14.6 Exercise 19

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 lightly damped circuit the energy, U, in the circuit decreases according to the following equation.

$\frac{d U}{d t} = −2 \beta U ,$ where $\beta = \frac{R}{2 L} .$

(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 $$

$Q \equiv \frac{U_{\text{begin}}}{\Delta U_{\text{one cycle}}} = \frac{1}{2 \pi R} \sqrt{\frac{L}{C}} .$

(Hint: For (b), to obtain Q, divide E at the beginning of one cycle by the change $\Delta E$ over the next cycle.)