Gauss's Law for Magnetism

Picture any closed surface, a sphere, a cube, a lumpy potato shape, sitting inside a magnetic field. Count the field lines piercing inward as negative and the ones piercing outward as positive. Gauss's law for magnetism says that sum is always zero, no matter the surface, no matter the field. This is written as the closed surface integral of B dot dA equals zero. It holds everywhere, for every magnet and every current configuration, with no exceptions.

The reason traces back to the shape of magnetic field lines themselves. Electric field lines can start on a positive charge and stop on a negative one, because isolated electric charges exist. Magnetic field lines have no such endpoints. A bar magnet's field loops from the north pole, out through space, and back in at the south pole, continuing inside the magnet to close the loop. Cut the magnet in half and you do not get an isolated north and an isolated south; you get two smaller magnets, each with its own matched pair of poles. Since no isolated magnetic charge, or monopole, has ever been found, no field line can begin or end in empty space. Any line that crosses into a closed surface must cross back out, so entries and exits always cancel.

This sits right next to Gauss's law for electric fields, and the contrast is the whole point of the topic. The electric version, closed surface integral of E dot dA equals Q_enclosed over epsilon₀, lets the flux be nonzero because charge can be enclosed. The magnetic version has no enclosed-charge term at all; it is flux equals zero, full stop. If you instead look at an open surface, like a single loop's worth of a flat disk, the flux need not be zero, and that nonzero, changing flux is exactly what drives Faraday's law in the next topic.