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
$\frac{E}{R}=\frac{E_1}{r_1}+\frac{E_2}{r_2}=\frac{E_1{r}_2+E_2{r}_1}{r_1{r}_2}$
$\therefore E=(\frac{E_1{r}_2+E_2{r}_1}{r_1{r}_2})R=\frac{E_1{r}_2+E_2{r}_1}{\left(r_1+r_2\right)}$
As, $E_2>E_1\Rightarrow E_1<E<E_2[$ For all values of $r_1$ and $r_2]$