A circuit consisting of a capacitor and a coil in series is connected to the mains. Varying the capacitance of the capacitor, the heat power genergated in the coil was increased `n=1.7` times. How much `(` in per cent `)` was the value of ` cos varphi` changed in the process ?

Options

A. `80%`
B. `25%`
C. `50%`
D. `30%`

Answer

D

Explanation

`A^("verage power")`
`(:P:) = (V^(2) R)/(Z^(2))`
In `LCR` circuit, `Z = sqrt(R^(2) + (X_(L) - X_(C ))^(2))`
`sqrt(R^(2) + (omega L - (1)/(omega C))^(2))`
For `C = C_(1) (:P_(1):) = (V^(2) R)/(R^(2) + (omega L - (1)/(omega C_(1)))^(2))`
For `C = C_(2), (:P_(2):) = (V^(2) R)/(R^(2) + (omega L - (1)/(omega C_(2)))^(2))`
Accoriding to problem, `(:P_(2):) = eta (:P_(1):)`
`:. (V^(2) R)/((omega L - (1)/(omega C_(2)))^(2) + R^(2)) = eta (V^(2) R)/((omega - (1)/(omega C_(1)))^(2) + R^(2))`
or
`(omega L - (1)/(omega C_(2)))^(2) + R^(2) = ((omega L - (1)/(omega C_(1)))^(2) + R^(2))/(eta)` ......(i)
Power factor, `cos phi = (R )/(Z)`
`:. cos phi = (R )/(sqrt((omega L - (1)/(omega C_(1)))^(2) + R^(2)))`
`cos phi_(1) = (R )/(sqrt((omega L - (1)/(omega C_(2)))^(2) + R^(2)))`
`:. (omega L - (1)/(omega C_(1)))^(2) + R^(2) (R^(2))/(cos^(2) phi_(1))`
similarly,
`(omega L = (1)/(omega C_(1)))^(2) + R^(2) = (R^(2))/(cos^(2) phi_(2))` .......(3)
Form eqns. (1),(2) and (3) we get
`:. (omega L - (1)/(omega C_(2)))^(2) + R^(2) = ((omega L - (1)/(omega C_(1)))^(2) + R^(2))/(eta)`
or `(R^(2))/(cos^(2) phi_(2)) = (R^(2))/(eta cos^(2) phi_(1))` or `cos^(2) phi_(2) = n cos^(2) phi_(1)`
`(cos phi_(2))/(cos phi_(1)) = sqrt(eta)` , `:. (cos phi_(2))/(cos phi_(1)) - 1 = sqrt(eta) - 1`
`:. (cos phi_(2) - cos phi_(1))/(cos phi_(1)) = (sqrt(eta) - 1)`
`:.` Percentage change in `cos phi` is
`((cos phi_(2) - cos phi_(1))/(cos phi_(1))) xx 100 = (sqrt(eta) - 1) xx 100`
On putting the value we get percentage change `30%`