Question

Two batteries of emf $E_{1}$ and $E_{2}$ $\left(E_{2} > E_{1}\right)$ and internal resistances $r_{1}$ and $r_{2}$ respectively are connected in parallel as shown in the figure. Then, which of the following statements is correct?

• A. The equivalent emf $E$ is smaller than $E_{1}$ .
• B. The equivalent emf $E=E_{1}+E_{2}$ .
• C. The equivalent emf $E$ is greater than $E_{2}$ .
• D. The equivalent emf $E$ of two cells is between $E_{1}$ and $E_{2}$ always.

Answer 1

Answer: (D)

The equivalent internal resistance of the two cells between $A$ and $B$ is
$R=\frac{r_{1} r_{2}}{r_{1} + r_{2}}$
If $E$ is the equivalent emf of the two cells in parallel, then
$\frac{E}{R}=\frac{E_{1}}{r_{1}}+\frac{E_{2}}{r_{2}}=\frac{E_{1} r_{2} + E_{2} r_{1}}{r_{1} r_{2}}$
$\therefore E=\left(\right.\frac{E_{1} r_{2} + E_{2} r_{1}}{r_{1} r_{2}}\left.\right)R=\frac{E_{1} r_{2} + E_{2} r_{1}}{\left(r_{1} + r_{2}\right)}$
As, $E_{2}>E_{1}\Rightarrow E_{1} < E < E_{2} [$ For all values of $r_{1} $ and $ r_{2}]$