9702/May Jun/12/2011/Q34

A source of electromotive force (e.m.f.) E has a constant internal resistance r and is connected to
an external variable resistor of resistance R.

As R is increased from a value below r to a value above r, which statement is correct?

A The terminal potential difference remains constant.
B The current in the circuit increases.
C The e.m.f. of the source increases.
D The largest output power is obtained when R reaches r.

Solution:
Answer: D

Since the battery has some internal resistance, the terminal potential difference of the battery (p.d. across its terminals – this is the p.d. available to the rest of the circuit) is less than the e.m.f E.

The voltage lost in the batter due to its internal resistance = Ir where I is the current in the circuit.
Current I = E / (R + r)
As R increases, the current I decreases. [B is incorrect] 

Terminal pd, V = E – (Ir) 
Since current I changes with the resistance R, the terminal p.d. does not remain constant as R is being changed. [A is incorrect]

The e.m.f. of the source is constant, it does not increase. [C is incorrect]

Output power in the load, P = I²R = E²R / (R + r)² 
E and r are kept constant while R is being varied. The largest output power is obtained when R reaches r. 

This can be proved by differentiating P (in the above equation) with respect to R and then equating to zero.
dP / dR = [E²(R + r)² – 2E²R(R + r)] / (R + r)⁴ 

For maximum power P, dP/dR = 0
E²(R + r)² – 2E²R(R + r) = 0               {divide by E²(R + r) on both sides,}
(R + r) – 2R = 0
R = r

Reference: PYQ - May/Jun 2011 Paper 12 Q34