In physics, Gauss's law for magnetism is one of Maxwell's equations, the four equations that underlie classical electrodynamics. It states that the magnetic field B has divergence equal to zero, in other words, that it is a solenoidal vector field. It is equivalent to the statement that magnetic monopoles do not exist. Rather than "magnetic charges", the basic entity for magnetism is the magnetic dipole. (Of course, if monopoles were ever found, the law would have to be modified, as elaborated below.)
Gauss's law for magnetism can be written in two forms, a differential form and an integral form. These forms are equivalent due to the divergence theorem.
Note that the terminology "Gauss's law for magnetism" is widely used, but not universal.^{[1]} The law is also called "Absence of free magnetic poles".^{[2]} (or some variant); one reference even explicitly says the law has "no name".^{[3]}
It is also referred to as the "transversality requirement"^{[4]} because for plane waves it requires that the polarization be transverse to the direction of propagation.
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The differential form for Gauss's law for magnetism is the following:
where
The integral form of Gauss's law for magnetism states:
where
The lefthand side of this equation is called the net flux of the magnetic field out of the surface, and Gauss's law for magnetism states that it is always zero.
The integral and differential forms of Gauss's law for magnetism are mathematically equivalent, due to the divergence theorem. That said, one or the other might be more convenient to use in a particular computation.
Due to the Helmholtz decomposition theorem, Gauss's law for magnetism is equivalent to the following statement:
This vector field is called the magnetic vector potential.
Note that there is more than one possible A which satisfies this equation for a given B field. In fact, there are infinitely many: Any field of the form φ can be added onto A to get an alternative choice for A, by the identity:
(see Vector calculus identities). This arbitrariness in A is called gauge freedom.
The magnetic field B, like any vector field, can be depicted via field lines (also called flux lines) that is, a set of curves whose direction corresponds to the direction of B, and whose areal density is proportional to the magnitude of B. Gauss's law for magnetism is equivalent to the statement that the field lines have neither a beginning nor an end: They either form a closed loop, or extend to infinity in both directions.
If magnetic monopoles were ever discovered to exist, then Gauss's law for magnetism would be disproved. Instead, the divergence of B would be proportional to the "magnetic charge density" ρ_{m}, as follows:
where is the vacuum permeability.
