• 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).

Further reading: