Pictures of Magnetic Field Lines

[lavinia]

I have looked in vain on the web for pictures of magnetic field lines for multiple linked current loops.
I would be happy just to see a picture of the field lines for a simple Hopf link but somewhere there must be pictures for the Borromean rings and other more complex links - and also braids.

It would be even better if there was a a way to understand the Biot-Savart operator well enough to draw these field lines myself.

Any references are appreciated.

 

[mfb]

As long as the rings are proper circular rings, you can use formulas for them instead of messing around with Biot-Savart. You "just" have to add the contributions from all rings, and then find some way to draw nice lines based on those calculated field vectors.

 

[lavinia]

As long as the rings are proper circular rings, you can use formulas for them instead of messing around with Biot-Savart. You "just" have to add the contributions from all rings, and then find some way to draw nice lines based on those calculated field vectors.

Thanks. I was worried about the fields inside the conductors. The field of one current loop passes through the other. The total field should be continuous though maybe not differentiable since outside the current loops it is curl free - but inside it may not be.

It seems that for arbitrary sets of current loops one can take the magnetic fields of each separately then add them together.

 

[mfb]

It seems that for arbitrary sets of current loops one can take the magnetic fields of each separately then add them together.

As long as the fields are not strong enough to influence the currents significantly.
The magnetic field inside can be more complicated. For DC currents, the influence of the ring where you are in should be roughly linear, starting at zero at the center and reaching the "outside" value (as given by the usual formulas) at the edge. This is not exact but it should give a good approximation. I guess you don't want to calculate a toroidal 3D integral of Biot-Savart...

 

[lavinia]

I guess you don't want to calculate a toroidal 3D integral of Biot-Savart...

No but the integral could be instructive.

I am interested in the geometry of the fields and how they define the cohomology ring of the three sphere minus the two solid tori. For a single current loop, the answer is in the linking number.

For two current loops I thought that one could take two magnetic fields. One gets the second by reversing the direction of the current in one of the loops while keeping the other the same. One would then show (hopefully) that the dual 1 forms to these two fields form a basis for the first de Rham cohomology group. Their geometry would then determine whether their wedge product is cohomologous to zero.

For instance, in the case of two circular current loops of the same radius that are parallel to the XY-plane, the field lines lie in planes that contain the z-axis. So the wedge product must be zero on the boundary of either loop.

 

[mfb]

No idea about the topology stuff.

For instance, in the case of two circular current loops of the same radius that are parallel to the XY-plane, the field lines lie in planes that contain the z-axis.

The individual lines yes, not sure about the combined magnetic field if the rings are not aligned (probably not).