Higer dims: Electromagnetic field lines and stereographic projection

In summary, the conversation discussed the observation that electromagnetic field lines resemble the stereographic projection of a 3D sphere onto a 2D surface. It was noted that in this comparison, the electric field represents longitude and the magnetic field represents latitude. There were also suggestions of a possible connection to Möbius transformations and spherical symmetry breaking between dimensions. However, it was acknowledged that there is no clear reason for the similarities. The conversation also discussed the similarity between the field lines and the path of electric current flowing through the Earth. Ultimately, the group was curious if there was a specific reason for these similarities or if it is simply a coincidence.
  • #1
Myslius
120
5
I've noticed that electromagnetic field lines are very similar to stereographic projection of 3D sphere on 2D surface. Pictures below.
14182_121_2.jpg

2jeaj3b.png


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?
 
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  • #2
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
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.
 

1. What are higher dimensions?

Higher dimensions refer to dimensions beyond the three spatial dimensions (length, width, and height) that we are familiar with. These dimensions are often represented mathematically and are used to explain complex concepts in fields such as physics and mathematics.

2. What is an electromagnetic field?

An electromagnetic field is a physical field that is created by electrically charged particles and is responsible for the interactions between those particles. It is made up of both an electric field and a magnetic field, which are intertwined and inseparable.

3. How are field lines used to represent electromagnetic fields?

Field lines are used to visually represent the direction and strength of an electromagnetic field. They are imaginary lines that follow the path of the field and are drawn perpendicular to the field's strength at any given point. The closer the lines are together, the stronger the field is at that point.

4. What is stereographic projection?

Stereographic projection is a method of projecting points from a higher dimensional space onto a lower dimensional space. In the context of electromagnetic field lines, it is used to visualize higher dimensional electromagnetic fields on a two-dimensional plane. This allows for a better understanding and analysis of these complex fields.

5. How are higher dimensions and stereographic projection related to electromagnetic field lines?

Higher dimensions and stereographic projection are used to explain and visualize the behavior of electromagnetic field lines in three-dimensional space. By representing these complex fields in a lower dimensional space, scientists can gain a better understanding of their properties and interactions. This is especially useful in studying and predicting the behavior of electromagnetic fields in various scenarios.

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