What is a parallel connection of cells? Obtain the equation for the equivalent emf of two cells connected in parallel.

(N/A) If the positive terminals of given cells are connected at one point and the negative terminals are connected at another point,then such a connection is called a parallel connection of cells.
As shown in the figure,two cells with emf $\varepsilon_{1}$ and $\varepsilon_{2}$ and internal resistances $r_{1}$ and $r_{2}$ respectively are connected in parallel between points $B_{1}$ and $B_{2}$.
The current in the cell with emf $\varepsilon_{1}$ is $I_{1}$ and in the cell with emf $\varepsilon_{2}$ is $I_{2}$.
The total current at junction $B_{1}$ is $I = I_{1} + I_{2}$.
Let the potentials at $B_{1}$ and $B_{2}$ be $V(B_{1})$ and $V(B_{2})$ respectively. The potential difference $V = V(B_{1}) - V(B_{2})$ across each cell is given by:
$V = \varepsilon_{1} - I_{1}r_{1} \implies I_{1} = \frac{\varepsilon_{1} - V}{r_{1}}$
$V = \varepsilon_{2} - I_{2}r_{2} \implies I_{2} = \frac{\varepsilon_{2} - V}{r_{2}}$
Substituting these into the total current equation:
$I = \frac{\varepsilon_{1} - V}{r_{1}} + \frac{\varepsilon_{2} - V}{r_{2}} = \left( \frac{\varepsilon_{1}}{r_{1}} + \frac{\varepsilon_{2}}{r_{2}} \right) - V \left( \frac{1}{r_{1}} + \frac{1}{r_{2}} \right)$
Rearranging for $V$:
$V \left( \frac{r_{1} + r_{2}}{r_{1}r_{2}} \right) = \left( \frac{\varepsilon_{1}r_{2} + \varepsilon_{2}r_{1}}{r_{1}r_{2}} \right) - I$
$V = \left( \frac{\varepsilon_{1}r_{2} + \varepsilon_{2}r_{1}}{r_{1} + r_{2}} \right) - I \left( \frac{r_{1}r_{2}}{r_{1} + r_{2}} \right)$
Comparing this with the equivalent circuit equation $V = \varepsilon_{eq} - Ir_{eq}$,we get:
$\varepsilon_{eq} = \frac{\varepsilon_{1}r_{2} + \varepsilon_{2}r_{1}}{r_{1} + r_{2}}$ and $r_{eq} = \frac{r_{1}r_{2}}{r_{1} + r_{2}}$.