Del dot is divergence, which you can visualize as field lines diverging outward from (or converging inward to) a point. Electrical field lines diverge from charge density, which means that field lines go to and from electrical charges. The zero in the corresponding magnetism equation means there are no magnetic charges or “monopoles”.
The curl (del cross) of electric fields is determined by the presence of changing (d/dt) magnetic fields, and vice versa. So when you have a changing electric field, it induces a circular magnetic field around it, and vice versa. Magnetic fields are also induced by currents J, or moving charges, which is shown in the real world by magnetic fields induced around electrical wires. You see this in electromagnets.
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