Question

A battery of e.m.f. E and internal resistance r is connected to an external resistance R the condition for maximum power transfer is

• A. $\displaystyle r
• B. $r>R$
• C. $r=1/R$
• D. $r=R$

Answer 1

Answer: (D)

$r=R$

We know that the current in the circuit $I=\frac{E}{R+r}$
and power delivered to the resistance R is
$P=I^{2}R=\frac{E^{2}R}{{(R+r)}^2}$
It is maximum when $\frac{dP}{dR}=0$
$\frac{dP}{dR}=E^2\left[\frac{{(r+R)}^2-2R(r+R)}{{(r+R)}^4}\right]=0$
or ${(r+R)}^2=2R(r+R)orR=r$