Two cells of the same emf E but internal resistance r1 and r2 are connected in series to an external resistor R (figure). What should be the value of R so that the potential difference across the terminals of the first cell becomes zero?

                     

Hint: The potential difference across the cells depends on the net current in the circuit.
Step 1: Applying Ohm's law,
Effective resistance = R + r1 + r2 and effective emf of two cells = E + E =2E, so the electric current is given by;
I=E+ER+r1+r2
Step 2: The potential difference across the terminals of the first cell and putting it equal to zero.
     V1=EI1r1=E2Er1+r2+Rr1=0
or  E=2Er1r1+r2+R1=2r1r1+r2+R
      r1+r2+R=2r1R=r1r2
This is the required relation.