6.1 Maximum power transfer theorem

1.1 Current in different situations

2.1 Expression for drift velocity
2.2 Mobility $\left( \mu \right)$

3.1 Temperature dependence of resistance
3.2 Resistivities of different materials
3.3 Limitations of ohm's law

5.1 Combination of cells

6.1 Maximum power transfer theorem
6.2 Kirchhoff's law

9.1 Comparison of emfs of two primary cells.
9.2 Determination of Internal resistance of a cell using potentiometer

10.1 Ammeter
10.2 Voltmeter

Current $(I)$ in the circuit, $$ \Rightarrow I = \frac{\xi }{{R + r}}$$ Power developed in resistance $R$, $$\begin{equation} \begin{aligned} \Rightarrow P = {I^2}R \\ \Rightarrow P = {\left( {\frac{\xi }{{R + r}}} \right)^2}R \\ \Rightarrow P = \frac{{{\xi ^2}R}}{{{{\left( {R + r} \right)}^2}}} \\\end{aligned} \end{equation} $$

Let us find for what value of $R$ maximum power is drawn from the battery. So, we will differentiate $P$ with respect to $R$. $$\begin{equation} \begin{aligned} \Rightarrow \frac{{dP}}{{dR}} = {\xi ^2}\frac{{\left[ {2R(R + r) - {{(R + r)}^2}} \right]}}{{{{(R + r)}^4}}} = 0 \\ \Rightarrow 2R(R + r) - {(R + r)^2} = 0 \\ \Rightarrow R = r \\\end{aligned} \end{equation} $$

So, to draw maximum power from the battery external resistance $R$ should be equal to the internal resistance $r$.