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).