Finding Quantitative Proof of Optimal Helmholtz Coil Separation

[locke]

I'm trying to track down a quantitative proof that the most uniform magnetic field between two Helmholtz coils occurs for a separation equal to their radii.

So far I've just been playing around with the Biot-Savart law and proving that B is identical at several trial points along the axis through the centre of the coils in this arrangement, but I can't figure out how to prove it for _every_ point along the axis.

Any help greatly appreciated.

 

[OlderDan]

I'm trying to track down a quantitative proof that the most uniform magnetic field between two Helmholtz coils occurs for a separation equal to their radii.

So far I've just been playing around with the Biot-Savart law and proving that B is identical at several trial points along the axis through the centre of the coils in this arrangement, but I can't figure out how to prove it for _every_ point along the axis.

Any help greatly appreciated.

If you are Java enabled, I suggest you begin by playing with the applet at this site

http://webphysics.davidson.edu/faculty/dmb/Helmholtz/HelmholtzCoils.html

Be sure to read carefully. It is not very long. The "time" in the graph is not time in this case, it is the radius of the coils. It is a graph applet that was no doubt created for general use and re-used here. Click on the Start the Exercise to see how the field along the axis of the coils varies as the radius varies. It should occur to you that when the curve gets flat in the middle, the field is uniform, so if you had a way to quantify the flatness of the middle of the curve, you would have an approach to the problem.

The flatness of curve can be quantified by calculating the radius of curvature of the curve. If you don't know how to do that, look here, especially at equation #5.

http://mathworld.wolfram.com/RadiusofCurvature.html

What value of the radius of curvature represents the most uniform field? What radius of the Helmholtz coil produces that condition?

 

[Galileo]

Try expanding the expression for the field as a taylor series about the point halfway the center of the coils.