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 ?

Explanation

When a solenoid is connected to a source of direct voltage.

Power consumed, `P_(1)=V_(0)^(2)/R`
When it is connencted to a sinusoidal voltage, effective impedance comes into role.

`Z=sqrt(R^(2)+X_(L)^(2))`
Now, power consumed will be
`P_(2)=V_(rms).I_(rms)cosphi = V_(0).V_(0)/Z.R/Z=(V_(0)^(2)R)/(Z^(2))`
Given, `P_(2)=P_(1)/eta`
`rArr (V_(0)^(2)R)/(Z^(2)) = (V_(0)^(2))/(etaR)rArretaR^(2)=Z^(2)=R^(2)+X_(L)^(2)`
`rArr X_(L)^(2)=(eta-1)R^(2)rArr = (2pifL)^(2)=(eta-1)R^(2)`
`therefore` Frequency of the oscillator `f=R/(2piL)sqrt(n-1)`
`=(44)/(2 xx (22)/7 xx 7 xx 10^(-3))sqrt(5-1) = 2 xx 10^(3)Hz = 2kHz`