Question

A charged $30\mu \text{F}$ capacitor is connected to a $27\text{mH}$ inductor. Suppose the initial charge on the capacitor is $6\text{mC}$. What is the total energy stored in the circuit initially? What is the total energy at later time?

Answer 1

Hint: "Total Energy in LC circuit"
Formula Used: $U=\frac{q^2}{2C}$
Given,
$L=27\text{mH}=27\times {10}^{-3}\text{H}$
$C=30\mu F=30\times {10}^{-6}\text{F}$

Initially there is no current in the circuit, energy in the inductor is zero.

Initial charge on the capacitor
$\left(q\right)=6mC=6\times {10}^{-3}\text{C}$

Total initial energy of $LC$ circuit is:
$U=\frac{q^2}{2c}$

Putting values,
$U=\frac{{(6\times {10}^{-3})}^2}{2\times 30\times {10}^{-6}}$
$U=0.6\text{J}$

As there is no resistance in the circuit, and given $C$ and $L$ are pure capacitor and inductor, then during oscillation of charge no thermal energy is dissipated $\left(i^{2}R\frac{1}{s}\right)$.

Thus total energy in the circuit remains constant. Thus, the energy in the system oscillates between the capacitor and the inductor but their sum or the total energy is constant in time

Final Answer: $0.6\text{J}$, same at later times.