## Geogebra App on Maximum Power Theorem

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

$P=I^2R$

and the current is given by

$I=E(R+r)$,

$P= E^2\times\frac{R}{(R+r)^2}=\frac{E^2}{\frac{r^2}{R}+R+2r}$

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

$\frac{r^2}{R}+R+2r$

is a minimum.

Differentiating the expression with respect to R, we get

$\frac{d(r^2/R+R+2r)}{dR}=-\frac{r^2}{R^2}+1$

When the denominator is a minimum,

$-\frac{r^2}{R^2}+1=0$, so

r = R.

## Geogebra Simulation of a Potentiometer

Some of the more challenging problems in the topic of electricity in the A-level syllabus are those involving a potentiometer. The solution involves the concept of potential divider and the setup can be used to measure emf or potential difference across a variety of circuits components. Basically, students need to understand the rule - that the potential difference across a device is simply a fraction of the circuit's emf, and that fraction is equal to the resistance of the device over the total resistance of the circuit.

$V_{device}=\frac{R_{device}}{R_{total}}*emf$

The intention of this Geogebra app is for students to practise working on their calculations, as well as to reinforce their understanding of the principle by which the potentiometer works.