A coil with inductance `L=0.70 H` and active resistance `r=20 Omega` is connected in series with an inductance `-` free resistance `R`. An alternating voltage with effective value `V=220 V` and frequency `omega=314 s^(-1)` is a applied across the terminals of this circuit. At what value of the resistance `R` will the maximum heat power be generated in the circuit ? What is it equal to ?

Explanation

`P=(V^(2)(R+r))/((R+r)^(2)+ omega^(2)L^(2))`
This is maximum where `R+r=omegaL` for
`P=(V^(2))/( R+r+((omegaL)^(2))/(R+r))=(V^(2))/( [sqrt(R+r)-(omegaL)/(sqrt(R+r))]^(2)+ 2 omegaL)`
Thus `R=omegaL- r ` for maximum power and `P_(max)=(V^(2))/(2 omega L)`
Substituting the values, we get `R=200Omega` and `P_(max)=.114 k W`