Gauss’s law for magnetism

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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]

1 Differential form
2 Integral form
3 In terms of vector potential
4 In terms of field lines
5 Modification if magnetic monopoles exist
6 References
7 See also

Differential form

The differential form for Gauss's law for magnetism is the following:

$\nabla\cdot\mathbf{B} = 0$

where

$\nabla \cdot$ denotes divergence,

B is the magnetic field.

Integral form

The integral form of Gauss's law for magnetism states:

$\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0$

where

S is any closed surface (a "closed surface" is the boundary of some three-dimensional volume; the surface of a sphere or cube is a "closed surface", but a disk is not),

dA is a vector, whose magnitude is the area of an infinitesimal piece of the surface S, and whose direction is the outward-pointing surface normal (see surface integral for more details),

The left-hand 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.

In terms of vector potential

Main article: Magnetic vector potential

Due to the Helmholtz decomposition theorem, Gauss's law for magnetism is equivalent to the following statement:

There exists a vector field A such that $\mathbf{B} = \nabla\times\mathbf{A}$

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 $\nabla$ φ can be added onto A to get an alternative choice for A, by the identity:

$\nabla\times \mathbf{A} = \nabla\times(\mathbf{A} + \nabla \phi)$

(see Vector calculus identities). This arbitrariness in A is called gauge freedom.

In terms of field lines

Main article: Field line

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.

Modification if magnetic monopoles exist

Main article: Magnetic monopole

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:

$\nabla\cdot\mathbf{B} = \rho_m$ (SI units, weber convention)^[4]

$\nabla\cdot\mathbf{B} = \mu_0 \rho_m$ (SI units, ampere·meter convention)^[5]

$\nabla\cdot\mathbf{B} = 4\pi\rho_m$ (cgs units)^[6]

where $\mu_0 \$ is the vacuum permeability.

References

^ Tai L. Chow (2006). Electromagnetic Theory: A modern perspective. Boston MA: Jones and Bartlett, p. 134. ISBN 0-7637-3827-1.
^ Jackson, John David (1999). Classical Electrodynamics, 3rd ed., New York: Wiley, Page 237. ISBN 0-471-30932-X.
^ Griffiths, David J. (1998). Introduction to Electrodynamics, 3rd ed., Prentice Hall, Page 321. ISBN 0-13-805326-X.
^ Jackson Classical Electrodynamics (1999), section 6.11, equation (6.150), page 273.
^ See, for example, arXiv:physics/0508099v1, eqn (4)
^ F. Moulin (2001). "Magnetic monopoles and Lorentz force" (pdf) 116B (8): 869–877.

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