# Applying Ampere's Law for two parallel wires

This seems a simple question but I'm not able to understand it properly. Say we have two parallel wires carrying same current I in different directions. We need to calculate the magnetic field at any point on the wires plane (not between the wires).

We know that the magnetic field (H) from one wire is I/2*pi*r where r is the distance from the wire. We can then use superposition to find the total magnetic field from both wires. The magnetic fields will tend to cancel each other but the net field will NOT be zero because the wires are not at the same distance from the test point.

Now, if we try to apply Ampere's law along a circular path that encloses both wires, the magnetic field will equal to zero since the current passing through the enclosed surface is zero (since both currents are equal but in opposite directions). Why does this method give us a different answer?

## 1 Answer

The magnetic field is not 0, but the integral around the contour is. The reason you cannot apply Ampere's law in this case is that the magnetic field is not homogeneous along the circle

• The left hand side of the law is: the line integral of H and the right hand side is the enclosed current. If the field is not homogeneous, I will not be able to pull H out of the left hand side. However, the right hand side can still be calculated and the total current will be zero. So regardless of what the left hand side is, H will still be equal to zero – Ahmad Jul 27 '16 at 3:44
• @Ahmad : Wrong. As Andrei said, the line integral (=sum) around the circle is zero, but this does not imply H must be zero at all points on the circle. At some points it will be +ve, at some points -ve, but the sum will be zero. – sammy gerbil Jul 27 '16 at 4:42
• @sammygerbil thanks a lot .. I got it now :).. if the line integral is zero, this does not imply that what's inside the integration is zero .. Many thanks.. I missed that point. – Ahmad Jul 27 '16 at 14:47