Shown n batteries connected to form a circuit. The resistances denote the internal resistances of the batteries which are related to the emf's as `r_(i)=k(epsilon)_(i)`where K is a constant. The solid dots represent the terminals of the batteries. Find (a)the current through the circuit and (b) the potential difference between the terminals of the ith battery.

Explanation

(a) Suppose the current is i in the indicated direction. Applying Kirchoff's loop law,
`(epsilon)_(1)-ir_(1)+(epsilon)_(2)-ir_(2)+(epsilon)_(3)-ir_(3)+….+(epsilon)_(n)-ir_(n)=0`
`or, i=((epsilon)_(1)+(epsilon)_(2)+(epsilon)_(3)+…+(epsilon)_(n))/(r_(1)+r_(2)+r_(3)+...+r_(n))`
`=((epsilon)_(1)+(epsilon)_(2)+(epsilon)_(3)+…+(epsilon)_(n))/(k(epsilon)_(1)+(epsilon)_(2)+(epsilon)_(3)+…+(epsilon)_(n))=1/k.`
`The potential difference between the terminals of the ith battery is
`(epsilon)_(i)-ir_(i)`
(epsilon)_(i)-(1/k)(k(epsilon)_(i))=0.`