In the absence of charges Maxwell's equations of electromagnetics are skew symmetric under interchange of the Electric and Magnetic field.
div B = 0
div E = 0
curl B = dE/dt
curl E = - dB/dt
Introducing a charge changes this as there are no observed sources for the Magnetic field.
(hence div B = 0 ). We have sources for the electric field (electrons and other charged particles) so with sources:
div B = 0
div E = e
curl B = j + dE/dt
curl E = - dB/dt
If a source for the magnetic field were to exist then it would be a magnetic monopole and
the symmetry would be restored to the equations. Lots of people have thought that this would be a good idea. Paul Adrian Maurice Dirac proved that if even one magnetic monopole existed then electric charge has to be quantized! (That's an awesome thing to prove). To date no one has found a magnetic monopole. They are expected in some GUT's and their absence was explained by Alan Guth using inflation.