Magnetic field/simple current loop

[granpa]

given a current loop, centered at the origin, and tilted 90 degrees so that it enters at x=1 and exits at x=-1, carrying a unit amount of current, and completely disregarding the z axis.

is the strength of the magnetic field at any point in the xy plane proportional to 1/(distance from 1,0)^2 - 1/(distance from -1,0)^2

in other words does it have an inverse square relation to the current passing through those two points.

I know there are better ways to calculate the net field but I'm looking to understand what is happening here at an intuitive level.

I need the whole field. not just the far field or some sort of approximation.

 

[granpa]

what I mean is:

can I find the contribution of each of the 2 currents separately by simply finding the vector which is at a right angle to the vector from that point to the current and making its magnitude equal to 1/(distance to the current)^2

and is adding those 2 vectors all that i need to do to calculate the total field

 

[Meir Achuz]

If it is a closed loop, B does not go like 1/r^2.

 

[granpa]

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/curloo.html#c2

 

[granpa]

ok. so evidently thet won't work.

thinking 3 dimensionally, if we look at only one of the 2 currents (due to current along line element dL) then we can see from the symmetry that its field is confined to a wedge extending from the origin through the endpoints of that line element.

I have no idea how to go about calculating the resulting field. I've never seen or done anything like it.

 

[Ben Niehoff]

To find the magnetic field of a finite loop, you will need to use elliptic functions. It's complicated.

However, it can be proven in general that

\vec B(\vec r) \propto \nabla \Omega(\vec r)

where \Omega(\vec r) is the (oriented) solid angle subtended by the current loop at the observation point.

 

[granpa]

any connection to the scalar potential of a magnetic field?

 

[Meir Achuz]

any connection to the scalar potential of a magnetic field?

Yes, Omega is the magnetic scalar potential, which happens to equal the solid angle subtended by a current loop of any shape.
For a circular current loop, the MSP can be found be either an elliptic integral, or by a Legendre polynomial, partial wave expansion.

 

[granpa]

so the msp isn't so much discontinuous as multi-valued (like an angle)?

 

[Ben Niehoff]

It's multivalued, or discontinuous if you make a branch cut.

However, in free space (in the absence of any currents), it is single-valued, as in this case (there is no J at the observation point).