Two cells of EMF $\varepsilon_1 = 2\ \text{V}$ (internal resistance $r_1 = 1\ \Omega$) and $\varepsilon_2 = 1.5\ \text{V}$ (internal resistance $r_2 = 2\ \Omega$) are connected in parallel, with like terminals joined together, so as to drive current through an external circuit in the same direction. Derive the expression for their equivalent EMF and equivalent internal resistance, and evaluate both for the given cells.

Two cells of EMF varepsilon₁ = 2V (internal resistance r₁ = 1Ω) and varepsilon₂ = 1.5V (internal resistance r₂

Example · Example 11

\varepsilon_1 = 2\ \text{V}

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For two unequal cells in parallel (Section 3.10.2), applying Kirchhoff's rules at the common node gives

\varepsilon_{\text{eq}} = \frac{\varepsilon_1 r_2+\varepsilon_2 r_1}{r_1+r_2} \qquad\qquad r_{\text{eq}} = \frac{r_1r_2}{r_1+r_2}

Substituting $\varepsilon_1=2\ \text{V}$ , $r_1=1\ \Omega$ , $\varepsilon_2=1.5\ \text{V}$ , $r_2=2\ \Omega$ :

$\varepsilon_{\text{eq}} = \frac{(2)(2)+(1.5)(1)}{1+2} = \frac{4+1.5}{3} = \frac{5.5}{3} \approx 1.83\ \text{V}$ …