Question

Find the net EMF and internal resistance if two cells of EMF E1 and E2 are connected in parallel having internal resistance R1 and R2

Answer 1

Let the potential across the load or the junction of the terminals of the batteries be  V.  Let the total current that flows through load from the batteries be I.  Let net EMF be E and the effective internal resistance be R.

V = E1 - i1 R1   =>   i1 = (E1 - V) / R1
V = E2 - i2 R2   =>   i2 = (E2 - V) / R2

     i1 + i2 = (E1 R2 + E2 R1)/(R1 R2) - V (R1 + R2)/(R1R2)

     V = (E1 R2 + E2 R1) / (R1+R2) - (i1+i2) (R1 R2)/(R1+R2)    --- (1)
we also have   V = E - (i1+i2) R     --- (2)

These two equations are true for all i1, i2..  Hence,
          E = (E1 R2 + E2 R1) / (R1+R2)
          R = (R1 R2)/(R1 + R2)