An inductor of inductance `2.0 mH` s connected across a charged capacitor of capacitance `5.0 mu F` and the resulting `LC` circuit is set oscillating at its natural frequency. Let `Q` denote the instantaneous charge on the capacitor and `I` the current in the circuit. It is found that the maximum value of charge `Q` is `200 mu C`.
a. When `Q = 100 mu C`, what is the value of `|(dI)/(dt)|`?
b. When `Q = 200 mu C`, 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|`?

Answer

`(a) 10^(4) A//s (b) 0 (c ) 2A (d) 100sqrt(3)muC`

Explanation

Given that
`Q=Q_(0) cosomegat`
`omega=1/sqrt(LC)=1/sqrt(2xx10^(-3)xx5xx10^(-6))=10^(4) rad//sec`
`(a) Q=Q_(0) cosomegat implies 100=200 cosomegat`
`cosomegat=1/2`
`since Q=Q_(0) cosomegat implies I=(dQ)/(dt)=Q_(0) omega sinomegat`
`(dI)/(dt)=-Q_(0)omega^(2) cosomegat`
`|(dI)/(dt)|=omega^(2)(Q_(0)cosomegat)=omega^(2)Q=10^(8)xx100xx10^(-6)`
`10^(4) Amp//sec`
`(b) Q=Q_(0)cosomegat`
`200=200 cosomegat implies cosomegat=1`
`sinomegat=0`
`I=(dQ)/(dt)=-Q_(0)omega sin omegat=0`
`(c ) l_(max)=Q_(0)omega=200xx10^(-6)xx10^(4)=2Amp`
`(d) I=-I_(0) sinomegat`
`I_(0)/2=-I_(0)sinomegat`
`sinomegat=sin (7pi)/6 implies omegat=(7pi)/6`
`cosomegat=cos(7pi)/6=-cospi/6=-(-sqrt(3))/2`
`Q=Q_(0)cosomegat=200muCxx(-sqrt(3)/2)`
`=-100sqrt(3)muC`
`|Q|=100sqrt(3) muC`