Maxwell-Gauss equation for electricity shows the path of ballistic or drift of hot charge carriers.
Maxwell-Gauss equation for magnetism shows the opposite of the normal vector to the path of curling electrons.
The Maxwell-Ampere equation describes curling electrons around a current. The current eddies out around surrounding molecules giving rise to this effect. The curl of the curl of the electron field can be seen as a current in some instances.
The Maxwell-Faraday equation the change in a curling electron field will create a looser electron circulation in the opposite direction to the electron field. This opposite circulation is due to conservation of angular momentum.
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