LC oscillation qualitative description

An ideal LC circuit has an inductor L and a capacitor C. Energy shuttles between the electric field of C (energy q²/2C) and the magnetic field of L (energy ½Li²).

Four stages in one period T

(a) t=0       (b) t=T/4     (c) t=T/2     (d) t=3T/4
 +|-|          | |          -|-|+         | |
 C   L        C   L         C   L         C   L
 q=+q0, i=0   q=0, i=max    q=-q0, i=0    q=0, i=max (reverse)
 all energy   all energy    all energy    all energy
 electric     magnetic      electric      magnetic

1. t = 0: capacitor fully charged (charge q0), no current. All energy is electric, q0²/2C.
2. 0 to T/4: capacitor discharges through L. Current grows, but the inductor opposes the rise (back emf). At T/4: q = 0, i is maximum, energy is all magnetic, ½L i0².
3. T/4 to T/2: current continues in the same direction because of the inductor, charging the capacitor with opposite polarity. At T/2: q = −q0, i = 0, energy is all electric again.
4. T/2 to T: the process repeats in reverse; at 3T/4 current is maximum in the opposite direction; at T the original state returns.

Graphs

 q  q0 |   .--.        .--.
       |  /    \      /    \
       |-/------\----/------\-- t
       |         \  /
       |          '--'
 i      current is a sine wave, T/4 ahead of q

• q = q0 cos ω0t and i = −ω0q0 sin ω0t, with ω0 = 1/√(LC) and period T = 2π√(LC).
• Electric energy U_E = (q0²/2C) cos²ω0t, magnetic energy U_B = (q0²/2C) sin²ω0t. They are out of phase, and U_E + U_B = q0²/2C is constant.
• In a real circuit resistance converts energy to heat, so the oscillations die out (damped).