"Pumping" Charges — Question 9

Question 9

A purely electrostatic loop cannot continuously heat a resistor

A proposed device sends positive charge Q around a closed stationary path using only a static electric field. It contains a resistor that would dissipate energy W>0 per circulation. No chemical, mechanical, or time-varying-field input is allowed.

1. Calculate the total work of the electrostatic field and test the proposed steady operation.

2. If a real pump is added, find its minimum work per circulation and average power when Q circulates once every T.

Question 9 - Solution

A purely electrostatic loop cannot continuously heat a resistor

(a) Method and calculation

$$W_E=Q\oint\vec E\cdot d\vec\ell=0.$$$$\Delta K+\Delta U=0\ \text{over a repeated cycle},\quad W_{\mathrm{input}}=W_{\mathrm{loss}}=W>0.$$The stated zero-input steady cycle is impossible.

(b) Interpretation and further calculation

$$W_{\mathrm{pump,min}}=W,\quad\mathcal E_{\min}=W/Q.$$$$I=Q/T,\quad P_{\mathrm{min}}=W/T=\mathcal E_{\min}I.$$

Physical check

Energy conservation requires a non-electrostatic source of work in a stationary resistive loop.