A source of sinusoidal emf with constant voltage is connected in series with an oscillating circuit with quality factor `Q=100`. At a certain frequency of the external voltage the heat power generated in the circuit reached the maximum value. How much `(` in per cent`)` should this frequency be shifted to decrease the power generated `n=2.0` times ?

Explanation

`P=(V^(2)R)/(R^(2)+(X_(L)-X_(C))^(2))`
At resonance `X_(L)=X_(C)implies omega_(0)=(1)/( sqrt(LC))`
Power generated will decrease `n` times when
`(X_(L)-X_(C))-( omegaL-(1)/( omegaC))^(2)=(n-1)R^(2)`
or ` omega-(omega_(0)^(2))/(omega)=+-sqrt(n-1)(R)/(L)=+-sqrt(n-1) 2 beta`
Thus `omegabar(+)2 sqrt(n-1)beta omega - omega_(0)^(2)=0`
`( omegabar+sqrt(n-1)beta)^(2)=omega_(0)^(2)+(n-1) beta^(2)`
or `(omega)/(omega_(0)=sqrt(1+(n-1)beta^(2)//omega_(0)^(2)))+-sqrt(n-1)beta//omega_(0)`
`(` taking only the positive sign in the first term to ensure positive valur for `(omega)/( omega_(0)).)`
Now `Q=(omega)/( 2 beta)=(1)/(2) sqrt(((omega_(0))/( beta))^(2) -1)`
`(omega_(0))/( beta)=sqrt(1+ 4 Q^(2))`
Thus `(omega)/( omega_(0))=sqrt(1+(n-1)/( ( 1+4Q)))+-sqrt(n-1)//sqrt(1+ 4 Q)`
For large `Q`
`|( omega-omega_(0))/( omega_(0))|=(sqrt( en -1))/( 2Q)-(sqrt(en -1))/(2Q)xx100%=0.5%`