Question

Which of the following statement(s) is/are correct?
This question has multiple correct options
A. If n identical cells are connected in series and then the battery thus formed is short circuited by a conducting wire, current through the wire will be independent of n.
B. If n identical cells are connected in parallel and then the battery thus formed is short circuited by a conducting wire, current through the wire will be directly proportional to n.
C. If n identical cells are connected in parallel and then the battery thus formed is short circuited by a wire having a constant resistance, current through the wire will directly increase as n increases.
D. None of the above.

Answer 1

When the cells are connected in series, then the net voltage and resistance are equal to the sum of various voltages and resistances respectively. When cells are connected in parallel to each other then the voltage remains the same for all the branches and the equivalent resistance is equal to resistance of one cell divided by the number of cells.
Formula used:
Ohm’s law for an electrical circuit is given
$V = IR$

Complete answer:
We are given identical cells. Let each cell have voltage E and let r be the internal resistance of each cell.
Let us first consider that these n cells are connected in series with each other. In series combination, we know that the total voltage is equal to sum of the voltages of the n cells which is given as
$V = nE$
The equivalent resistance is equal to the sum of the resistances. Therefore, we have
$R = nr$
Hence, the current in the circuits can be obtained by using the Ohm’s law in the following way.
$I = \dfrac{V}{R} = \dfrac{{nE}}{{nr}} = \dfrac{E}{r}$
This means that the current in series combination is independent of the number of cells in the circuit.
Next, let us consider that these n cells are connected in parallel with each other. In parallel combination of these cells, we know that the total voltage is equal to the voltage of one cell which is given as
$V = E$
The equivalent resistance is equal to the resistance of one cell divided by the total number of cells. Therefore, we have
$R = \dfrac{r}{n}$
Hence, the current in the circuits can be obtained by using the Ohm’s law in the following way.
$I = \dfrac{V}{R} = \dfrac{E}{{\dfrac{r}{n}}} = \dfrac{{nE}}{r}$
This means that the current in parallel combination is directly proportional to the total number of cells connected in parallel with each other.
$I \propto n$
Now if the parallel combination of cells is connected to a constant resistance R’ in series then current is given as
$I = \dfrac{E}{{R' + \dfrac{r}{n}}} = \dfrac{{nE}}{{nR' + r}}$
Here the current increases with an increase in the value of n.

Based on the above discussion, we can say that the correct answers are options A, B and C.

Note:
It should be noted that r is the internal resistance of the cells which are used in the above circuit. It is the resistance offered by the electrodes and electrolyte present inside the cell and causes a small voltage drop inside the battery.