Gauss’s law for magnetism

In physics , Gauss’s law for magnetism is one of four Maxwell’s equations that underlie classical electricity . It states that the deviation of the magnetic field B is equal to zero, in other words, that it is a solenoid vector field . This is equivalent to the statement that magnetic monopoles do not exist. Instead of “magnetic charge”, the basic unit for magnetism is the magnetic dipole . (If monopoles are ever found, the law would have to be modified, as explained below.)

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 .

The name “Gauss’s law for magnetism” is not universally used. The law is also called the “absence of free magnetic poles “; [2] One reference even explicitly states that the law has “no name”. [3] This is also known as the “transversality requirement” [4] because plane waves require that the polarization is transverse to the direction of propagation.

Differential form

The differential form of Gauss’s law for magnetism is:

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

where means deviation , and b is the magnetic field .

Integral form

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

\oiint{\displaystyle \textstyle _{S}} {\displaystyle \mathbf {B} \cdot \mathrm {d} \mathbf {S} =0}

where S is any closed surface (see image at right), and dS is a vector whose magnitude is the area of ​​a minimal piece of surface S , and whose direction is the surface normal to outward (see the surface integral for more details). see) )

The left 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.

Because of the divergence theorem, the integral and differential forms of Gauss’s law for magnetism are mathematically equivalent. That said, one or the other may be more convenient to use in a particular calculation.

In this form the law states that for each volume element in space, there is an equal number of “magnetic field lines” entering and exiting the volume. No total “magnetic charge” can form at any point in space. For example, the south pole of a magnet is exactly as strong as the north pole, and a free-floating south pole is not allowed without north poles (magnetic monopoles). Conversely, this is not true for other fields, such as an electric field or a gravitational field , where the total electric charge or mass can build up in a volume of space.

Vector potential

Because of 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}}.

The vector field A is called the magnetic vector potential .

Note that there is more than one possible A that satisfy this equation for a given B field. In fact, there are infinitely many: any field of the form can be added on to obtain an alternative option for a , by identity (see vector calculus identities ):

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

Since the curl of the gradient is the zero vector field :

\nabla \times \nabla \phi ={\boldsymbol {0}}

This arbitrariness in A is called gauge freedom.

Field line

The magnetic field B can be represented through field lines (also called flux lines )—that is, a set of curves whose direction coincides with the direction of B , and whose area density is proportional to the magnitude of B. . Gauss’s law for magnetism is equivalent to the statement that field lines have no beginning and no end: each one either forms a closed loop, winds forever, never on its own. do not add back at all, or extend to infinity.

Modifications if magnetic monopoles exist

If magnetic monopoles were discovered, Gauss’s law for magnetism states that the deviation of B would be proportional to the magnetic charge density m , which is consistent with Gauss’s law for electric fields. For zero magnetic charge density ( m = 0 ), the basic form of Gauss’s law of magnetism is the result.

The revised formula is not standard in SI units ; In one variation, magnetic charge has Weber ‘s units , in another it has units of ampere – meter.

units	the equation
CGS units [7]
SI units (Weber convention) [8]
SI units ( ampere – meter convention) [9]

where μ ₀ is the vacuum permeability.

So far, examples of magnetic monopoles have been disputed in widespread search,  although some papers report examples matching that behavior.

History

The idea of ​​non-existence of magnetic monopoles was originated by Petrus Peregrinus de Maricourt in 1269. His work greatly influenced William Gilbert, whose 1600 work De Magnet further expanded the idea. This law was re-introduced by Michael Faraday in the early 1800s, and it later made its way into the electromagnetic field equations of James Clerk Maxwell.

Numerical calculation

In numerical computation, the numerical solution may not satisfy Gauss’s law for magnetism due to discretization errors of numerical methods. However, in many cases, for example, for magnetohydrodynamics, it is important to preserve accurately (up to machine accuracy) Gauss’s law for magnetism. Violation of Gauss’s law for magnetism at the discrete level would introduce a strong nonphysical force. Taking energy conservation into account, the violation of this condition leads to a non-conservative energy integral, and the error is proportional to the deviation of the magnetic field. ^[12]

There are several methods of preserving Gauss’s law for magnetism in numerical methods, including divergence-cleaning techniques,  the constrained transport method, potential-based formulation ^[15] and de Rum complex-based finite element methods  Are included.  where stable and structure-preserving algorithms are constructed on unstructured meshes with finite element difference forms.