Problem
6. [17 pts total \( ] \) In an oscillating LC circuit, \( C=30.0 \mu \mathrm{F} \). (a) If the period of the circuit is to be \( 5.50 \mathrm{~ms} \), what should the inductance of the inductor be? [6 pts] (b) If the charge on the capacitor is \( 10.0 \mu \mathrm{C} \) and there is no current at \( t=0 \), write down an explicit expression for the charge on the capacitor as a function of time \( q(t) \), giving the values of any variables you use (other than \( t \) ). [3 pts] (c) What is the maximum current? [3 pts] (d) At what time is the energy stored in the magnetic field of the inductor equal to the energy stored in the electric field of the capacitor? [ \( 5 \mathrm{pts} \) ]
Solution
Step 1 ：1. T = 5.50\times10^{-3}\mathrm{s}, C = 30.0\times10^{-6}\mathrm{F}, T=2\pi\sqrt{LC}, L=?
Step 2 ：2. L = \frac{T^2}{4\pi^2C}
Step 3 ：3. L=1.0522\times10^{-3}\mathrm{H}
Step 4 ：4. q(t) = Q_0\cos(\omega t), Q_0 = 10.0\times10^{-6}\mathrm{C}, \omega = \frac{1}{\sqrt{LC}}, q(0)=10.0\times10^{-6}\mathrm{C}, q(t)=?
Step 5 ：5. q(t)=10.0\times10^{-6}\cos(\frac{1}{\sqrt{1.0522\times10^{-3}\times30.0\times10^{-6}}}t)
Step 6 ：6. I(t) = -\frac{d}{dt}q(t), I_{max} = \frac{d}{dt}(|Q_0\cos(\omega t)|), Q_0 = 10.0\times10^{-6}\mathrm{C}, \omega = \frac{1}{\sqrt{LC}}
Step 7 ：7. I_{max} = 56.8762\times10^{-6}\mathrm{A}
Step 8 ：8. \frac{1}{2}LI_{max}^2 = \frac{1}{2}CQ_{max}^2, t=?
Step 9 ：9. t = \frac{1}{2\omega}\arccos(\frac{1}{2})
