"Pumping" Charges — Question 7

Question 7

A moving charged belt links mechanical motion to source current

One face of an insulating belt carries positive surface charge density sigma. Its width is w and speed v. All transported charge is deposited on a terminal at voltage V above the pickup terminal. Assume lossless steady transfer.

1. Derive the delivered current and required mechanical power.

2. Find the opposing electric force on the belt from energy balance and explain its dependence on speed.

Question 7 - Solution

A moving charged belt links mechanical motion to source current

(a) Method and calculation

$$dQ=\sigma(wv\,dt)\Rightarrow I=\sigma wv.$$$$P_{\mathrm{mechanical}}=VI=\sigma wvV.$$

(b) Interpretation and further calculation

$$Fv=P_{\mathrm{mechanical}}\Rightarrow F=\sigma wV.$$$$\frac{dF}{dv}=0\quad\text{at fixed }\sigma,w,V.$$The force is independent of speed in this ideal model, while power grows linearly with speed.

Physical check

Power divided by speed gives force; emf itself has units of energy per charge, not force.