Battery shown in figure has e.m.f. E and internal resistance r. Current in the circuit can be varied by sliding the contact J.If at any instant current flowing through the circuit is I, potential difference between terminals of the cell is V, thermal power generated in the cell is equal to
$\eta$ fraction of total electrical power generated in it, then which of the following graphs is correct?

 

Solution:

Terminal voltage $V=E-Ir.$  Hence, the graph between V and I will be a straight line having negative slope and positive intercept.
Thermal power generated in the external circuit, $P=EI-I^{2}r.$ Hence, graph between P and I will be a parabola passing through origin.
Also at an instant, thermal power generated in the cell = I²r and total electrical power generated in the cell
= Ei. Hence the fraction $\mathrm{\eta }=\frac{{\mathrm{I}}^2\mathrm{r}}{\mathrm{EI}}=\left(\frac{\mathrm{r}}{\mathrm{E}}\right)\mathrm{I};$ so $\eta \propto I.$ means graph between $\eta$ and I will be a straight line passing through origin.