GeoGebra link: https://www.geogebra.org/m/hscshcj8

This simulation demonstrates the power dissipated in a variable resistor given that the battery has an internal resistance (made variable in this app as well).

Since the power dissipated by the resistor is given by

and the current is given by

,

This power will be a maximum if the expression for the denominator

is a minimum.

Differentiating the expression with respect to R, we get

When the denominator is a minimum,

, so

*r* = *R*.