Gauss's Law for Magnetism
Also called Maxwell's second equation, No magnetic monopoles
Gauss's law for magnetism states that the magnetic flux through any closed surface is exactly zero. It is the formal statement that magnetic field lines always close on themselves, because isolated magnetic poles do not exist.
Printed on the AP Physics C: Electricity and Magnetism sheet, and the zero on the right is the content. Every field line entering a closed surface leaves it again.
AP Physics C: E&M reaches it through EK 12.1.A.3: magnetic field lines must form closed loops, as described by Gauss's law for magnetism. EK 12.1.A.1.i supplies the reason, that magnetic fields can be produced by magnetic dipoles or combinations of dipoles but never by monopoles. EK 12.1.A.3.i places it in context: Maxwell's equations are the collection of equations that fully describe electromagnetism, and Gauss's law for magnetism is Maxwell's second equation.
Read it against its electric twin. Gauss's law for electricity gives , which is nonzero whenever a net charge sits inside. The magnetic version has no such term because there is nothing to enclose. Cut a bar magnet in half and you get two magnets, never a lone north pole.
It is not a calculating tool. Where the electric law delivers a field under symmetry, this one delivers zero on every surface and so constrains rather than computes. Its work is done when it tells you a proposed field is impossible.