Gauss's Law for Magnetism
Magnetic field lines never start or end.
Lesson at a glanceShow the key points
- No isolated magnetic poles have been found; the smallest source is a dipole.
- ∮B⃗·dS⃗ = 0 (36.25).
- Magnetic field lines are closed loops.
Electric field lines begin and end on charges. Magnetic field lines never begin or end at all, and that one fact is a law of nature.
From poles to a flux law
Coulomb’s law E = q/4πε₀r² leads to Gauss’s law ∮E⃗·dS⃗ = qinside/ε₀. In exactly the same way, the pole law B = (μ₀/4π)m/r² would give ∮B⃗·dS⃗ = μ₀minside, the net pole strength enclosed.
But no isolated magnetic pole has ever been found. Cut a magnet in half and you get two magnets, each with both poles. The smallest source of magnetic field is a dipole, whose net pole strength is zero. So the net pole strength inside any closed surface is zero, and:
Draw a closed surface round one charge of an electric dipole: field lines only leave it, so the flux is not zero. Draw one round one end of a bar magnet: lines leave through part of the surface and enter through another part, because inside the magnet they continue from S back to N. As many go in as come out.
- 1No isolated magnetic poles: the smallest magnetic source is a dipole.
- 2∮B⃗·dS⃗ = 0 for every closed surface (36.25).
- 3Magnetic field lines form closed loops; electric field lines start and end on charges.
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