Gauss's law for magnetism
/ gowss /
Gauss's law for magnetism is the formal statement of a simple truth: magnetic field lines never begin or end; they always form closed loops. Whatever magnetic field flows into any closed surface must flow back out again, so the net magnetic 'flow' through any closed surface is exactly zero. It answers, are there magnetic 'sources' and 'sinks' the way electric charges are sources of electric field?
The law says the total magnetic flux through any closed surface is zero: the net flux of B through a closed surface = 0. Compare this with Gauss's law for electricity, where the flux through a closed surface equals the enclosed charge divided by epsilon_0. The electric version is nonzero because you can trap electric charge inside; the magnetic version is zero because there is no magnetic charge to trap. This is the mathematical way of saying magnetic monopoles do not exist: you cannot enclose a lone north pole, because every north pole comes married to a south pole.
This is one of the four Maxwell's equations, the compact set that describes all of electricity and magnetism. Its content is quietly profound: it is the reason a cut magnet always yields two smaller complete magnets, never a separated pole, and it is why every field line you draw around a magnet must eventually close on itself.
Draw any imaginary closed bag around one pole of a bar magnet. Every field line that pokes out through the bag somewhere else pokes back in: the ins and outs cancel exactly, so the net magnetic flux is zero.
Field lines have no ends, so the net flux through any closed surface is zero.
If a true magnetic monopole were ever discovered, this law would need a source term, exactly like Gauss's law for electricity. So far, despite long searches, none has been found.