Question

In the given diagram `E_1` and `E_2` are the emfs of cells, `r_1` and`r_2` are the internal resistances of cells. `R` is the external resistance. What are the values of `I_1` and `I_2`?

Answer 1

Applying Kirchhoff's voltage law in the loop containing `E_1,r_1 and R` we get

`R(I_1+I_2)+r_1I_1-E_1=0`

`=>12(I_1+I_2)+0.5*I_1-6=0`

`=>24I_1+24I_2+I_1-12=0`

#=>25I_1+24I_2=12.... [1]#

Applying Kirchhoff's voltage law in the loop containing `E_2,r_2 and R` we get

`R(I_1+I_2)+r_2I_1-E_2=0`

`=>12(I_1+I_2)+1*I_2-10=0`

`=>12I_1+12I_2+I_2-10=0`

#=>12I_1+13I_2=10.... [2]#

Multiplying [1] by 12 and [2] by 25 we get

`300I_1+288I_2=144...[3]`

#300I_1+325I_2=250.... [4]#

Subtracting [3] from [4] we get

`37I_2=106`

`=>I_2=106/37A~~2.86A`

Inserting `I_2=106/37` in [2]

`=>12I_1+13*106/37=10`

`I_1=(370-13*106)/(37*12)A~~-2.27A`