Find the current flowing current flowing thorugh the resistance `R` in the circuit shown in Fig. The internal resistances of the batteries are negilble.

Explanation

Indicate the currents in all the branches using charge conservation as shwon in the figure. Applying loop rule, `-Delta varphi = 0` in the loops `1A781, 1B781` and `B456B` respectively we get
`xi_(0) = (i_(0) - i_(1)) R_(1)` ....(1)
`i_(3) R_(3) + i_(2) R_(2) - xi_(0) = 0` ....(2) and
`(i_(1) - i_(3)) R - xi - i_(3) R_(3) = 0` ...(3)
Solving Eqs. (1). (2) and (3), we get teh sougth current
`(i_(1) - i_(3)) = (xi (R_(2) + R_(3)) + xi_(0) R_(3))/(R (R_(2) + R_(3)) + R_(2) R_(3))`