Question

In a potentiometer experiment, the balancing length of a cell is 560 cm. When an external resistance of $10\Omega$ is connected in parallel to the cell, the balancing length changes by 60 cm. The internal resistance of the cell is
$A. 1.4 \Omega$
$B. 1.6 \Omega$
$C. 0.12 \Omega$
$D. 1.2 \Omega$

Answer 1

To solve this question, first find the emf of the cell when only the cell is connected. Then, find the emf of the cell when a resistor is connected in parallel to the cell. Compare these two equations and obtain an expression. Now, evaluate this expression to obtain an equation for internal resistance of the cell. Substitute the values in this equation and find the internal resistance of the cell.

Complete step-by-step solution:
Let the EMF of the cell be $\varepsilon$
The internal resistance be r
The balancing length when only cell is connected be ${l}_{1}$
The balancing length when resistance is connected be ${l}_{2}$
Given:
${l}_{1}= 560cm$
${l}_{2}={l}_{1}-60=560-60=500cm$
$R= 10\Omega$

When only cell is connected, emf is given by,
$E= k{l}_{1}$ …(1)
Where, k is a constant

When a resistor is connected to the cell, then the emf is given by,
$E= k{l}_{2}\left( \dfrac {R+r}{R}\right)$ …(2)
Where, k is a constant

From equation. (1) and (2) we get,
$k{l}_{1}=k{l}_{2}\left(\dfrac {R+r}{R}\right)$

Cancelling k on both the sides we get,
${l}_{1}={l}_{2}\left(\dfrac {R+r}{R}\right)$
$\Rightarrow \dfrac {{l}_{1}}{{l}_{2}}= \dfrac {R+r}{R}$
$\Rightarrow R \dfrac {{l}_{1}}{{l}_{2}}= R+r$
$\Rightarrow r= R \dfrac {{l}_{1}}{{l}_{2}}-R$
$\Rightarrow r= R \left(\dfrac {{l}_{1}}{{l}_{2}}-1\right)$

Substituting values in above equation we get,
$r= 10 \left( \dfrac {560}{500}-1 \right)$
$\Rightarrow r= 10 \left (\dfrac {560-500}{500} \right)$
$\Rightarrow r= 10 \times \dfrac {60}{500}$
$\Rightarrow r= 10 \times \dfrac {6}{50}$
$\Rightarrow r= 10 \times 0.12$
$\Rightarrow r= 1.2\Omega$

Thus, the internal resistance of the cell is $1.2\Omega$.

So, the correct answer is option D i.e. $1.2\Omega$.

Note:
Students have to keep in mind that the emf of a cell is always greater compared to the potential difference across the cell. Also, the internal resistance of the cell is not constant. It increases with the increase in external resistance connected across the terminals of the cell. Internal resistance of a new cell is always low but it increases with the continuous use of the cell. Internal resistance of the cell is inversely proportional to the temperature of the cell.