Shown n batteries connected to form a circuit. The resistances denote the internal resistances of the batteries which are related to the emf 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.

Options

A. 1/k, 0
B. k, 0
C. `k^2`, 1
D. 2/k, 0

Answer

A

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.`