Forced Oscillations — Question 9

Question 9

Resonant charge amplitude measures resistance when L and C are known

A driven circuit has L=.100 H and C=100 microfarads. At its natural frequency the measured steady charge amplitude is 2.00 mC with source peak 10.0 V.

1. Infer the resistance and current amplitude.

2. Find the damped natural frequency after the source is disconnected, if the circuit remains closed.

Question 9 - Solution

Resonant charge amplitude measures resistance when L and C are known

(a) Method and calculation

$$\omega_0=1/\sqrt{LC}=316.228\,s^{-1},\quad Q=V_p/(R\omega_0).$$$$\boxed{R=15.8114\,\Omega,\quad I_p=\omega_0Q=.632456\,A}.$$

(b) Interpretation and further calculation

$$\alpha=R/(2L)=79.0569\,s^{-1},\quad\omega_d=\sqrt{\omega_0^2-\alpha^2}.$$$$\boxed{\omega_d=306.186\,s^{-1}}.$$

Physical check

The forced frequency equals the source frequency; after disconnection the remaining motion has the damped natural frequency.