An LC circuit consists of an inductor with inductance L and a capacitor with capacitance C . At time t = 0 , the capacitor is fully charged with charge Q 0 and the current in the circuit is zero. Wh

Question

An LC circuit consists of an inductor with inductance L and a capacitor with capacitance C . At time t = 0 , the capacitor is fully charged with charge Q 0 ​ and the current in the circuit is zero. What is the instantaneous current in the inductor when the energy stored in the electric field of the capacitor is equal to the energy stored in the magnetic field of the inductor?

TestHub · Accepted Answer

In an LC circuit, the total energy is conserved: $U_{t o t a l} = U_{E} + U_{B} = \frac{Q _{0}^{2}}{2 C}$ .

When the energy stored in the electric field ( $U_{E}$ ) is equal to the energy stored in the magnetic field ( $U_{B}$ ), then $U_{E} = U_{B} = \frac{U _{t o t a l}}{2}$ . So, $U_{B} = \frac{1}{2} L I^{2} = \frac{1}{2} (\frac{Q _{0}^{2}}{2 C})$ . This simplifies to $L I^{2} = \frac{Q _{0}^{2}}{2 C}$ , which gives $I^{2} = \frac{Q _{0}^{2}}{2 L C}$ . Therefore, the instantaneous current is $I = \frac{Q _{0}}{2 L C}$ .

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