The Electrical-Mechanical Analogy — Question 7

Question 7

A constant driving voltage shifts the electrical equilibrium

A mass-spring system under constant force F0 has equilibrium $x_e=F_0/k$. Its analogue obeys $L\ddot q+q/C=V_0$, with positive constant V0 and no resistance.

1. Find the electrical equilibrium and the equation for displacement from it.

2. If released from q=0 and i=0, find the extrema of capacitor charge.

Question 7 - Solution

A constant driving voltage shifts the electrical equilibrium

(a) Method and calculation

$$q_e/C=V_0\Rightarrow\boxed{q_e=CV_0}.$$$$y=q-q_e\Rightarrow\boxed{L\ddot y+y/C=0}.$$

(b) Interpretation and further calculation

$$y(0)=-CV_0,\quad\dot y(0)=0\Rightarrow |y|_{\max}=CV_0\quad(V_0>0).$$$$\boxed{q_{\min}=0,\quad q_{\max}=2CV_0}.$$

Physical check

The equilibrium shifts but the small-oscillation frequency remains unchanged in this linear model.