An oscillating circuit consisting of a capacitor with capacitance `C=10 muF,` a coil with inductance `L=6.0muH` and active resistance `R=10Omega.` The mean power that should be fed to the circuiit to maintain undamped harmonic oscillations with an external driving power with 50Hz and a `V_(m)` of 280 V is

Options

A. 3.8 W
B. 48 W
C. 3 mW
D. 48 mW

Answer

A

Explanation

`R= 10 Omega L = 6 muH,C=10muH`

Impedence of the circuit,
`Z=sqrt(R^(2)+(X_(c)-X_(L))^(2))`
Capacitive reactance, `X_(c)=(1)/(omegaC) = (1)/(2pifC)`
`X_(c)=(1)/(2xx314xx50xx10xx10^(-6))`
`=318.47 Omega`
Inductive reactance `X_(L)= omegaL = 2pifL`
`X_(L)=2xx314 xx 50 xx 6 xx 10 ^(-6)`
`= 18.84 xx 10^(-4) Omega`
Resistance, R `=10Omega`
`Z= sqrt(10^(2)+(318.4-18.84xx10^(-4))^(2))`
`Z = sqrt(100+(318.39)^(2))=sqrt(101477.36)`
`Z=318.55 Omega`
rms current, `i_(rms)= (V_(rms))/(Z) = (V_(0)//sqrt(2))/(Z)`
`i_(rms)= (280)/(sqrt(2)xx318.55)=0.621 A`
As energy is only dissipated in resistance as heat and no power dissipates in capacitor and inductor, so average power dissipates as heat
`p= i_(rms)^(2).R`
`p=(0.621)^(2)xx10`
`p= 3.8 W`
3.8 W power should be fed to maintain an undamped oscillation.