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 ______ $$\Omega$$.

Solution :

Let emf of each cell be $$E$$ and internal resistance be $$r$$.

When cells are connected in series :

Equivalent emf,

$$E_s = 2E$$

Equivalent internal resistance,

$$r_s = 2r$$

Current through external resistance $$R = 5\ \Omega$$ :

$$I_s = \frac{2E}{5+2r}$$

When cells are connected in parallel :

Equivalent emf,

$$E_p = E$$

Equivalent internal resistance,

$$r_p = \frac{r}{2}$$

Current :

$$I_p = \frac{E}{5+\frac{r}{2}}$$

Given currents are equal :

$$\frac{2E}{5+2r} = \frac{E}{5+\frac{r}{2}}$$

Cancelling $$E$$ :

$$\frac{2}{5+2r} = \frac{1}{5+\frac{r}{2}}$$

Cross multiplying :

$$2\left(5+\frac{r}{2}\right) = 5+2r$$

$$10+r = 5+2r$$

$$r = 5\ \Omega$$

Final Answer :

$$5\ \Omega$$