For the two cells having same EMF $E$ and internal resistance $r$, the current passing through the e

Question

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

Professor PC Thomas Classes · Answer

When the two identical cells are connected in series, the total EMF is $2E$ and the total internal resistance is $2r$. The current is $I_s = \frac{2E}{R + 2r}$. When connected in parallel, the effective EMF remains $E$ and the effective internal resistance is $r/2$. The current is $I_p = \frac{E}{R + r/2}$. Since $I_s = I_p$, we equate them: $\frac{2E}{R + 2r} = \frac{E}{R + r/2} \implies 2(R + r/2) = R + 2r \implies 2R + r = R + 2r \implies r = R$. Given the external resistor is $R = 6 \Omega$, the internal resistance is $r = 6 \Omega$.