# Simplified Optimization of the Magnetic Configuration of HEMP-Thrusters
## 2 Materials and Methods
### Analytical Estimate  
Throughout the paper, the symmetry of the cylindrical device is used. A HEMP-Thruster consists of several stages of ring magnets with repelling orientation to each other. In this paper, one stage will be studied, but the extension to more stages can be performed. The natural coordinate system consists of the radial \(r\), azimuthal \(\varphi\), and axial \(z\) coordinates. Figure 2 shows the (\(r\),\(z\))-plane of the magnetic geometry for a typical stage of a HEMP-Thruster. The \(z\)-axis is the symmetry axis. A rotation around the \(z\)-axis yields the three-dimensional stage. Terms proportional to \(\partial_{\varphi}\) and the magnetic field component \(B_{\varphi}\) are zero due to the symmetry. Vector quantities are denoted in bold letters. The starting point for the optimization is to find the ratio of length \(L\) to radius \(R\) where the magnetic flux \(\mathbf{B}\) is homogeneous between both ends of the magnet. In other words, we look for an optimal geometry of \(L\) and \(R\), where the field at the origin is as parallel as possible along the \(z\)-axis. The end regions form, together with other stages, the cusp regions where the wall contact happens. Hence, the condition of a homogeneous field in the middle of one stage supports the design goal to limit the wall contact as much as possible to the cusps. For the analytical analysis, the magnetization \(\mathbf{M}\) of the magnet is assumed constant. In the magnetostatic case, [PERSON]'s equations reduce to  
\[\
abla\times\mathbf{H} =0\Leftrightarrow\mathbf{H}=-\
abla\Phi_{m}\] \[\
abla\mathbf{B} =0\,,\]  
where \(\Phi_{m}\) is a magnetic potential, \(\mathbf{B}\) is the magnetic induction and \(\mu_{0}\) is the vacuum permeability. \(\mathbf{B}\) is linked with the magnetic field \(\mathbf{H}\) and \(\mathbf{M}\) by  
\[\mathbf{B}=\mu_{0}(\mathbf{H}+\mathbf{M})\,.\]  
The two equations for the magnetic field produce a Poisson equation for the magnetic potential  
\[\
abla^{2}\Phi_{m}=\
abla\mathbf{M}\,,\]  
where the sources are the divergence of the magnetization. This equation can be treated in the same way as the electrostatic Poisson equation [13]. Instead of charges, the divergence of the magnetization is the source on the right-hand side. In this application, the sources of the potential are the front faces of the magnets, where the constant magnetization changes at the boundary to the surrounding media. For an infinitesimally thin cylinder, the calculation of the potential on the axis of the stage (\(r=0\)) can be simplified to  
\[\Phi_{m}(z)=\frac{\mu_{0}MR}{4\pi}\left(\frac{1}{\sqrt{R^{2}+\left(z+L/2 \right)^{2}}}-\frac{1}{\sqrt{R^{2}+\left(z-L/2\right)^{2}}}\right), \tag{1}\]  
Figure 2: (\(r\),\(z\))-plane of the magnetic geometry for a typical stage of a HEMP-Thruster. The center is denoted by the origin (O). The radius \(R\) and the half-length \(L/2\) are also shown. where \(R\) and \(L\) are the radius and length of the magnetic stage, respectively. Here, the front faces are reduced to a point source. A Taylor's series expansion of Equation (1) at the origin (\(z=0\)) up to the fifth order results in the numerator  
\[\Phi_{m}(z)\propto 16\,\big{(}L^{5}-20\,L^{3}R^{2}+30\,LR^{4}\big{)}z^{5}\] \[+4\,\big{(}L^{7}+2\,L^{5}R^{2}-32\,L^{3}R^{4}-96\,LR^{6}\big{)}z^{3}\] \[+\big{(}L^{9}+16\,L^{7}R^{2}+96\,L^{5}R^{4}+256\,L^{3}R^{6}+256\, LR^{8}\big{)}z\,.\]  
The calculation is performed with the open-source mathematics software system SageMath [14]. A corresponding notebook can be found on GitHub [15], where the data is made public. In order to have a magnetic field \(\mathbf{H}\), which is parallel to the symmetry axis up to the highest order, the fifth- and first-order terms of the potential shall remain. The \(z^{3}\) term must vanish, i.e., the polynomial in front needs to be zero, so that the resulting field is parallel up to the fourth order.