Gauss's law for magnetism

Gauss's law for electricity reads ∮E⃗⋅dA⃗ = q_enc/ε₀: draw any closed surface, sum E⃗⋅dA⃗ over it with the outward normal, and the total counts the charge inside. Write the same statement for the magnetic field and the right-hand side is zero: ∮B⃗⋅dA⃗ = 0, for every closed surface, in every field, at every instant. Nothing has been left out. The equation is empty on the right because no experiment has isolated the magnetic counterpart of charge. Term by term, B⃗⋅dA⃗ is positive where the field leaves the surface and negative where it enters, so the law says the two tallies always match. This is not a statement about symmetric geometry or about weak fields: it holds for a surface of any shape, drawn around anything — a magnet, a coil, half a transformer, a patch of empty space in a radio wave.

A field line ends where flux is created. Shrink a closed surface onto a point and ∮B⃗⋅dA⃗ = 0 says nothing is created there, so magnetic field lines have no ends: each closes on itself or runs off to infinity. That is the entire topology of a dipole. Outside a bar magnet the lines arc from the north end round to the south end; inside the iron they carry on, from south to north, and join up. Test it. Take a bar of 2.0 cm² cross-section carrying B = 0.80 T along its axis, and draw a closed surface that swallows the north half, cutting straight across the middle. The cut face has an outward normal pointing back toward the south end while B⃗ there points the other way, so that face contributes −0.80 × 2.0 × 10⁻⁴ = −1.6 × 10⁻⁴ Wb. The law then fixes the rest with no calculation of the external field at all: the flux leaving through everything outside the iron is +1.6 × 10⁻⁴ Wb. Move the cut, or reshape the surface, and the two numbers change together.

Cut the bar in half at that surface and you do not end up with a north pole in one hand and a south pole in the other. Each piece is a complete magnet with two poles of its own, and the pattern survives all the way down, because the poles were never objects. Every magnetic field we can make comes from a current loop or from an intrinsic magnetic moment — the Bohr magneton that sets its scale is 9.27 × 10⁻²⁴ J T⁻¹ — and both are dipoles. A current loop returns as much field as it sends out, so the divergence of its field is zero exactly, not approximately. In a permanent magnet the aligned atomic moments add to a magnetization whose uncompensated surface currents run around the bar; 'north pole' names the face those lines leave from, not a lump of magnetic charge sitting there. Break the magnet and the surface currents simply reroute around the new faces.

Flatten a closed surface into a pillbox straddling the boundary between two materials, faces of area ΔA on either side. Let the thickness go to zero so the rim contributes nothing, and the law leaves (B₂⊥ − B₁⊥)ΔA = 0: the component of B⃗ normal to an interface is continuous, whatever sits on the two sides. Note what does jump. For a linear material H⃗ = B⃗/μ, so crossing from air into iron with μr = 5000 at normal incidence leaves B⊥ untouched and drops H⊥ by a factor of 5000. Apply the same argument to the walls of a tube of field lines and you get B₁A₁ = B₂A₂: pinch the bundle into a smaller cross-section and the field there is stronger in exact proportion. That is the working rule for magnetic circuits. An iron core of cross-section 4.0 cm² at B = 1.2 T carries Φ = 1.2 × 4.0 × 10⁻⁴ = 4.8 × 10⁻⁴ Wb, and across an air gap where fringing spreads that same flux over 4.6 cm² the field is (4.8 × 10⁻⁴)/(4.6 × 10⁻⁴) = 1.04 T, about 13 % lower.

Three consequences you will use constantly. First, the magnetic flux through an open surface depends only on its boundary curve: two surfaces spanning the same loop — a flat disc and a deep bag — together bound a closed volume whose net flux is zero, so their fluxes are equal. That is what makes 'the flux through the circuit' in Faraday's law well defined; otherwise you would have to say which soap film you meant. Second, ∇⋅B⃗ = 0 everywhere is exactly the condition for B⃗ to be the curl of something, since ∇⋅(∇ × A⃗) = 0 for any A⃗. The vector potential exists for that reason alone, and the machinery of magnetostatics and radiation is built on it. Third, take a plane wave B⃗ = B⃗₀ cos(k⃗⋅r⃗ − ωt). Then ∇⋅B⃗ = −k⃗⋅B⃗₀ sin(k⃗⋅r⃗ − ωt), and requiring that to vanish at every position and time forces k⃗⋅B⃗₀ = 0. The magnetic field of a plane electromagnetic wave has no component along the direction of travel; transversality is not an extra assumption, it is this equation.

The zero is an experimental result, so it is worth knowing how hard it has been pushed. Searches have looked for trapped poles in rock, sediment and lunar samples, for monopoles made in accelerator collisions — MoEDAL sits at the LHC for this purpose — and for a single pole drifting through a superconducting ring, where it would leave a permanent step in the trapped flux. Blas Cabrera recorded one such event in 1982; it never repeated, and no detector has seen another. Cosmic-ray searches now hold any monopole flux below roughly 10⁻¹⁵ per square centimetre per second per steradian. Were one found, the law would read ∮B⃗⋅dA⃗ = qₘ, enc with magnetic charge measured in webers, ∇⋅B⃗ = ρₘ would replace the zero, and Faraday's law would gain a magnetic-current term. One reason to keep looking: Dirac showed in 1931 that a single pole of strength g would force every electric charge to be a multiple of one unit, through the condition eg = nh. A pole of h/e = 4.14 × 10⁻¹⁵ Wb would account for the charge quantisation we already observe.