Question

Two non-ideal batteries are connected in parallel. Consider the following statements.
$(i)$The equivalent emf is smaller than either of the two emfs.
$(ii)$The equivalent internal resistance is smaller than either of the two internal resistances.

• A. Both (i) and (ii) are correct.
• B. (i) is correct but (ii) is wrong.
• C. (ii) is correct but (i) is wrong.
• D. Both (i) and (ii) are wrong.

Answer 1

Answer: (C)

Let two cells o f emf 's ${\epsilon }_1$ and ${\epsilon }_2$ and of internal resistance $r_1$ and $r_2$ respectively are connected in parallel

The equivalent emf is given by
${\epsilon }_{eq}=\frac{{\epsilon }_1{r}_2+{\epsilon }_2{r}_1}{r_1+r_2}\dots \left(i\right)$
The equivalent internal resistance is given by
$\frac{1}{r_{eq}}=\frac{1}{r_1}+\frac{1}{r_2}$
or $r_{eq}=\frac{r_1{r}_2}{r_1+r_2}\dots \left(ii\right)$
Let us consider, two cells connected in parallel are of same emf $\epsilon$ and same internal resistance r
From equation $\left(i\right)$, we get
${\epsilon }_{eq}=-\frac{\epsilon r+\epsilon r}{r+r}=\epsilon$
From equation $\left(ii\right)$, we get
$r_{eq}=\frac{r^2}{r+r}=\frac{r}{2}$