LC circuit analogy, equal maximum energies

An LC circuit (an ideal inductor L and capacitor C, with no resistance) oscillates electrically in exactly the same mathematical way as a mass on a spring oscillates mechanically.

Differential equation

   +q |----|-q        Capacitor C discharges through L.
      |    |          At any instant: q = charge, i = dq/dt.
      C    L
      |____|

By Kirchhoff's voltage law, (voltage across C) + (voltage across L) = 0:

q/C + L di/dt = 0
L d²q/dt² + (1/C) q = 0
d²q/dt² + ω0² q = 0 ,   ω0 = 1/√(LC)

For a mass on a spring: m d²x/dt² + kx = 0, i.e. d²x/dt² + ω0² x = 0 with ω0 = √(k/m). The two equations have the same form, so q varies simple harmonically: q = q0 cos(ω0 t).

So the LC circuit is the electrical analogue of SHM.

Maximum energies

1. Charge: q = q0 cos ω0t, so the current is i = dq/dt = −q0 ω0 sin ω0t.
2. Energy in the capacitor: U_E = q²/2C = (q0²/2C) cos² ω0t. Its maximum value is q0²/2C (when i = 0).
3. Energy in the inductor: U_B = ½Li² = ½L q0² ω0² sin² ω0t. Its maximum value is ½L q0² ω0² (when q = 0).
4. Put ω0² = 1/LC: (U_B)max = ½L q0² /(LC) = q0²/2C.

(U_E)max = q0²/2C = (U_B)max

Hence the maximum energy stored in the electric field of the capacitor equals the maximum energy stored in the magnetic field of the inductor.

Also U_E + U_B = (q0²/2C)(cos² ω0t + sin² ω0t) = q0²/2C = constant. Energy is exchanged between the two fields while the total stays the same, just as KE and PE are exchanged in SHM.