The electric power transferred by a cell to an external resistance is maximum when the external resistance is equal to _____________.

(r = internal resistance)

Let the external resistance be R and the value of current is i
The value of current in the circuit is calculated as,
$i=\frac{E}{R+r}$

power $P = i^{2}R$

Therefore, $P = {\left(\frac{E}{R+r}\right)}^{2}R$

To find the maximum value of P, we differentiate P w.r.t R and find the value of R for which $\frac{dP}{dR}$= 0. 

This gives us R = r for maximum power.

Hence, the power will be maximum when the value of external resistance is equal to the value of internal resistance.