Obtain the equivalent emf and internal resistance of a parallel combination of 2 cell of emf E1 and E2 and internal resistance R1 and R2.

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Solution

We have two cells E1 and E2 with internal resistance r1 and r2 connected in parallel with a resistor R.
Let, I1 and I2 be the current through E1 and E2 respectively.
By applying KCL(Kirchhoff's Current Law),
I = I1 + I2
Let there resultant EMF is Eeff and resultant resistance be Req.
By KVL(Kirchhoff's Voltage Law)
Potential across R = Eeff - IReq
Potential across upper branch = E1 - Ir1
Potential across Lower Branch = E2 - I*r2
After solving we will get
**Eeff = (E1/r1 + E2/r2)Req
and
Req = r1r2/(r1 +r2)

Eequa=(E1R2+E2R2)/(R1+R2)**

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