LC Oscillations, Quantitatively — Question 1

Question 1

Two nonzero initial conditions determine the LC phase

An ideal LC loop has $L=.500$ H, $C=2.00\,\mu F$, $q(0)=6.00\,\mu C$, and $i(0)=8.00$ mA, with $i=\dot q$.

1. Find q(t), its amplitude, and its phase in the form $Q\cos(\omega t-\delta)$.

2. Find peak current and total energy.

Question 1 - Solution

Two nonzero initial conditions determine the LC phase

(a) Method and calculation

$$\omega=1/\sqrt{LC}=1000\,s^{-1},\quad q=q_0\cos\omega t+(i_0/\omega)\sin\omega t.$$$$\boxed{q=(10.0\,\mu C)\cos(1000t-.927295),\quad\delta=\tan^{-1}(8/6)}.$$

(b) Interpretation and further calculation

$$I_{\max}=\omega Q=.0100\,A,\quad U=Q^2/(2C).$$$$\boxed{U=25.0\,\mu J}.$$

Physical check

A positive initial current requires a negative phase in the cosine argument.