The total $emf =E+E=2 E$
Total resistance $=R+r_{1}+r_{2}$
$\therefore $ Current flowing through the circuit
$i=\frac{2 E}{R+r_{1}+r_{2}}$
According to question $E=i r_{1}$
$\Rightarrow \, i=\frac{E}{r_{1}}$
$\therefore \, \frac{E}{r_{1}}=\frac{2 E}{R+r_{1}+r_{2}}$
$R=r_{1}-r_{2}$
