Question

For the batteries shown in fig. `R_1, R_2 and R_3` are the internal resistance of `E_1, E_2, and E_3,` respectively Then, which of the following is / are correct?

A. The equivalent internal resistance R of the system is given by `(R_1R_2R_3)//(R_1R_2 +R_2R_3+R_3R_1).`
B. If `E_3 = (E_1R_2 +E_2R_1)//(R_1+R_2),` the equivalent emf of the batteries will be equal to `E_3`
C. The equivalent emf to the battery is equal to `E = (E_1 +E_2 +E_3)//3`
D. The equivalent emf of the battery not only depends upon values of `E_1,E_2, and E_3` but also depends upon values of `R_1, R_2 and R_3.`

Answer 1

Correct Answer - A::B::D
We know that the equivalent internal resistance is
`R_(eq) = (1)/(sum(1)/(2)) = (1)/((1)/(R_(1)) +(1)/(R_(2)) +(1)/(R_(3)))`
`=(R_(1)R_(2)R_(3))/(R_(1)R_(2)+R_(2)R+3+R_(3)+R_(1))`
and the equivalent emf is
`E_(eq)=(sum(E )/(R ))/(sum(1)/(R ))=((E_(1))/(R_(1))+(E_(2))/(R_(2))+(E_(3))/(R_(3)))/((1)/(R_(1))+(1)/(R_(2))+(1)/(R_(3)))`
Use `E_(3) = (E_(1)R_(2) + E_(2)R_(1))/(R_(1) + R_(2))` to get
`E_(eq) = (E_(1)R_(2) +E_(2)R_(1))/(R_(1)+R_(2)) = E_(3)`