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|`?

Explanation

`L = 2.0 mH = 2.0 xx 10^(-3) H`
`C = 5.0 mu F = 5.0 xx 10^(-6) F`
`Q_(max) = 200 mu C = 200 xx 10^(-6) C`
In an `LC` circuit, energy transfer continues from inductance to capacitance and vice versa.
a. By kirchhoff's law in an `LC` circuit
`(Q)/(C) + L(dI)/(dt) = 0 rArr (dI)/(dt) = - (Q)/(LC)`
`:. |(dI)/(dt)| = (Q)/(LC) = (100 xx 10^(-6))/(2.0 xx 10^(-3) xx 5.0 xx 10^(-6)) = 10^(4) A s^(-1)`
b. When `Q = 200 mu C`, the entire energy of circuit resides in capacitance. That is, no energy is stored in inductance.
`:. (1)/(2) LI^(2) = 0 rArr I = 0`
c. Maximum value of `I` is given by
`(1)/(2) LI_(max)^(2) = (Q_(max)^(2))/(2C) rArr I_(max) = (1)/(sqrt(LC)) Q_(max)`
or `I_(max) = (1)/(sqrt((2.0 xx 10^(-3))xx(5.0 xx 10^(-6)))) xx 200 xx 10^(-6) = 2A`
d. Given `I = I_(max)/(2) = (2)/(2) = 1A`
Then again from the conservation of energy,
`(1)/(2) LI^(2) + (Q^(2))/(2C) = (1)/(2) LI_(max)^(2), (Q^(2))/(2C) = (1)/(2) L(L_(max)^(2) - I^(2))`
`Q = sqrt(LC(I_(max)^(2) - I^(2))`
`= sqrt((2.0 xx 10^(-3) xx 5.0 xx 10^(-6))(2^(2) - 1^(2))`
`= 10^(-4) sqrt(3) C = 1.732 xx 10^(-4) C = 1.732 mu C`