We have to show that in a damped LC circuit the fraction of the energy lost per cycle of oscillation, , is given to a close approximation by . The quantity is often called the Q of the circuit (for “quality”). A “high-Q” circuit has low resistance and a low fractional energy loss per cycle .

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The differential equation giving the variation of charge q as a function of time in a damped LC circuit is

The solution of this differential equation is

where

.

Assuming that

,

current can be approximated by the function

The total electromagnetic energy at any instant in the LC circuit is the sum of electrical energy stored in the capacitor and the magnetic energy stored in the inductor. That is

The dissipation of energy in the LC circuit in the resistor R is given by the relation

The total dissipation of energy over one cycle will therefore be given by the integral

The quantity is called the Q of the circuit.

Thus