Constant total energy in the LC circuit

Since there is no resistor in the ideal LC circuit, there is no dissipation: the total energy is constant in time. At every instant it is the sum of the electrical energy in the capacitor and the magnetic energy in the inductor.

Total energy of the LC circuit

$\ev{E_\text{tot} = \frac{Q^2}{2C} + \frac{L\,i^2}{2} = \text{costante}}$

The motion is a continuous transfer between the two forms, exactly as in a pendulum energy passes from potential to kinetic and back:

• When $Q$ is at its maximum ($Q_0$), the current is zero ($i = 0$): all the energy is electrical, stored in the charged capacitor. This is the analogue of a pendulum at rest at its highest point.
• A quarter period later, $Q = 0$ and the current is at its maximum ($i_0$): all the energy is magnetic, stored in the field of the inductor. This is the analogue of the pendulum flying through its lowest point.

And so on, indefinitely. The constant value of $E_\text{tot}$ also lets us relate the two amplitudes: equating the energy at the two extreme instants, $Q_0^2/(2C) = L\,i_0^2/2$, from which $i_0 = Q_0/\sqrt{LC} = \omega\,Q_0$.

Collegamenti

Argomenti: Electromagnetic induction Concetti: LC and RLC oscillations · Conservation of mechanical energy

Esercizi collegati: Problem — Natural frequency of an LC circuit · Problem — Capacitor of an AM radio receiver · Worked exercise — Forced RLC, find the generator’s EMF