A one microfarad condenser is charged to \(50 \mathrm{~V}\). The charging battery is then disconnected and a \(10 \mathrm{mH}\) coli is connected across the capacitor so that LC oscillations occur. What is the maximum current in the coil? Assume that the circuit contains no resistance.

\(\frac{\mathrm{q}}{\mathrm{C}}=-\mathrm{L} \frac{\mathrm{di}}{\mathrm{dt}}=(-\mathrm{L}) \frac{\mathrm{dq}}{\mathrm{dt}}=\frac{\mathrm{di}}{\mathrm{dq}} \Rightarrow \int_{\mathrm{Q}_0}^0 \mathrm{qdq}\) \(=(-\mathrm{LC}) \int_0^{\mathrm{i}} \max \mathrm{idi} \)
\( \left.\Rightarrow \frac{\mathrm{q}^2}{2}\right|_{\mathrm{Q}_0} ^0=\left.(-\mathrm{LC}) \frac{\mathrm{i}^2}{2}\right|_0 ^{\mathrm{i}^{\mathrm{max}}} \Rightarrow 0-\mathrm{Q}_0^2=\) \((-\mathrm{LC})\left(\mathrm{i}_{\max }{ }^2\right) \)
\( \Rightarrow \mathrm{i}_{\max }=\frac{\mathrm{Q}_0}{\sqrt{\mathrm{LC}}}=\frac{\mathrm{CV}}{\sqrt{\mathrm{LC}}}=50 \sqrt{\frac{1 \times 10^{-6} \mathrm{~F}}{10 \times 10^{-3} \mathrm{H}}} \)
\( =50 \times 10^{-2} \mathrm{~A} \)
\( \Rightarrow \mathrm{i}=0.5 \mathrm{~A}\)