Gauss's law for magnetism
In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics.
The law is also called "Absence of free magnetic poles".
The integral form of Gauss's law for magnetism states:
The integral and differential forms of Gauss's law for magnetism are mathematically equivalent, due to the divergence theorem.
Due to the Helmholtz decomposition theorem, Gauss's law for magnetism is equivalent to the following statement:[4][5] The vector field A is called the magnetic vector potential.
The magnetic field B 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: Each one either forms a closed loop, winds around forever without ever quite joining back up to itself exactly, or extends to infinity.
The modified formula for use with the SI is not standard and depends on the choice of defining equation for the magnetic charge and current; in one variation, magnetic charge has units of webers, in another it has units of ampere-meters.
[10] This idea of the nonexistence of the magnetic monopoles originated in 1269 by Petrus Peregrinus de Maricourt.
In the early 1800s Michael Faraday reintroduced this law, and it subsequently made its way into James Clerk Maxwell's electromagnetic field equations.
Violation of Gauss's law for magnetism on the discrete level will introduce a strong non-physical force.
[11] There are various ways to preserve Gauss's law for magnetism in numerical methods, including the divergence-cleaning techniques,[12] the constrained transport method,[13] potential-based formulations[14] and de Rham complex based finite element methods[15][16] where stable and structure-preserving algorithms are constructed on unstructured meshes with finite element differential forms.
Left: Some examples of closed surfaces include the surface of a sphere, surface of a torus, and surface of a cube. The magnetic flux through any of these surfaces is zero.
Right: Some examples of non-closed surfaces include the disk surface , square surface, or hemisphere surface. They all have boundaries (red lines) and they do not fully enclose a 3D volume. The magnetic flux through these surfaces is not necessarily zero .