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(A) The total emf of the series combination is $E_{eq} = E + E = 2E$.The total internal resistance is $r_{eq} = r_{1} + r_{2}$.The current $I$ flowing

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$r_{1} - r_{2}$

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(A) The total emf of the series combination is $E_{eq} = E + E = 2E$.The total internal resistance is $r_{eq} = r_{1} + r_{2}$.The current $I$ flowing through the circuit is given by $I = \frac{2E}{r_{1} + r_{2} + R}$.The potential difference $V$ across the source with internal resistance $r_{1}$ is given by $V = E - I r_{1}$.Given that $V = 0$,we have $E - I r_{1} = 0$,which implies $E = I r_{1}$.Substituting the expression for $I$:$E = \left( \frac{2E}{r_{1} + r_{2} + R} \right) r_{1}$$1 = \frac{2 r_{1}}{r_{1} + r_{2} + R}$$r_{1} + r_{2} + R = 2 r_{1}$$R = 2 r_{1} - r_{1} - r_{2}$$R = r_{1} - r_{2}$

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