13.5.2 Maximum Efficiency

Everybody wants a battery with zero internal resistance. But in real life, we get real batteries. Suppose we are stuck with a particular battery with an internal resistance r. We are interested in how the efficiency of our circuit is affected by the value of our external resistance R.

The efficiency of the circuit can be written as

$\displaystyle \eta =\frac{{\text{power dissipated in }R}}{{\text{power dissipated in }R\text{ and }r}}=\frac{{{{I}^{2}}R}}{{{{I}^{2}}R+{{I}^{2}}r}}=\frac{R}{{R+r}}$

So clearly, the larger the external resistance, the higher the efficiency. As the internal resistance becomes a smaller fraction of the total resistance, the power wasted in the internal resistance also becomes a smaller fraction of the total power. Take special notice that when $R=r$ , the efficiency is exactly 50%!