For the two cells having same EMF E and internal resistance r, the current passing through the external resistor 6Ω is same when both the cells are connected either in parallel or in series. The value of internal resistance r is ____ Ω .

Two identical cells (each with EMF $$E$$ and internal resistance $$r$$) give the same current through an external resistance of $$6\Omega$$ whether connected in series or parallel. We need to find $$r$$.

Current in series connection.

In series, the total EMF is $$2E$$ and total internal resistance is $$2r$$:

$$ I_{series} = \frac{2E}{2r + 6} $$

Current in parallel connection.

In parallel, the equivalent EMF is $$E$$ and equivalent internal resistance is $$\frac{r}{2}$$ (two identical resistances $$r$$ in parallel):

$$ I_{parallel} = \frac{E}{\frac{r}{2} + 6} = \frac{2E}{r + 12} $$

Set the currents equal and solve.

$$ \frac{2E}{2r + 6} = \frac{2E}{r + 12} $$

Cancel $$2E$$ from both sides:

$$ \frac{1}{2r + 6} = \frac{1}{r + 12} $$

$$ 2r + 6 = r + 12 $$

$$ r = 6 \, \Omega $$

The correct answer is Option (3): 6 $$\Omega$$.