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

Power consumed `P_(1) = (V_(0)^(2))/R `
`Z =sqrt( R^(2) + X_(L)^(2))`
Power consumed
`P_(2) = V_(rms) i_(rms) cos phi`
` =V_(0) (V_(0))/(Z). R/(Z)`
`= (V_(0)^(2))/(Z^(2)`
Given `P_(2) = (p_(1))/eta`
`(V_(0)^(2))/(Z^(2)) = (V_(0)^(2))/(etaR) implies etaR^(2) = Z^(2) = R^(2) + X_(L)^(2)`
`X_(L)^(2) = (eta -1) R^(2)`
`f = (R)/(2pi L) sqrt(beta -1)`
`= (4)/(22 xx (22)/(7)xx 7xx 10^(-3)) sqrt(5 -1)`
`= 2xx 10^(3) Hz = 2kHz`

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