Which one of the following equations is the correct equation for the electrical circuit shown in the figure ?

Options

A. `E_(1)-(i_(1)+i_(2))R+i_(1)r_(1)=0`
B. `E_(1)-(i_(1)+i_(2))R-i_(1)r_(1)=0`
C. `E_(2)-i_(2)r_(2)-E_(1)-i_(1)r_(1)=0`
D. `E_(2)-(i_(1)+i_(2))R+i_(2)R_(2)=0`

Answer

B

Explanation

We consider three closed loops in the circuit.
Applying Kirchhoff's second law viz `sum dv=0` to the closed loops, we get
(1) For the loops AEFBA,
`-(i_(1)+i_(2))R+E_(2)-i_(2)r_(2)=0`
or `E_(2)-(i_(1)+i_(2))R-i_(2)r_(2)=0`
(2) For the loop CEFDC,
`-(i_(1)+i_(2))R-i_(1)r_(1)+E_(1)=0`
or `E_(1)-(i_(1)+i_(2))R-i_(1)r_(1)=0`
(3) For the loop BACDB,
`E_(2)-i_(2)r_(2)+i_(1)r_(1)=E_(1)=0`
or `E_(2)-E_(1)-i_(2)r_(2)+i_(1)r_(1)=0`
Thus we find equation (b) is the correct equation.