# Equation for $B$ field inside Hemholtz Coils with Magnetic "Cores"

1. If I added two disks of material with high $$\mu$$ to the two coils in a Helmholtz Coil (such mumetal - as shown in the picture), would the resulting equation for the B field between the coils be $$\overrightarrow{\textbf{B}}=\left( \frac{4}{5} \right)^{\frac{3}{2}}\frac{\mu_r \mu_0 n I}{R}~?$$

2. Would the disks of material have to be disks, or could they be rings, and still get the same boost in $$\mu$$?

I think a thin flat ferromagnetic disk of low coercivity disk that is excited by a current running around its perimeter has an almost negligible effect on its environment because the induced poles on its flat sides are too close to each other; you know, the internal "demagnetizing field". This is why magnets are made into the shape of a long thin rod so that their poles are far apart or made into a toroid so that there are no induced poles at all. Another way to think of it is that the $$\bf B$$-field will be essentially the same on either side of the disk because its normal component must be continuous, and when the disk is thin what goes in is what goes out. Internally, it will have next to zero $$\bf H$$ with a large permeability $$\mu$$ so their product is essentially the constant external bias $$\bf B_0$$, the same as outside as given by the loop.
This is just a computational confirmation that @hyportnex is correct in their answer's reasoning. Adding mu-metal disks only slightly increases the magnetic field strength at the centre of the coils, $$B_{centre}$$, while making it impossible to analytically calculate either the strength or position variation.
Using Finite Element Method Magnetics (FEMM), I created a pair of Helmholtz coils 1 metre in diameter, 2 cm thick, 0.5 m apart, each with 10 amperes running through 400 turns of AWG 18 wire. With no mu-metal disks (as shown on the left), both the Helmholtz coil formula and the simulation give a field at the centre of the coils of $$B_{centre}=0.007193\,\mathrm{T}$$.
If a 48 cm radius, 2 cm thick mu-metal disk is inserted in each coil (as shown on the right), the the central field only increases to $$B_{centre}=0.007962\,\mathrm{T}$$. Adding the disks also changes the field at the centre from convex (i.e. a local maximum in both $$r$$ and $$z$$) to a saddle shape (with a local minimum in $$z$$).
Note: The figures have the $$z$$ axis of the coils along their left edge, with radial distance $$r$$ horizontal. To emphasize the small changes near the centre of the coils, the colours show the $$B$$ values with a scale range of ±1% of the value at the centre of the coils. i.e. Light blue means $$B<0.99 B_{centre}$$ and purple means $$B>1.01 B_{centre}$$. The black lines are flux lines.