# Uniform magnetic field within two Helmholtz coils

I understand that the magnetic field produced by a single Helmholtz coil is approximately :

$$B=N\frac{\mu_0IR^2}{2(R^2+x^2)^{3/2}}$$

Where $$x$$ is the distance from the centre of the coil to the point of interest along the central axis of the coil, R is the radius of the coil, N is the number of turns in the coil, I, is the current in the coil and $$\mu_0$$ is the permeability of free space (in vacuum).

The question I have is, how do you calculate the separation distance between two Helmholtz coils, such that the magnetic field between them is uniform.

On my attempt, I used the principle of super-position, with the aid of the following diagram:

I said the following must be true:

$$B_{constant} = \frac{\mu_0NIR^2}{2}[\frac{1}{(R^2+x^2)^{3/2}}+\frac{1}{(R^2 +(d-x)^2)^{3/2}}]$$

I am wondering now if this approach will work, and also if there is an easier method as there seems to be a lot of algebra involved and a potential polynomial in d of degree 12.

You don't require the magnetic field to be uniform - you require that the correction to the magnetic field amplitude for small $\delta x$ shift from the centre is at most $\mathcal{O}(\delta x^3)$.
The formula you use is correct. Try calculating the maximum of the field intensity along the axis of symmetry for two equal coils (in your configuration) with current $I$ separated by distance $d$. Then require $\frac{\partial^2 B}{\partial x^2} = 0$.
Added a few plots of the magnetic field $B$ versus the distance $x$ on the axis for various configurations. Note the 'flatness' around the maximum for $\frac{d}{R} = 1$
• But surely the requirement would be that $\frac{\partial{B}}{\partial{x}} = 0$ such that B doesn't change with x? Nov 8, 2017 at 17:08
• Yes, but this is the requirement that the field has a maximum. Which is at $d/2$ whether or not $d = R$ as in Helmholtz configuration. The key difference is that in generic $d/R$ ratio the $B_{xx}$ doesn't necessarily vanish - only if $d/R = 1$ it does - try writing it out! Nov 8, 2017 at 18:01
• In fact, by symmetry, the deviations from the central field are proportional to $\delta x^4$. (Since the magnetic field is obviously an even function of $\delta x$, its power series can only contain even terms.) Nov 8, 2017 at 18:24