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A series circuit consisting of a capacitor with capacitance with capacitance `C`, a rasistance `R`, and a coil with inductance `L` and negligible active resistance is connected to an oscillator whose frequency can be varied without changing the voltag amplitude. Find the frequency at which the voltage amplitude is maximum `(a)` across the capacitor , `(b)` across the coil. |
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Answer» `(a) v_(c)=(1)/( omegaC) (V_(m))/( sqrt(R^(2)+(omegaL-(1)/( omegaC))^(2)))` `=( V_(m))/( sqrt( (omegaRC)^(2)+ ( omega^(2) LC-1)^(2)))=(V_(m))/( sqrt(((omega^(2))/( omega_(0)^(2))-1)^(2)+ 4 beta^(2) omega^(2) // omega_(0)^(4)))` `=(V_(m))/( sqrt(((omega^(2))/( omega_(0)^(2))-1-( 2 beta)/( omega_90)^(2))^(2)+(4 beta^(2))/( omega_(0)^(2))-( 4 beta^(4))/( omega_(0)^( 4))))` This is maximum when `omega^(2)=omega_(0)^(2)- 2 beta^(2) = (1)/( LC)-(R^(2))/( 2L^(2))` `(b)` `V_(L)=I_(m) omegaL=Vm(omegaL)/( sqrt(R^(2)(omegaL-(1)/( omegaL))^(2)))` `=( V_(m)L)/( sqrt((R^(2))/(omega^(2))+(L-(1)/( omega^(2)C))^(2)))=(V_(m)L)/(sqrt(L^(2)-(1)/( omega^(2))((2L)/(C)-R^(2))+(1)/( omega^(4)C^(2))))` `=(V_(m)L)/( sqrt(((1)/( omega^(2)C)-(L- ( CR^(2))/( 2)) )^(2)+L^(2)- ( L- (1)/( 2) CR^(2))^(2)))` This is maximum when `(1)/( omega^(2)C)=L-(1)/( 2) CR^(2)` or ` omega^(2) =( 1)/( LC-(1)/(2) C^(2) R^(2))=(1)/( (1)/( omega_(0)^(2))-( 2 beta^(2))/( omega_(0)^(4)))` `=( omega_(0)^(4))/( omega_(0)^(2)-2 beta^(2)) `or ` omega=( omega_(0)^(2))/( sqrt( omega_(0)^(2)-2 beta^(2)))` |
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