Gauss's law for magnetism

For an electric dipole, a closed surface round the positive charge has net outward flux, q/ε₀, while a surface enclosing no net charge has as many lines entering as leaving.

Magnetic field lines are continuous closed loops, so any closed surface, even one round a single pole of a magnet or an end of a solenoid, has as many lines entering as leaving.

The flux through a small area ΔS is ΔφB = B·ΔS. Gauss's law for magnetism: the total magnetic flux through any closed surface is zero, ΣB·ΔS = 0.

Why: no isolated magnetic pole (monopole) has ever been found, so B has neither sources nor sinks. The most basic magnetic unit is a current loop or dipole, and every magnetic effect can be built from arrangements of these.

Example 5.3: magnetic lines cannot spread out from a point or from one plate, cannot cross, and a static field line can form a closed loop only round a region carrying current. Lines inside a toroid are correct; lines of a solenoid or between pole pieces that stay perfectly straight at the ends break Ampere's law, so the ends must fringe.

Example 5.4: if monopoles existed, the closed-surface flux would equal μ₀q_m, where q_m is the magnetic charge enclosed.

Example 5.4: no element of a wire feels a force from its own field, though it can feel one from other elements of the same wire (zero for a straight wire). A system with zero net charge can still have a magnetic moment, as paramagnetic atoms do.