Find the current through the 10V battery in the circuit shown.

First we define the current variables I_{1}, I_{2},
and I_{3} as shown. The we apply Kirchhoff's rules.

Junction rule at A:

I_{1} + I_{2} = I_{3}

Loop rule: Top left triangular loop:

-(10 V) + (2W )I_{3 }= 0

From this we immediately obtain:

I_{3} = 5A

Loop rule: whole square, going counter-clockwise from B:

10 V - 12 V +(2W )I_{2} = 0

From this we get:

I_{2} = 1 A

Finally, from the junction equation we find:

I_{1} = I_{3} - I_{2} = 5 A - (1 A) = 4A

So the current through the 10 V battery is 4 A to the left.

Check the answer by writing a loop equation for the bottom triangle.