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While studying 'Parallel grouping of batteries with internal resistance' I came across its derivation and found that the equivalent emf to be
Σ(E/r)/(Σ(1/r))
Why didn't we take the equivalent emf as just Σ(E/r) from the first step as we had the opportunity to do so. Why do we have to bring the Σ(1/r) to the numerator?
This is because if you check the original circuit used in the derivation, you will find that the external resistance R is in series with the internal resistances r (which are in parallel with each other). Hence if you calculate the net resistance you will find that it comes out to be the one in the denominator. To compare with I =E/R, you need to have to get to net resistance in the denominator so as to get the net Emf in the numerator. To adjust this, they have multiplied and divided by ∑ (1/r) to get the actual expression as the net resistance comes out to be R + 1/ ∑ (1/r). As r is in parallel with one another which gives it's net resistance parallel component as 1/ ∑(1/r). While R is in series with 1/ ∑ (1/r) which gives the total net resistance as R + 1/ ∑ (1/r). Hope my answer satisfies your needs.
