Objectives:

• Identify and understand that a source of e.m.f. has an internal resistance
• To understand the term terminal p.d. and how it involves ‘lost volts’
• Be able to select and use the equations  $\epsilon = I (R + r)$ , and  $\epsilon = V + Ir$

The power supplied by the cell must equal the power delivered to the resistor R and the power wasted in the cell due its internal resistance;

$Power \ supplied = power \ delivered \ to \ R + power \ wasted \ in \ r$

$P_{\epsilon} = P_{R} + P_{r}$

Since two of the equations for power are;  $P = IV$  and  $P = I^{2}R$ , we can write;

$I \epsilon = I^{2}R + I^{2}r$

$I \epsilon = I^{2}(R + r)$

$\epsilon = I(R + r)$

$\frac{\epsilon}{(R + r)} = I$

by squaring everything we get;

$\frac{\epsilon^{2}}{(R + r)^{2}} = I^{2}$

Multiplying both sides by R gives;

$\frac{\epsilon^{2}R}{(R + r)^{2}} = I^{2}R$

Since  $P_{R} = I^{2}R$ , the right hand side of this equation is the power used by resistor R;

$P_{R} = \frac{\epsilon^{2}R}{(R + r)^{2}}$

And because of this the peak of the ‘power verses resistance’ curve is at R = r, as can be seen below;

Maximum power is delivered when the load resistance (of the component) is equal to the internal resistance (When the load is matched to the source).