A very long circuit

Electricity and Magnetism

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 $\frac{r}{R}=\frac{1}{2}.$