Find the current through the 10V battery in the circuit shown.
First we define the current variables I1, I2, and I3 as shown. The we apply Kirchhoff's rules.
Junction rule at A:
I1 + I2 = I3
Loop rule: Top left triangular loop:
-(10 V) + (2W )I3 = 0
From this we immediately obtain:
I3 = 5A
Loop rule: whole square, going counter-clockwise from B:
10 V - 12 V +(2W )I2 = 0
From this we get:
I2 = 1 A
Finally, from the junction equation we find:
I1 = I3 - I2 = 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.