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|`?

Options

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 of its maximum value, what is the value of `|Q|`?

Explanation

This is a problem `L-C` oscillations.
Charge stored in the capacitor oscillates simple harmonically as
`Q=Q_0sin(omegat+-phi)`
Here `Q_0=`maximum value of `Q=200muC=2xx10^-4C`
`omega=1/sqrt(LC)=1/sqrt((2xx10^-3)(5.0xx10^-6))=10^4s^-1`
Let at `t=0, Q=Q_0` then
`Q(t)=Q_0cosomegat` ........i
`I(t)=(dQ)/(dt)=-Q_0omegasinomegat` and ........ii
`(dI(t))/(dt)=-Q_0omega^2cosomegat` ............iii
a. `Q=100muC`
or `Q_0/2` at `cosomegat=1/2`
or `omegat=pi/3`
At `cosomegat=1/2,` from eqn iii:
`|(dI)/(dt)|=(2.0xx10^-4C)(10^4s^-1)^2(1/2)`
`|(dI)/(dt)|=10^4A//s`
b. `Q=200muC` or `Q_0` when `cosomegat=1,` i.e. `cosomegat=0,2pi`........
At the time `I(t)=-Q_0omegasinomegat`
or `I(t)=0 (sin0^2=sin2pi=0)`
c. `I(t)=-Q_0omegasinomegat`
`:.` Maximum value of `I` is `Q_0omega`
`I_max=Q_0omega=(2.0xx10^-4)(10^4)`
`I_max=2.0A`
d. From energy conservation
`1/2LI_max^2=1/2LI^2+1/2Q^2/C`
or `Q=sqrt(LC(I_max^2-I^2))`
`I=I_max/2=1.0A`
`:. Q=sqrt((2.0xx10^-3)(5.0xx10^--6)(2^2-1^2))`
`Q=sqrt3xx10^-4C`
or `Q=1.732xx10^-4C`