A potentiometer has uniform potential gradient across it. Two cells connected in series (i) to support each other and (ii) to oppose each other are balanced over 6 m and 2 m respectively on the potentiometer wire. The e.m.fs of the cells are in the ratio of
(A) 1:2
(B) 1:1
(C) 3:1
(D) 2:1

Hint : The cells are connected in series and balanced over 6 m and 2 m respectively on the potentiometer wire. When cells are connected to support each other, the equivalent emf becomes e1 + e2, and when connected to oppose each other, the equivalent emf becomes e1 - e2. Now, since the potentiometer has uniform potential gradient across it, therefore
 $\dfrac{{{e_1} + {e_2}}}{2} = \dfrac{{{e_1} - {e_2}}}{6} = k$ (constant)

Complete step-by-step solution:
Two cells are connected in series and balanced over 6 m and 2 m respectively on the potentiometer wire.
Let the e.m.fs of the cells be e1 and e2 respectively.
When cells are connected to support each other, the equivalent emf becomes e1 + e2. When cells are connected to oppose each other, the equivalent emf becomes e1 - e2.
Now, since the potentiometer has uniform potential gradient across it, therefore
 $\dfrac{{{e_1} + {e_2}}}{2} = \dfrac{{{e_1} - {e_2}}}{6} = k$ (constant)
$ \Rightarrow {e_1} + {e_2} = 2k$ and ${e_1} - {e_2} = 6k$
Adding these equations we get,
$2{e_1} = 8k$
$ \Rightarrow {e_1} = 4k$
$ \Rightarrow {e_2} = 2k - 4k = - 2k$
Therefore, $\dfrac{{{e_2}}}{{{e_1}}} = \dfrac{{2k}}{{4k}} = \dfrac{1}{2}$

Hence, the correct answer is option (A) 1:2.

Additional Information: A potentiometer is more sensitive than voltmeter. Its accuracy can be increased by increasing the length of wire. It can measure internal resistance and emf of the cell.Potentiometer is non portable and we have to make calculations to find the result as it's not showing direct reading.

Note: Always remember If the cross section area of wire is not uniform then gradient will not be constant. Current should pass through wire as per required time to protect wire from heat up and its resistance and gradient will not change.By using a long wire and adjusting a small gradient it can measure extremely small pd with accuracy.