Building Helmholtz Coils?

A Helmholtz coil's design is based on the fact that for two *thin* wire round loops axially separated by 1 radius, the first, second, and third derivatives of B wrt z and r are zero at the center of the assembly. This means you have to move pretty far away from center before B changes significantly. If that is what you want (a fairly uniform B), then you want to approximate a thin wire round loop for each coil. The easiest thing you can do is make the width and height of each wire bundle forming the loop about the same. If you wanted to make absolutely sure the Helmholz coil criterion is satisfied *exactly*, you'd want each winding to be an H. coil itself (separated by radius). This means the assembly will look like an X from the side, but with the center open. Generally, the first solution is what people do. You get a stronger B that way that's still pretty uniform.

Before building anything, I suggest you perform a mathematical exercise to show the above using the Biot Savart Law to find B(z) on axis, and the fact that divergence of B=0 to find the radial derivatives. This will give you the insight needed to figure things like this out for yourself.