I want to apply the method of images to quasi-stationary calculate the 2D magnetic field distribution of a magnetic dipole and a half space with certain electrical properties (permeability, conductivity). The setup is illustrated in the sketch below:

System setup

The two conductors L1 and L2 are carrying a sinusoidal AC current in opposite directions. This represents an approximation of a magnetic dipole, thereafter referred to as the magnetic dipole. The conductors are surrounded by air. However, there is also a half space (yellow colored) with different permeability ($\mu_h$) and conductivity ($\kappa_h$) values.

For the case $\kappa_h = 0 S/m$ method of images provides an analytical solution for the calculation of magnetic field distribution in the upper half space:

$\textbf{H} = \textbf{H}_D(0,h) - \eta \cdot \textbf{H}_D(0,-h)$

where $\textbf{H}_D(x_0,y_0)$ is the magnetic field distribution of a magnetic dipole at position $(x_0,y_0)$ in air and $\eta = \frac{\mu_h - \mu_0}{\mu_h + \mu_0}$

Now I am wondering if the method of images can be applied to the case of a conductive half space ($\mu_h = \mu_0$ and $\kappa_h > 0 S/m$) as well. In this case occurring eddy currents will affect the field distribution in the upper half space. Does anybody know how to consider these? Can the method of images be applied? Does anybody know specific literature which covers this topic? Any comments will be greatly appreciated!


If the frequency or conductivity is high enough that you can treat the conducting area as a perfect electrical conductor, then yes, you can use the method of images to find the resulting field.

This paper has that scenario, for a circular current, shown in figure 8.

current loop image

This web page has better figures which represent the dipole as a dipole moment vector and how that is flipped to create its image when near a perfect electrical conductor.

dipole moment images

In the case of the perfect conductor, and an even slightly oscillating magnetic field, there is no penetration into the conductor so I don't think its permeability matters. In the case of a slightly conducting half space it's going to get more complicated and I have no idea whether you can use the method of images, but you can check out this paper which gives a solution you could implement numerically for arbitrary permeability and conductivity.


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