Helmholtz coil approximations

My friend Joss Ives got this blog post rolling with this tweet:

When measuring the radius of a Helholtz coil which has some decent thickness in the r-direction, do you measure to inside, outside or middle—
Joss Ives (@jossives) April 20, 2012

What I decided to do to check this out was to integrate the field for a true coiled wire. The first thing I did was figure out the parametric equation for a coil. Here’s what I came up with:

$f(u)=\left[x(u),y(u),z(u)\right]=\left[\left(1+\frac{u}{200}\right)\cos(u),\left(1+\frac{u}{200}\right)\sin(u),\frac{1}{20}\sin\left(\frac{u}{4}\right)\right]$

and here’s what that curve looks like:

plot of the coil parametric function

For that picture I had u go from 0 to 16 pi or 8 round trips.

Now that I had a mathematical expression for the coil, I could go about determining the exact magnetic field for it, rather than assuming it was 8 perfect circular loops. The field is given by:

$\vec{B}(\vec{r})=\frac{\mu_0}{4\pi}\int\frac{\vec{dl}\times\left(\vec{r}-\vec{r'}\right)}{\left|\vec{r}-\vec{r'}\right|^{3/2}}$

The trick to getting this to work was to figure out $\vec{dl}$ . I got that by calculating the directional derivative of the parametric curve, f’(u). I wasn’t, at first, convinced that f’(u) would be right, so I thought about an easier integral first, namely, the length of the parametric curve. For that, you need $\left|\vec{dl}\right|$ which is given by $\sqrt{x'(u)^2+y'(u)^2+z'(u)^2}$ , also known as the length of f’(u). For the vector form of dl, I needed a normalized unit vector in the direction of dl times its length, but that’s just the same as the original vector, f’(u).

Ok, then I was off and running. I could figure out r and r’ from the coordinates, and I just need to ask Mathematica to numerically integrate all three components of the field. Here’s the code that did that:

Mathematica syntax for integrating the magnetic field

You give bfield the location where you want to calculate the field.

To answer Joss’ question, I decided to look at the field strength on the axis, and compare it to the field for an 8-turn coil with an adjustable radius. That latter field is given by:

$B=\frac{\mu_0}{4\pi}\frac{(2\pi R)R}{\left(R^2+z^2\right)^{3/2}}$

I’ve purposely factored the numerator to show where the length of the coil comes into play. What I did was to calculate the coil field at several points on the axis and then fit that data to the equation above. The result was:

fitting the coil field to a simple circular wire field

I also show the value of R for the best fit along with the error on that. What’s interesting is how close the best value of R is to the midpoint of the coil:

$R_\text{mid}=1+\frac{1}{2}\frac{16\pi}{200}=1.12566$

However, it’s important to realize that the length of the two coils I’m comparing are not the same. The best fit length is:

$\text{length}_\text{best fit}=16\pi R_\text{best}=56.47$

while the length of the spiraled coil is given by integrating the dl mentioned above to get 56.61.

Thanks, Joss, that was fun!

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About Andy "SuperFly" Rundquist

Associate professor of physics at Hamline.

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6 Responses to Helmholtz coil approximations

ProfeJMarie (Janet) says:

April 22, 2012 at 10:15 am

But what do these coils DO? Are they changing the world? Are they helping cure cancer? Huh? Huh?

- Andy "SuperFly" Rundquist says:
  
  April 22, 2012 at 6:07 pm
  
  For those who haven’t guessed, that’s my better half joking about questions I always get from family about research I do. Basically, if it’s not medical research, it isn’t worth anything, or so it feels like sometimes.
  
Joss Ives says:

April 22, 2012 at 1:18 pm

Clever work Andy. It looks like I know into whose ear I should place a bug!

I’m still trying to decide if I find it surprising that there was such good agreement of the radius with the midpoint of the coil given that you only used 8 coils.

- Andy "SuperFly" Rundquist says:
  
  April 22, 2012 at 6:08 pm
  
  I might go do a longer coil to check, though the integration takes a while so I can’t just whip it out. It would be interesting to see effects from non-integer coils, too.
  
  - Joss Ives says:
    
    April 22, 2012 at 6:25 pm
    
    I was also thinking about non-integer coils. Things like the effect of an extra half-coil as a function of the number of coils.
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