Clarification of HW6.1

One of you asked:

Dr. LeClair, I was curious about number one. The problem seems pretty straight forward, I would hesitate based on its straight forward reading even calling it a problem but I wanted to know more what you wanted to know from this problem and what you expected to see in our answers. Do you want us to do this purely on research and give you an answer that goes into detail on what we find, since you said, "Can you suggest ways in which two short coils or current rings might be arranged to achieve good uniformity over a limited region?" Or would you like to see a derivation that leads us to the final conclusion? My point being that I only found out what the answer is through researching and I want to know whether you would like to see me spit out a derivation to show that I know what I found is true, or would you like me to just explain where B is most uniform and why that is the case?

I guess to be really specific, a uniform field could be defined by stipulating that the spatial derivatives of the field along the axial direction vanish in the middle. Say z is the direction along the coils' axes, then dB/dz=0 and d²B/dz² = 0 halfway between the two for a very uniform field.

There is one special spacing of the coils that makes that true. So, call the distance between them (say) 2b, find the field at an arbitrary point z between them. Find the first two derivatives, and set both equal to zero at z=b. This will give you a value for b in terms of the coil radius R. Look at http://en.wikipedia.org/wiki/Helmholtz_coil for the actual construction and spacing.

So I guess ideally what I'd like to see is this - a condition that defines uniform field (derivatives zero) and a calculation that specifically finds the spacing for which it is fulfilled. I would probably give mostly full credit for researching the answer and providing a solution, but without a solid reason (i.e., derivation), it wouldn't be the full story or full credit.