How to find the potential of a field that has regions of non-zero curl

[kated]

Homework Statement

I want to find the potential of a field at a point where the curl of the field is zero but there are regions(for example far away from the point that interest us) that have non zero curl.

Relevant Equations

E=-gradV
curl E = Ω ≠ 0

We know that in electrostatics, there is path independency for line integral of E, so E is a conservative field and thus we have E=-gradV. Integrating this from ro(reference point of our choice) to the point r we are studying, along a random path, we get the solution of the above equation, e.g. the potential V. But, there are some areas (possibly far away from
the point r that interests us) where the curl E is not zero, e.g., curl E = Ω ≠ 0 , with Ω
being a vector field perpendicular to the plane, which only exists in these areas, then
how would we solve again Α=gradΛ, to find the scalar function Λ at point r? (how will the previous (usual)
solution change/get corrected?) Consider the (unusual) case where Ω is a static
(time-independent) field.

 

[Orodruin]

You cannot construct a global potential in this situation. The best you can do is to integrate the field in a region that is simply connected and excludes the region with non-zero curl.

 

[kated]

What do you mean by excluding the region with non-zero curl? Doesn't it affect the result? If for example I have an infinite wire with width=d(y=0, y=d) with B a magnetic field that exist only inside the wire and is perpedicular to the plane, and we want to find the potential at a point r far away and above from the wire then if we choose a path starting below the wire where B=0 and ending at r(again B=0) but the path goes through the wire doesnt this affect the result? I mean if we do the same process we do when we are talking about conservative fields isnt the result 0 which does not satisfy the system? How can we correct this?

 

[Orodruin]

What do you mean by excluding the region with non-zero curl? Doesn't it affect the result? If for example I have an infinite wire with width=d(y=0, y=d) with B a magnetic field that exist only inside the wire and is perpedicular to the plane, and we want to find the potential at a point r far away and above from the wire then if we choose a path starting below the wire where B=0 and ending at r(again B=0) but the path goes through the wire doesnt this affect the result?

Of course it does. The region where the curl is non-zero is then not simply connected, thereby violating the explicit requirement:

The best you can do is to integrate the field in a region that is simply connected and excludes the region with non-zero curl.

I mean if we do the same process we do when we are talking about conservative fields isnt the result 0 which does not satisfy the system? How can we correct this?

You can't. In your non-simply-connected region, it is generally not possible to write the field as the derivative of a potential because the region is not simply connected.