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 $\varepsilon_{1}$ and $\varepsilon_{2}$ and of internal resistance $r_{1}$ and $r_{2}$ respectively are connected in parallel

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