Electric Potential Energy — Question 8

Question 8

Friction changes required actuator work without changing electrostatic path independence

A charge q moves quasistatically from x=0 to L and back along a track in a uniform field $E\hat{\mathbf x}$. Kinetic friction has constant magnitude f in either direction. Let q,E,L,f be positive.

1. Find the electric, friction and actuator work on the outward leg.

2. Find those works on the return and the total actuator work for the closed trip.

Question 8 - Solution

Friction changes required actuator work without changing electrostatic path independence

(a) Method and calculation

$$W_{e,\to}=qEL,\quad W_{f,\to}=-fL.$$$$\Delta K=0\Rightarrow W_{{\mathrm{ext}},\to}=fL-qEL.$$

(b) Interpretation and further calculation

$$W_{e,\leftarrow}=-qEL,\quad W_{f,\leftarrow}=-fL,\quad W_{{\mathrm{ext}},\leftarrow}=fL+qEL.$$$$W_{e,\mathrm{loop}}=0,\quad W_{\mathrm{ext,loop}}=2fL,\quad Q_{\mathrm{heat}}=2fL.$$

Physical check

The friction work has the same negative sign on both legs, while the conservative electric work reverses sign.