Find the potential difference `V_(a)-V_(b)`in the circutis shown in figure

Explanation

In circuit ab baa
` E_2+iR_2+iR_3 = 0 `
` iR_2 + i_1 R_3 = -E_2`
` rArr iR_2 R_1 + iR_3 R_1 = -E_2 R_1 ` ……..(1)
` and i_1 R_3 + E_1- (i-i_1)R_1 = 0 `
` E_1 = i_1(R_1 + R_3) - iR_1 `
` rArr (E_1) (R_2) = iRi(R_2) + (i_1)(R_2)(R_3) - i(R_1) (R_2) ..........(2) `
` Adding (1) and (2) , we get `
` (i_1)(R_1)(R_2) + i_1(R_3)(R_1) + i_1 (R_2)(R-3)`
` = E_1R_2 - E_2 R_1`
` rArr i_1 ((R_3)(R_1) + (R_2)(R_1) + (R_2)(R_3))`
` (E_1)(R_2) - (E_2)(R-1) `
` i_1 = ((E_1)(R_2) - (E_2)(R_1)/(R_3)(R_1) + (R_2)(R_1) + (R_2)(R_3))`
` V_a - V_b = ((E_1)(R_2) - (E_2)(R_1)/(R_3)(R_1) + (R_2)(R_1) + (R_2)(R_3)) R_3)`
` = ((E_1)(R_2)(R_3)-(E_2)(R_1)(R_3)/(R_3)(R_1) + (R_3)(R_1) + (R_2)(R_3))`
`= [ (E_1/R_2 - E_2/R_1)/((1/R_2)+(1/R_1)+(1/R_3))]`
(b) Same as a .