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 = 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, B, C, D

Explanation

(a) `10^(4)A//s`" " (b) Zero
(c ) `2.0 A`" " (d) `1.732xx10^(-4)C`
This is a problem of `L-C` oscillations
Charge stored in the capacitance oscillates simple harmonically as `Q=Q_(0)sin (omega t pm phi)`
Here `Q_(0)`=maximum value of
`Q=200 muC=2xx10^(-4) C`
`omega=1/(sqrt(LC))=1/(sqrt((2xx10^(-3)H)(5.0xx10^(-6)F)))=10^(4)s^(-1)`
Let at `t=0, Q=Q_(0)` then ......(1)
`Q(t)=Q_(0)cos omegat`
`I=(dQ)/(dt)=-Q_(0)omega sin ometgat` and .....(2)
`(dI)/(dt)=-Q_(0)omega^(2)cos(omegat).....(3)`
(a) when `Q=100 muC` or `(Q_(0))/2, cos omega t=1/2`
From Eq. (3):
`|(dI)/(dt)|=(2.0xx10^(-4)C)(10^(4)s^(-1))^(2)((1)/(2))`
`|(dI)/(dt)|=10^(4) A//s`
(b) When `Q=200 mu C` or `Q_(0)` then `cos(omegat)=1 i.e., omega t=0,2 pi....`
or `I=0`
(c ) `I_(max)=Q_(0)omega=(2.0xx10^(-4)C)(10^(4)s^(-1))`
`I_(max)=2.0A`
(b) From energy conservation.
`(1)/(2)LI_(max)^(2)=(1)/(2)LI^(2)+(1)/(2)(Q^(2))/(C )` or `Q=sqrt(LC(I_(max)^(2)-I^(2)))`
`I=(I_(max))/(2)1.0A`
`therefore Q=sqrt((20xx10^(-3))(5.0xx10^(-6))(2^(2)-I^(2)))`
`Q=sqrt3xx10^(-4) C` or `Q=1.732xx10^(-4) C`