There are several ways to obtain Gauss`s law for magnetism in numerical methods, including divergence cleaning techniques,[13] constrained transport method,[14] potential-based formulations,[15] and finite element methods based on the de Rham complex,[16][17] where structure-preserving stable algorithms are constructed on unstructured meshes with finite element differential forms. Gauss`s law of magnetism is a physical application of Gauss`s theorem (also known as the divergence theorem) in calculus, discovered independently by Lagrange in 1762, Gauss in 1813, Ostrogradsky in 1826 and Green in 1828. Gauss`s law for magnetism simply describes a physical phenomenon that a magnetic monopoly does not exist in reality. This law is therefore also called „absence of free magnetic poles“. The name „Gauss`s law for magnetism“[1] is not commonly used. The law is also called „absence of free magnetic poles“; [2] One reference even explicitly states that the law has „no name“. [3] It is also known as the „transversality requirement“[4] because for plane waves, it requires that the polarization be perpendicular to the direction of propagation. The differential form of Gauss`s law for magnetism is: In physics, Gauss`s law for magnetism is one of the four Maxwell equations underlying classical electrodynamics. It indicates that the magnetic field B has zero divergence,[1] in other words, that it is a solenoid vector field. This means that magnetic monopoles do not exist. [2] Instead of „magnetic charges“, the basic unit of magnetism is the magnetic dipole. (If monopolies were ever found, the law would have to be changed, as detailed below.) In numerical computation, the numerical solution may not satisfy Gauss`s law for magnetism due to discretization errors in numerical methods.

However, in many cases, for example for magnetohydrodynamics, it is important to maintain the Gaussian law for magnetism accurately (down to the accuracy of the machine). Violation of Gauss`s law for discrete plane magnetism results in a strong non-physical force. As for energy saving, the violation of this condition leads to a non-conservative energy integral, and the error is proportional to the divergence of the magnetic field. [12] However, as Pierre Curie pointed out in 1894, magnetic monopoles may be conceivable. The introduction of fictitious magnetic charges into Maxwell`s equations can give Gauss`s law for magnetism the same appearance as Gauss`s law for electricity, and mathematics can become symmetric. Because of Helmholtz`s decomposition theorem, Gauss`s law for magnetism is equivalent to the following statement:[5][6] People had long noticed that when a magnetic bar is divided into two parts, two small magnets with their own south and north poles are formed. This can be explained by the law of the Ampere circuit: the magnetic bar consists of many rings of circular current, each essentially a magnetic dipole; Macroscopic magnetism is based on the alignment of microscopic magnetic dipoles. Since a small ring of current always produces an equivalent magnetic dipole, there is no way to create a free magnetic charge. So far, no magnetic monopole has been found in experiments, although many theorists believe that a magnetic monopole exists and are still looking for it. The integral and differential forms of Gauss`s law for magnetism are mathematically equivalent due to the divergence theorem. In other words, one or the other might be more convenient to use in a particular calculation.

The magnetic field B can be represented by field lines (also called flow lines) – that is, a set of curves whose direction corresponds to the direction of B and whose surface density is proportional to the size of B. Gauss`s law for magnetism is equivalent to asserting that field lines have no beginning or end: each forms a closed loop, winds eternally without ever connecting exactly to itself, or extends to infinity. If magnetic monopoles were discovered, then Gauss`s law for magnetism would indicate that the divergence of B would be proportional to the magnetic charge density ρm, analogous to Gauss`s law for the electric field. The zero net magnetic charge density (ρm = 0) is the original form of Gauss`s law of magnetism. The left side of this equation is called the net flux of the magnetic field of the surface, and Gauss`s law for magnetism states that it is always zero. Gauss`s law, one of two statements describing electric and magnetic fluxes. Gauss`s law for electricity states that the electric flux Φ on each closed surface is proportional to the net electric charge q trapped by the surface; That is, Φ = q/ε0, where ε0 is the electric permittivity of free space and has a value of 8.854 × 10–12 square kulombs per newton per square meter. The law implies that isolated electrical charges exist and that similar charges repel each other while different charges attract.

Gauss`s law for magnetism states that the magnetic flux B on any closed surface is zero; that is, div B = 0, where div is the divergence operator. This law is consistent with the observation that isolated magnetic poles (monopoles) do not exist. Gauss`s law for magnetism states that there are no magnetic monopoles and that the total flux through a closed surface must be zero. This page describes the integral and differential forms of the time domain of Gauss`s law for magnetism and how the law can be derived. The frequency range equation is also given. At the bottom of the page, a brief history of Gauss`s law for magnetism is provided. Gauss`s law for magnetism can be written in two forms, a differential form and an integral form. These forms are equivalent because of the divergence theorem. this is Gauss`s law for magnetism in differential form.

Gauss`s law can be derived using the Biot–Savart law, which is defined as follows: Gauss`s law for magnetic fields in the differential form can be derived using the divergence theorem. The divergence theorem states: Gauss`s law for magnetic fields in integral form is given by: The equation can also be written in the frequency domain as follows: S {displaystyle scriptstyle S} B ⋅ d S = 0 {displaystyle mathbf {B} cdot mathrm {d} mathbf {S} =0}. ∇ ⋅ B = 0 {displaystyle nabla cdot mathbf {B} =0} The law in this form states that for every solid element in space, there are exactly the same number of „magnetic field lines“ entering and leaving the volume.

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