In a potentiometer experiment, it is found that no current passes through the galvanometer when the terminals of the cell are connected across 52cm of the potentiometer wire. If the cell is shunted by a resistance of 5$\Omega$, a balance is found when the cell is connected across 40cm of the wire. Find the internal resistance of the cell.
A. $0.5 \Omega$
B. $0.6 \Omega$
C. $1.2 \Omega$
D. $1.5 \Omega$

Hint: A potentiometer is a device that is used in the measurement of internal resistance of a cell. If a resistance is connected in series with the internal cell resistance then, the current in the circuit reduces due to the increased resistance and so the potential difference modifies accordingly.

Formula used:
If $l_0$ is the length (of null point) when only cell E (with resistance r) is connected and l is the length when R shunt is connected then,
$r = (\dfrac{l_0 - l}{l})R$

Complete step by step answer:
A potentiometer consists of a long wire that has some resistance per unit length (when this is multiplied by the length, we get total resistance of the wire). When the balance is reached, the potential difference of the battery (or total resistance battery + shunt) will be proportional to the balance length.

Initially, the potential difference is E from the cell and with r being its internal resistance. As a shunt R is connected in series, the resistance becomes r+ R, therefore the current in the circuit derived from an emf E will reduce to:
$I = \dfrac{E}{r + R}$
Therefore, the potential drop across R becomes,
$V= (\dfrac{E}{r + R}) R$
Which gives us:
$\dfrac{E}{V} = 1+ \dfrac{r}{R}$
$\Rightarrow$ $r= (\dfrac{E}{V} - 1) R$

Now, in a potentiometer the potential difference in proportional to the balance length so we can write:
$\dfrac{E}{V} = \dfrac{l_0}{l}$
Therefore, substituting this value to find r, we get:
$r = (\dfrac{l_0 - l}{l})R$

Now, we're given that $l_0 = 52$ cm, l= 40 cm and R= 5$\Omega$.
So we may write:
$r = (\dfrac{52 - 40}{40})5 \Omega$.
We get the value of r as:
$r = (\dfrac{12}{8}) \Omega$.
$\Rightarrow$ $r = 1.5 \Omega$.

So, the correct answer is “Option D”.

Note:
Note the values if the lengths carefully as it might cause confusion. To avoid confusion try finding the ratio of E and V, then what goes in the denominator there, will be in the denominator of the expression for r.