Work, Energy, and Emf — Question 3

Question 3

Equal charge in and out does not give unit energy efficiency

A cell of constant emf E and resistance r is charged by quantity Q at current magnitude I, then discharged by the same Q at the same I. Assume $Ir<E$ and no other loss.

1. Find input energy, recovered energy and total heat.

2. Find round-trip efficiency and its first-order approximation for $Ir/E\ll1$.

Question 3 - Solution

Equal charge in and out does not give unit energy efficiency

(a) Method and calculation

$$W_{\mathrm{in}}=(E+Ir)Q,\quad W_{\mathrm{recovered}}=(E-Ir)Q.$$$$W_{\mathrm{heat}}=W_{\mathrm{in}}-W_{\mathrm{recovered}}=2IrQ.$$

(b) Interpretation and further calculation

$$\eta=\frac{E-Ir}{E+Ir}=\frac{1-x}{1+x},\quad x=Ir/E.$$$$\eta=1-2x+O(x^2).$$

Physical check

Chemical energy returns to its initial value after equal charge cycling, but irreversible heating remains.