the applied voltage. An additional capacitor `C'`, when joined with the capacitor `C` present in the circuit, makes the power factor of the circuit unity. The capacitor `C'` must have been connected in:

Options

A. series with C and has a magnitude `(C)/((omega^(2)LC-1))`
B. series with C and has a magnitude `((1-omega^(2)LC))/(omega^(2)L)`
C. parallel with C and has a magnitude `((1-omega^(2)LC))/(omega^(2)L)`
D. parallel with C and has a magnitude `(C)/((omega^(2)LC-1))`

Answer

C

Explanation

Power factor an AC circuit containing L, C and R connected in series is given by
`cosphi=(R)/(sqrt(R^(2)+[omegaL-(1)/(omegaC)]^(2))`
When an additional capacitance `C'` is joined in parallel with capacitor C, then it makes power factor of circuit unit i.e.
`cosphi=1 implies (R)/(sqrt(R^(2)+(omegaL-(1)/(omega(C+C')))^(2)))=1`
`implies omegaL=(1)/(omega(C+C'))implies C+C'=(1)/(omega^(2)L)`
`implies C'=(1-omega^(2)LC)/(omega^(2)L)`
`implies C'=(1-omega^(2)LC)/(omega^(2)L)`