(a) Two cells of emf E₁ and E₂ have their internal resistances r₁ and r₂ , respectively.
Deduce an expression for the equivalent emf and internal resistance of their parallel combination when connected across an external resistance R. Assume that the two cells are supporting each other.
(b) In case the two cells are identical, each of emf E = 5 V and internal resistance r = 2 $Ω$ , calculate the voltage across the external resistance R = 10 $Ω$ .

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Detailed Solution

Current drawn from cell -1
𝐼1 = (𝐸1 − 𝑉) / 𝑟1
Current drawn from cell -2
𝐼2 = (𝐸2 – 𝑉) / 𝑟2
Resultant current 𝐼 = 𝐼1 + 𝐼2
On solving
$\begin{array}{l} ∴ I = \frac{E_{1} r_{2} + E_{2} r_{1}}{r_{1} r_{2}} - V (\frac{r_{2} + r_{1}}{r_{1} r_{2}}) \\ ∴ V = \frac{E_{1} r_{2} + E_{2} r_{1}}{r_{1} r_{2}} - I (\frac{r_{1} r_{2}}{r_{2} + r_{1}}) \\ V = E_{eq} - {Ir}_{eq} \\ E_{eq} = \frac{E_{1} r_{2} + E_{2} r_{1}}{r_{1} + r_{2}} \\ r_{eq} = \frac{r_{1} r_{2}}{r_{2} + r_{1}} \end{array}$
Current through R $r_{eff} = \frac{r_{1} r_{2}}{r_{1} + r_{2}} = \frac{2 \times 2}{2 + 2} = 1 Ω$
P.D across R $= \frac{5}{11} \times 10 = 4 .54 volt$