Find the potential difference `varphi_(A) - varphi_(B)` between points `A` and `B` of the circuit shown in Fig.

Explanation

First of all, make the charge distribution, as shwon in the figure.
In the loop 12341, using `-Delta varphi = 0`
`(q_(1))/(C_(1)) - xi_(1) + (q_(1) - q_(2))/(C_(3)) = 0` ....(1)
Similarly, in the loop 61456, using `-Delta varphi = 0`
`(q_(2))/(C_(2)) + (q_(2) - q_(1))/(C_(3)) - xi_(2) = 0` ....(2)
From Eqs. (1) and (2) we have
`q_(2) - q_(1) = (xi_(2) C_(2) - xi_(1) C_(1))/((C_(2))/(C_(3)) + (C_(1))/(C_(3)) + 1)`
Hence, `varphi_(A) - varphi_(B) = (q_(2) - q_(1))/(C_(3)) = (xi_(2) C_(2) - xi_(1) C_(1))/(C_(1) + C_(2) + C_(3))`