A cell of E.M.F. E and internal resistance r supplies currents for the same time t  through external resistance R₁  and R₂ respectively.  If the heat produced in both cases is the same then the internal resistance is

$ r = \sqrt{R_1R_2}$

$ H_1 = (\frac{E}{r+R_1})^2 R_1 t$

$ H_2 = (\frac{E}{r+R_2})^2 R_2 t$

$ H_1 = H_2$

$ \Rightarrow R_1(r+R_2)^2 = R_2(r+R_1)^2$

$ r^2(R_1 - R_2) - R_1 R_2 ( R_1 - R_2)=0$

$ since , R_1 -R_2 \text{ is not equal to zero}$

$\Rightarrow r = \sqrt{R_1R_2}$