Question

Two identical cells, when connected either in parallel or in series gives same current in an external resistance $5 \Omega$ The internal resistance of each cell will be__ $\Omega$

Hint:

In problems involving identical cells, equating the total currents in series and parallel connections leads to solving for the internal resistance of the cells.

Answer 1

Correct Answer: 5

The correct answer is 5.

$i=\frac{2\epsilon }{5+2r}\dots$(1)
$i=\frac{\epsilon }{\frac{r}{2}+5}\dots$ (2)
Equating (1) and (2)
$\frac{2\epsilon }{5+2r}=\frac{\epsilon }{\frac{r}{2}+5}$
$\Rightarrow r+10=5+2r$
$r=5$

Answer 2

Let the internal resistance of each cell be \( r \), and the emf of each cell be \( \epsilon \). For parallel connection, the total current is: \[ i = \frac{2\epsilon}{5 + 2r} \quad \text{(1)} \] For series connection, the total current is: \[ i = \frac{\epsilon}{\frac{r}{2} + 5} \quad \text{(2)} \] Equating (1) and (2) gives: \[ \frac{2\epsilon}{5 + 2r} = \frac{\epsilon}{\frac{r}{2} + 5} \] Cross-multiplying: \[ 2\epsilon \left( \frac{r}{2} + 5 \right) = \epsilon \left( 5 + 2r \right) \] Simplifying: \[ r + 10 = 5 + 2r \] Thus: \[ r = 5 \, \Omega \] The internal resistance of each cell is \( r = 5 \, \Omega \).