# Higer dims: Electromagnetic field lines and stereographic projection

1. Oct 29, 2013

### Myslius

I've noticed that electromagnetic field lines are very similar to stereographic projection of 3D sphere on 2D surface. Pictures below.

In such comparison, electric field represents longitude and magnetic field represents latitude. For more visualization see: http://youtu.be/wFVlzRE_3-o?t=10m20s

Any insights on such observation? Is there any particular reason why this is the case?

2. Nov 26, 2013

### klausen

This has also been on my mind.

There might be a connection to Möbius transformations: http://en.wikipedia.org/wiki/Möbius_transformation
especially since they relate to the lorentz transformations, which are central to EM: http://en.wikipedia.org/wiki/Möbius_transformation#Lorentz_transformations

However, this is neither a clear relation nor a definate reason for the similarities.

Electric current flowing the path of least resistance through the earth has a similar form:
http://www.epa.gov/esd/cmb/GeophysicsWebsite/pages/reference/_img/fig270.jpg

I have a feeling it has to do with spherical symmetry breaking between dimensions. The reason being that this behaviour of field lines is given with Greens theorem/2D general Stokes theorem (or simply the boundry theorem) which is a relation between adjacent dimensions. And the fact that there is no magnetic field for spherical symmetry.

Someone wiser please let us know!

3. Nov 26, 2013

### Jano L.

Electromagnetic field of which sources do you have in mind? If we take two point charges of opposite sign, the electric field lines will look similar to projection of Earth's meridian on your picture. The similarity is in that in both cases, we have two singularities and lines that connect them. However, the analogy does not seem exact. The projections of meridians are circles in plane, but the electric field lines are not.