Two cells of emf $\displaystyle \mathrm{E}_{1}$ and $\displaystyle \mathrm{E}_{2}$ and internal resistances $\displaystyle \mathrm{r}_{1}$ and $\displaystyle \mathrm{r}_{2}$ are connected in parallel, with their terminals of the same polarity connected together. Obtain an expression for the equivalent emf of the combination.

$\displaystyle V \equiv V(B_1) - V(B_2) = \varepsilon_1 - I_1 r_1$$\displaystyle V \equiv V(B_1) - V(B_2) = \varepsilon_2 - I_2 r_2$$\displaystyle I = I_1 + I_2$$\displaystyle I = \dfrac{\varepsilon_1 - V}{r_1} + \dfrac{\varepsilon_2 - V}{r_2}$$\displaystyle I = \left(\dfrac{\varepsilon_1}{r_1} + \dfrac{\varepsilon_2}{r_2}\right) - V\left(\dfrac{1}{r_1} + \dfrac{1}{r_2}\right)$$\displaystyle V = \dfrac{\varepsilon_1 r_2 + \varepsilon_2 r_1}{r_1 + r_2} - I\,\dfrac{r_1 . r_2}{r_1 + r_2}$$\displaystyle \because\ V = \varepsilon_{eq} - I r_{eq}$$\displaystyle \varepsilon_{eq} = \dfrac{\varepsilon_1 r_2 + \varepsilon_2 r_1}{r_1 + r_2}$