Gauss’s Law for Magnetism
The second of Maxwell’s four equations — and the simplest:
$$\nabla \cdot \mathbf{B} = 0$$
In words: the divergence of the magnetic field is zero. Everywhere. Always.
What it says
Compare with Gauss’s law for electricity: $\nabla \cdot \mathbf{E} = \rho / \varepsilon_0$.
The electric field has sources (positive charges) and sinks (negative charges). The magnetic field has neither. There are no magnetic charges — no “magnetic monopoles.”
This means:
- Every magnetic field line that enters a region must also leave it
- Field lines never start or end — they always form closed loops
- Draw any closed surface: the net magnetic flux through it is exactly zero
$$\oint \mathbf{B} \cdot d\mathbf{A} = 0$$
Compare electric field lines (which start and end on charges) with magnetic field lines (which always form loops). Any closed surface around the dipole has zero net magnetic flux.
The deepest asymmetry in physics
Why $\nabla \cdot \mathbf{B} = 0$ instead of $\nabla \cdot \mathbf{B} = \rho_m / \mu_0$ (with magnetic charges $\rho_m$)?
Nobody has ever found a magnetic monopole. If you could isolate a north pole without a south pole, this equation would need a right-hand side. Physicists have searched — Paul Dirac showed in 1931 that monopoles would explain why electric charge comes in discrete units — but none have been observed.
This equation is the formal statement of an empirical fact: nature, as far as we can tell, doesn’t have magnetic charges.
If one were ever discovered, it would be one of the biggest events in physics. This is the equation that would change.
Equation #2 of 4
The two Gauss’s laws together describe the divergence of the electromagnetic field:
| Equation | Says |
|---|---|
| $\nabla \cdot \mathbf{E} = \rho / \varepsilon_0$ | Electric charges exist (sources/sinks of $\mathbf{E}$) |
| $\nabla \cdot \mathbf{B} = 0$ | Magnetic charges don’t exist (no sources/sinks of $\mathbf{B}$) |
The remaining two equations describe the curl — how the fields rotate. That’s where things get dynamic: changing fields create other fields.