Magnetic Field Lines and Why They Close

Iron filings around a bar magnet curve into loops because the field itself has no beginning or end. Outside the magnet, lines leave the north pole and arc around to the south pole. That looks like the pattern would stop there, but it doesn't: inside the magnet, the field keeps going, running from the south pole back up to the north pole, closing the loop. Every single field line is a closed curve with no endpoints anywhere, inside or outside the material.

This closure is forced by a basic fact: no one has ever isolated a single magnetic pole. Electric field lines start on positive charges and end on negative ones because isolated positive and negative charges exist separately. Magnetism has no equivalent. Every source of magnetic field, from a bar magnet to a current loop, is a dipole, a north and south bound together. Cut a bar magnet in half and you might expect a lone north piece and a lone south piece. Instead you get two smaller magnets, each with its own complete north and south pole. Keep cutting, down to individual atoms, and you still never isolate a pole. This is why flux through any closed surface drawn around a magnet, enclosing both poles or just part of it, always comes out to zero: whatever field lines go in must come back out.

This sets up the next idea directly. Gauss's law for electric fields works because isolated charge exists, so flux through a closed surface measures enclosed charge. Magnetism's version, Gauss's law for magnetism, says the flux is always zero, because there is no magnetic charge to enclose. If an isolated monopole ever turned up, its field would radiate outward like a point charge's, straight lines with real endpoints, and that zero-flux law would break.