# Modelling a Multipole Ring Magnet

I am trying to come up with a realistic model of a multipole ring magnet to use in a simulation.

I'm considering a 3 pole pair ring magnet pictured below, where red represents the north poles and blue the south. (The color gradients don't imply anything physical)

Here is another depiction of the same magnet

My model is to split each pole pair, like shown in the first image, and replace it with a current carrying coil of wire projected intot he 2D plane. The current density of each coil alternates to represent the changes in polarity.

$$J_m=\left(\frac{2 \left(r-R_{\text{in}}\right)}{R_{\text{out}}-R_{\text{in}}}-1\right)\begin{cases} -1 & \frac{1}{2}\leq (\frac{2 \pi \theta }{3} \bmod 1)<1 \\ 1 & \text{True} \end{cases}$$

Now I can solve for the magnetic field of this model

$$\nabla\times\left(\frac{1}{\mu}\nabla\times A(x,y)\right)=\nabla\times M=J_m$$

My $$B$$ field solution looks like this

It seems reasonable to me, but I'm wondering if anyone has any opinion if this is valid model, or how it could be improved.

• Do you use FEM in 3D or in 2D? – Alex Trounev Dec 1 '19 at 11:26
• Should have clarified, I'm using 2D FEM – user2757771 Dec 1 '19 at 11:42
• Is the 2D ring magnet model realistic? – Alex Trounev Dec 1 '19 at 11:46
• Thats the opinion I'm looking for. I created this model based on the simple assumtion that a bar magnet behaves similarly to a coil and am using that to produce an equivalent current model for the magnet. I'm looking for an opinion if it's a good model for simulating a cross section of a ring magnet. – user2757771 Dec 1 '19 at 11:49
• Where is the bar magnet model published? – Alex Trounev Dec 1 '19 at 11:52

Figure 2 shows the distribution of the vector potential in the plane $$z=0$$ (left), and the magnetic field in the plane $$y=0$$ (right). To test the 3D FEM, we used the integral equation for the vector potential (it is in all books on the theory of electromagnetic fields starting with Maxwell) $$\vec {A}=\frac {\mu_0}{4\pi}\int{\frac{\vec {j}}{r}dV}. (1)$$ On the upper face of the magnet in the center we have from 3D FEM $$B_z=-0.00521774$$, and from eq.(1) $$B_z=-0.00528921$$. This is a good coincidence. The disadvantage of this model is the current distribution in Fig. 1, which we chose to describe the magnetization. Ideally, this should be a thin layer (surface current), but this is not acceptable for 3D FEM. We can also replace the magnet with a set of rectangular loops and use the exact formulas for the magnetic field from the article M. Misakian, “Equations for the magnetic ﬁeld produced by one or more rectangular loops of wire in the same plane,” J. Res. Natl. Inst. Stand. Technol., vol. 105, pp. 557– 564, 2000.This algorithm is easily generalized to a ring magnet. Now consider the field of a ring magnet composed of 6 bars. We can use 3D FEM to simulate the vector potential and magnetic field of a multipole magnet. Figure 3 shows the geometry of the current region ("coil"), the distribution of current (red) and magnetic field (blue). In Figure 4 shows the distribution of the magnetic field in the plane of symmetry of the ring $$z=0$$, on the upper side $$z = 0.5$$, and above the ring at $$z = 0.75$$.