Constructing a Tri-Axial Square Helmholz Coil

[rushingwind]

Homework Statement

I'm a summer REU student tasked with building a tri-axial square helmholtz coil setup. My frame will be made of aluminum while the wire used will be copper. I've been researching for days, having actually never studied these coils before, and I'm having serious trouble figuring out the correct dimensions, turns, necessary wire thickness, and current.

The center of the field will contain a MOT. I'm looking to cancel out slightly more than 1 gauss. Presently, I don't quite understand how to apply Gauss's Law (or if I'm doing it correctly/incorrectly), as I've only taken Introductory Physics (I have, however, taken Calculus, but not Calc II).

The Attempt at a Solution

So far, this is the setup I've been able to come up with: (I've edited this after finally being able to see my workspace). I'm very limited in space, so the sides of each square set must be 18in (about .46m), 16in (about .41m), and 14in (about .36m). That brings my calculations for spacing between pairs to .25m, .2m, and .19m, respectively.

Running 2A of current through each coil (can't go below 1A, or above 5A), these are the number of turns I calculated:

18in/.46m -- 15 turns
16in/.41m -- 13 turns
14in/.36m -- 12 turns

I need to cancel out slightly more than 1 gauss at the center of this tri-axial configuration, and my math shows me that it does... If I'm doing this right. I've also calculated that I need at least 14 gauge copper wire for this setup.Disclaimer: I appreciate any pointers because I actually have no idea what I'm doing, even after days of research. I don't know if my work is on the right track or completely off, as this is a little beyond my current level of physics education! Any help (or pointers in the right direction!) would be greatly appreciated.

Thank you in advance.

 

[TSny]

Gauss' law isn't relevant here. The magnetic fields of the coils can be calculated using the Biot-Savart law. Using that law I found an expression for the field produced at the midway point between a pair of square coils if the distance between the coils is the optimal distance of .5445 L, where L is the length of a side of a coil. I find that the B-field (in Gauss) produced at that point is

B = (.001629)*I*N/L where I is the current, N is the number of turns, and L is the length of a side of a coil.

This seems to agree fairly closely with your figures.

The currents in the two coils should be in the same direction.

The direction of the field will be along the axis of the pair of coils. Thus, with three pairs of coils whose axes are mutually perpendicular, you can produce fields in any direction at the midpoint by independently adjusting the currents in the 3 pairs of coils.

Here's a document that contains a picture of a tri-axial square helmholtz coil
http://www.laboratorio.elettrofisico.com/pdf/Misura/mis_01_helmholtz_coils.pdf

 

[rushingwind]

Thank you so much, TSny. I was having trouble trying to apply Gauss' Law... no wonder! I feel more confident about the numbers I came up with now. Thank you so much for the link to the document, and your guidance.

 

[marcusl]

Helmholtz coils are designed to produce a uniform magnetic field at the center, specifically one in which the quadratic change in field along the axis (i.e., the second derivative) is zero. The spacings of your first and third coils are correct to produce the Helmholtz condition, but the second spacing is inconsistent with its coil dimension. Here is a reference that may be helpful:
http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&sqi=2&ved=0CE8QFjAC&url=http%3A%2F%2Fdigitalcommons.unl.edu%2Fcgi%2Fviewcontent.cgi%3Farticle%3D1043%26context%3Dphysicsrudd&ei=yyLpT5SFJo2K8QSk8dGCDg&usg=AFQjCNGp8OwoTtpXteHoXfO7WiNyQoXTMg
A Google search on square Helmholtz coils will produce many others.

 

[TSny]

Here is a reference that may be helpful:
http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&sqi=2&ved=0CE8QFjAC&url=http%3A%2F%2Fdigitalcommons.unl.edu%2Fcgi%2Fviewcontent.cgi%3Farticle%3D1043%26context%3Dphysicsrudd&ei=yyLpT5SFJo2K8QSk8dGCDg&usg=AFQjCNGp8OwoTtpXteHoXfO7WiNyQoXTMg

That's an interesting reference. I wasn't aware that increasing the separation beyond the "Helmholtz distance" of .5445L increases the distance of uniformity of the field along the axis.

Thanks.

 

[rushingwind]

Thank you so much everyone. I really appreciate the references and the help. I'm going to start building two of the coils tomorrow in the shop room. You guys are a lifesaver! :)