Chapter 14, Problem 83CP

In a damped oscillating circuit the energy is dissipated In the resistor. The Q-factor Is a measure of the persistence of the oscillator against the dissipative loss. (a) Prove that for a ligt1y danced circuit the energy, U, in the circuit decreases according to the following equation.

$\frac{du}{dt}=-2\beta u,where\beta =\frac{R}{2L}$

(b) Using the definition of the Q-factor as energy divided by the loss over the next cycle, prove that Q-factor of a lightly damped oscillator as defined in this problem is

$Q=\frac{U_{begin}}{△U_{onecycle}}=\frac{1}{R}\sqrt{\frac{L}{C}}$c

(Hint: For (b), to obtain Q, divide E at the beginning of one

cycle by the change E over the next cycle.)