Question

The figure shows a potentiometer in which R is a variable resistance and AB is uniform wire of length 100cm having resistance r. In an experiment when R=0 the null point is obtained at 30cm from end A. For another value of R, the null point is obtained at 60cm from A. Find the A value R in the second case.

Answer 1

Hint : As a very first step, one could read the question well and hence note down the important points from it. Then analyse the circuit well and hence consider the two cases. Then one could find the current in both cases and then equate them to get the value of the external resistance value that is asked.

Formula used:
Current,
$I={\displaystyle \frac{V}{R+r}}$

Complete step-by-step solution:
In the question, we are given two cases of null points for the potentiometer of variable resistance R and AB uniform wire of length 100cm with resistance r. In the first case we have the null point at 30cm from end A and in the second case we have the null point at 60cm from end A. We are supposed to find the resistance R for the second case.
The internal resistance of 100cm wire is r, then the internal resistance for 1cm would be 0.1r.
In the first case, external resistance is zero and the null point is obtained at 30cm from end A and the corresponding internal resistance would be 0.3r.
The current here would be given by ohm’s law as,
$I={\displaystyle \frac{V}{R+r}}={\displaystyle \frac{10}{0+0.3r}}={\displaystyle \frac{100}{3r}}$………………………………………..(1)
For the second case with some value of R, the null point is at 60cm from end A, the corresponding internal resistance would be 0.6r.
The current here would be,
$I={\displaystyle \frac{V}{R+r}}={\displaystyle \frac{10}{R+0.6r}}$……………………………………………………………… (2)
Equating (1) and (2),
$100(R+0.6r)=10\left(0.3r\right)$
$\Rightarrow 10R+6r=0.3r$
$\Rightarrow 10R=-5.7r$
$\therefore R=0.57r$
Therefore, we found the value of R to be 0.57r.

Note: A potentiometer can be considered as a three terminal resistor that has a sliding as well as rotating contact which has the capability to form an adjustable voltage divider. If any two of these three terminals are used, it acts as a variable resistor or a rheostat.