Question

The n identical cells, each of EMF $\varepsilon $ and internal resistance r, are joined in series to form a closed circuit. The potential difference across any one cell is:
A. Zero
B. $\varepsilon $
C. $\dfrac{\varepsilon }{n}$
D. $\dfrac{n-1}{n}\varepsilon $

Answer 1

Hint: Here, the emf of each cell is the same, it means they will send out the same potential difference when connected to an external circuit. Also, according to Ohm's law current is defined as the ratio of total EMF to the total resistance(r),\[i=\dfrac{emf}{r}\]. The net EMF of a series circuit is \[n\varepsilon \] and the net internal resistance is $nr$. Now to find out the potential across any one cell we write the equation of voltage (according to Kirchhoff’s voltage law) through that cell.

Complete step by step answer:
We know that in series combination current remains same and in case of parallel combination the potential difference remains same. The emf of each is $\varepsilon $, so the net emf of the cells will become $n\varepsilon $. Also, the resistances are in series combination and so the net resistance is given by:$r+r+r+r..........n \, times=nr$.
According to Ohm's law current is defined as the ratio of total EMF to the total resistance, \[i=\dfrac{V}{r}\]
Therefore the current in the circuit is $i=\dfrac{n\varepsilon }{nr}=\dfrac{\varepsilon }{r}$.
Drawing the equivalent circuit,

The potential across any cell can be given as,
${{V}_{a}}-{{V}_{b}}=-\varepsilon +ir\\
\Rightarrow{{V}_{a}}-{{V}_{b}} =-\varepsilon +\dfrac{\varepsilon }{r}\times r\\
\Rightarrow{{V}_{a}}-{{V}_{b}} =-\varepsilon +\varepsilon \\
\therefore{{V}_{a}}-{{V}_{b}} =0\\$
Therefore the potential difference across any one cell is zero.

Note:One should know to write the potential across two points in a circuit using Kirchhoff's voltage law equation. EMF or the electromotive force is defined as a potential difference across a cell when the cell is not in use and is the maximum potential difference across the cell. The potential difference of a cell is defined as the difference in potential between the two terminals of the cell when the cell is being used, that is when current passes through the cell. Internal resistance of a cell is defined as the opposition to the flow of current by the cell.