A solenoid with inductance `L=7 m H` and active resistance `R=44 Omega` is first connected to a source of direct voltage `V_(0)` and then to a source of sinusoidal voltage with effective value `V=V_(0)` . At what frequency of the oscillator will be power consumed by the solenoid be `eta=5.0 ` times less than in the former case ?

Answer

2

Explanation

In case of direct current, `P_(DC)=V_(M)^(2)/R`
The impedance of `LR` circuit is
`Z=sqrt(R^(2)+X_(L)^(2))=sqrt(R^(2)+omega^(2)L^(2))`
`:. I_(M)=V_(M)/Z=V_(M)/sqrt(R^(2)+omega^(2)L^(2))`
`:. P_(AC)=I_(M)^(2)R=(V_(M)^(2)R)/(R^(2)+omega^(2)L^(2))`
According to the problem, `P_(DC)=eta P_(AC)`
or `V_(M)^(2)/R=eta (V_(M)^(2)R)/(R^(2){1+((omegaL)/R)^(2)})` or `1/R=eta/(R{1+((omegaL)/R)^(2)})`
or `omega=R/L sqrt(eta^(2)-1) rArr 2piv=R/L sqrt(eta^(2)-1)`
`v=R/(2pi L)sqrt(eta^(2)-1)=2kHz`