# A very long circuit

A large number of AA batteries with emf $$\mathcal{E}=1.5\, \mbox{V}$$ and resistors with resistance $$R$$ form an infinite circuit, as shown in the figure. The internal resistance of the batteries is $$r$$. Thevenin's theorem tells us that this infinite circuit can be replaced by a single source with emf $$\mathcal{E}_\text{eq}$$, connected in series with an equivalent resistor $$R_\text{eq}$$. Determine the equivalent emf $$\mathcal{E}_\text{eq}$$ in volts if $$\frac{r}{R}=\frac{1}{2}.$$

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