A potentiometer is 10 m long. It has a resistance of 20 Ω. It is connected in series with a battery of 3 V and a resistance of 10 Ω. What is the potential gradient along with wire?

A potentiometer is 10 m long. It has a resistance of 20 Ω. It is connected in series with a battery of 3 V and a resistance of 10 Ω. What is the potential gradient along with wire?
Solution:

In figure, AB is a potentiometer wire.
length of potentiometer wire, $L$ = 10 m
resistance of potentiometer wire, $R_{AB}$ = 20 Ω
resistance in series, $R$ = 10 Ω
potential gradient , $k$ = ?
total current flowing through the wire is,
$$\begin{array}{rl}I& =\frac{E}{R+R_{AB}}\\ & =\frac{3}{30}\\ \therefore I& =0.1A\end{array}$$ Now,
potential drop across AB is,
$$\begin{array}{rl}V& =I{R}_{AB}\\ & =0.1\times 20\\ \therefore V& =2V\end{array}$$ So, the potential gradient along with the wire is, $$\frac{V}{L}=\frac{2}{10}=0.2V/m$$