Two cells of $emfs$ $E_1$ and $E_2$ and internal resistances $r_1$ and $r_2$ are connected in parallel. The $emf$ and internal resistance of the equivalent source is
Medium
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In parallel connection
${{\rm{E}}_{{\rm{eq}}}} = \frac{{\Sigma \frac{{{{\rm{E}}_1}}}{{{{\rm{r}}_1}}}}}{{\Sigma \frac{1}{{{{\rm{r}}_1}}}}} \Rightarrow \frac{{\frac{{{{\rm{E}}_1}}}{{{{\rm{r}}_1}}} + \frac{{{{\rm{E}}_2}}}{{{{\rm{r}}_2}}}}}{{\frac{1}{{{{\rm{r}}_1}}} + \frac{1}{{{{\rm{r}}_2}}}}}{{\rm{E}}_{{\rm{eq}}}} = \frac{{{{\rm{E}}_1}{{\rm{r}}_2} + {{\rm{E}}_{2{\rm{r}}}}}}{{{{\rm{r}}_1} + {{\rm{r}}_2}}}$
$\frac{1}{{{r_{eq}}}} = \Sigma \frac{1}{{{r_1}}} = \frac{1}{{{r_1}}} + \frac{1}{{{r_2}}} \Rightarrow {r_{eq}} = \frac{{{r_1}{r_2}}}{{{r_1} + {r_2}}}$
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