An inductor of inductance `2.0mH` is connected across a charged capacitor of capacitance `5.0muF` and the resulting `L-C` circuit is set oscillating at its natural frequency. Let `Q` denote the instantaneous change on the capacitor and `i` the current in the circuit. It is found that the maximum value of `Q` is `200muC` .
(a) When `Q=100muC` , what is the value of `|di//dt|` ?
(b) When `Q=200muC` , what is the value of `i` ?
(c)Find the maximum value of `i`
(d) When `i` is equal to one-half its maximum value, what is the value of `|Q|` ?

Explanation

`C=5muhF` , `L=2mH` , `Q_(0)=200muC`
`omega+(1)/(sqrt(LC))=(1)/(sqrt(2xx10^(-2)xx5xx10^(-6)))=10^(4)rad//sec`
(a) `Q=Q_(0)cosomegat`
`i=-(dQ)/(dt)=Q_(0)omegasinomegat`
`(di)/(dt)=Q_(0)omega^(2)cosomegat=omega^(2)Q`
`=(10^(4))^(2)(100xx10^(-6))=10^(4)A//sec`
(b) When `Q=Q_(0)` i.e. `cosomegat=1` , `sinomegat=0`
`i=Q_(0)omegasinomegat=0` OR When `Q=Q_(0)` , whole energy resider in capacitor, i.e. `i=0`
(c) `i=_(0)omegasinomegat`
`i_(max)=Q_(0)omega=(200xx10^(-6)xx(10^(4))=2xx10^(4)A` OR Total energy `=(Q_(0)^(2))/(2C)=(1)/(2)Li_(max)^(2)`
`i_(max)=(Q_(0))/(sqrt(LC))=Q_(0)omega`
(d) `i=Q_(0)omegasinomegat=i_(max)sinomegat`
When `i=(1)/(2)i_(max)`
`(i_(max))/(2)=i_(max)sinomegat impliessinomegat=(1)/(2)`
`cosomegat=(sqrt(3))/(2)`
`Q=Q_(0)cosomegat=(sqrt(3))/(2)Q_(0)=(sqrt(3))/(2)xx200`
`=100sqrt(3)muC`