Question

The cirucit shown in fig. extends to the right into infinity. Each battery has the electromotive force V(unknown) and the internal resistance r(known). Each resistor has the resistance 4r. The reading of the idal ammeter shown in the diagram is I. Find the value of V in terms of I and r.

Answer 1

Let net emf and net resistance across CD be E and R, respectively.
Then net emf and resistance across AB will also be E and r.

`(1)/(R ) = (1)/(r ) +(1)/((R+4r))` or `(R+4r) = (4r^2)/(R )`
and `E=((E )/(R+4r)+(V)/(r ))/((1)/(R+4r)+(1)/(r ))`
`=(Er =-V(R+4r))/(r+R +4r) = (Er +V(R+4r))/(r+5r)`
or `E(R+5r) =Er +V(R+4r)`
or `E(R+4r)= V (R +4r) = V(R +4r) or E =V` (ii)
Also `I = (E )/(R +4r) = (ER)/(4r^2) = (VR)/(4r^2)`
or `R = (4r^2I)/(V)` (iii)
form (i) and (iii), `(4r^2I)/(V) +4r = (4r^2)/(4r^2I ) V = (V)/(I)`
or `4r^(2)I+4rV = (V^(2))/(I)`
or `V^(2) - 4rIV - 4r^(2)I^(2) = 0 or V = 2Ir (1+sqrt2)`