Question

In an experiment to measure the internal resistance of a cell by a potentiometer, it is found that the balance point is at a length of 2 m, when the cell is shunted by a $5\;\Omega$ resistance; and is at a length of 3 m, when the cell is shunted by a $10\;\Omega$ resistance. The internal resistance of the cell is
A. $1\;\Omega$
B. $1.5\;\Omega$
C. $10\;\Omega$
D. $15\;\Omega$

Answer 1

The above problem can be solved using the concept of the potentiometer. The different variables and terms are used to resolve the given situation and the appropriate measurement for the variables' magnitude. The cell's internal resistance is calculated by considering the mathematical formula relating to the length of the balance point and the total length of the shunt wire with the resistances at the specific point.

Complete step by step answer:
Given:
The length from the balance point is, ${L_1} = 2\;{\rm{m}}$.
The shunt resistance is, ${R_1} = 5\;\Omega$.
The length of the shunt is, ${L_2} = 3\;{\rm{m}}$.
The resistance for the shunted cell is, ${R_2} = 10\;\Omega$.
Write the mathematical formula for the ratio of lengths as,
$\dfrac{{{L_1}}}{{{L_2}}} = \dfrac{{{R_1}\left( {{R_2} + r} \right)}}{{{R_2}\left( {{R_1} + r} \right)}}$
Here, r is the internal resistance.
Solve by substituting the values in the above equation as,

\dfrac{{{L_1}}}{{{L_2}}} = \dfrac{{{R_1}\left( {{R_2} + r} \right)}}{{{R_2}\left( {{R_1} + r} \right)}}\\\ \dfrac{{2\;{\rm{m}}}}{{3\;{\rm{m}}}} = \dfrac{{5\;\Omega \times \left( {10\;\Omega + r} \right)}}{{10\;\Omega \times \left( {5\;\Omega + r} \right)}}\\\ 2\;{\rm{m}} \times \left[ {10\;\Omega \times \left( {5\;\Omega + r} \right)} \right] = 3\;{\rm{m}} \times \left[ {5\;\Omega \times \left( {10\;\Omega + r} \right)} \right]\\\ r = 10\;\Omega \end{array}$$ Therefore, the internal resistance of the cell is $$10\;\Omega $$ **So, the correct answer is “Option C”.** **Note:** Try to understand the meaning of the potentiometer, along with the working of the potentiometer. The potentiometer is an arrangement that can be used to find the unknown value of resistances and the cell's internal resistance. The potentiometer reading is taken with the element known as the jockey wire. That indicates the points like a neutral point; this neutral point is also known as the Balance point. Moreover, the potentiometer is also used to determine the unknown values of potential differences. There are several applications for the potentiometer and its concepts are vital for the field of electrical research and analysis.