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In physics, **Gauss's law for magnetism** is one of the four Maxwell's equations that underlie classical electrodynamics. It states that the magnetic field **B** has divergence equal to zero,^{ [1] } in other words, that it is a solenoidal vector field. It is equivalent to the statement that magnetic monopoles do not exist.^{ [2] } Rather than "magnetic charges", the basic entity for magnetism is the magnetic dipole. (If monopoles were ever found, the law would have to be modified, as elaborated below.)

**Physics** is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

**Maxwell's equations** are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. An important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (*c*) in a vacuum. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum of light from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon.

A **magnetic field** is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. The effects of magnetic fields are commonly seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. They exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field vary with location. As such, it is described mathematically as a vector field.

- Differential form
- Integral form
- Vector potential
- Field lines
- Modification if magnetic monopoles exist
- History
- Numerical computation
- See also
- References

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.

In vector calculus, the **divergence theorem**, also known as **Gauss's theorem** or **Ostrogradsky's theorem**, is a result that relates the flow of a vector field through a surface to the behavior of the tensor field inside the surface.

The name "Gauss's law for magnetism"^{ [1] } is not universally used. The law is also called "Absence of free magnetic poles";^{ [2] } 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.

In particle physics, a **magnetic monopole** is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole. A magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring theories, which predict their existence.

In physics, a **plane wave** is a special case of wave or field: a physical quantity whose value, at any moment, is constant over any plane that is perpendicular to a fixed direction in space.

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

where ∇ · denotes divergence, and **B** is the magnetic field.

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

where *S* is any closed surface (see image right), and d**A** 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).

In mathematics, **infinitesimals** are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small. The word *infinitesimal* comes from a 17th-century Modern Latin coinage *infinitesimus*, which originally referred to the "infinity-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it usually must be compared to another infinitesimal object in the same context. Infinitely many infinitesimals are summed to produce an integral.

In mathematics, a **surface integral** is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate over its scalar fields, and vector fields.

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.

**Flux** describes any effect that appears to pass or travel through a surface or substance. A flux is either a concept based in physics or used with applied mathematics. Both concepts have mathematical rigor, enabling comparison of the underlying mathematics when the terminology is unclear. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In electromagnetism, flux is a scalar quantity, defined as the surface integral of the component of a vector field perpendicular to the surface at each point.

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.

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

Due to the Helmholtz decomposition theorem, Gauss's law for magnetism is equivalent to the following statement:^{ [5] }^{ [6] }

The vector field **A** 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):

since the curl of a gradient is the zero vector field:

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: Each one either forms a closed loop, winds around forever without ever quite joining back up to itself exactly, or extends to infinity.

If magnetic monopoles were discovered, then Gauss's law for magnetism would state the divergence of **B** would be proportional to the * magnetic charge density**ρ*_{m}, analogous to Gauss's law for electric field. For zero net magnetic charge density (*ρ*_{m} = 0), the original form of Gauss's magnetism law is the result.

The modified formula in SI units is not standard; in one variation, magnetic charge has units of webers, in another it has units of ampere-meters.

Units | Equation |
---|---|

cgs units^{ [7] } | |

SI units (weber convention)^{ [8] } | |

SI units (ampere-meter convention)^{ [9] } | |

where *μ*_{0} is the vacuum permeability.

So far, no magnetic monopoles have been found, despite extensive search.^{ [10] }

This idea of the nonexistence of magnetic monopoles originated in 1269 by Petrus Peregrinus de Maricourt. His work heavily influenced William Gilbert, whose 1600 work De Magnete spread the idea further. In the early 1800s Michael Faraday reintroduced this law, and it subsequently made its way into James Clerk Maxwell's electromagnetic field equations.

In numerical computation, the numerical solution may not satisfy Gauss's law for magnetism due to the discretization errors of the numerical methods. However, in many cases, e.g., for magnetohydrodynamics, it is important to preserve Gauss's law for magnetism precisely (up to the machine precision). Violation of Gauss's law for magnetism on the discrete level will introduce a strong non-physical force. In view of energy conservation, violation of this condition leads to a non-conservative energy integral, and the error is proportional to the divergence of magnetic field^{ [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.

In physics the **Lorentz force** is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge *q* moving with a velocity **v** in an electric field **E** and a magnetic field **B** experiences a force of

**Magnetohydrodynamics** is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magnetofluids include plasmas, liquid metals, salt water, and electrolytes. The word "magnetohydrodynamics" is derived from *magneto-* meaning magnetic field, *hydro-* meaning water, and *dynamics* meaning movement. The field of MHD was initiated by Hannes Alfvén, for which he received the Nobel Prize in Physics in 1970.

An **electric field** surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as **E**-field. The electric field is defined mathematically as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strength. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

An **electric potential** is the amount of work needed to move a unit of charge from a reference point to a specific point inside the field without producing an acceleration. Typically, the reference point is the Earth or a point at infinity, although any point can be used.

In physics, specifically electromagnetism, the **magnetic flux** through a surface is the surface integral of the normal component of the magnetic field **B** passing through that surface. The SI unit of magnetic flux is the weber (Wb), and the CGS unit is the maxwell. Magnetic flux is usually measured with a fluxmeter, which contains measuring coils and electronics, that evaluates the change of voltage in the measuring coils to calculate the measurement of magnetic flux.

In physics, **Gauss's law**, also known as **Gauss's flux theorem**, is a law relating the distribution of electric charge to the resulting electric field. The surface under consideration may be a closed one enclosing a volume such as a spherical surface.

In physics, specifically electromagnetism, the **Biot–Savart Law** is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. The Biot–Savart law is fundamental to magnetostatics, playing a role similar to that of Coulomb's law in electrostatics. When magnetostatics does not apply, the Biot–Savart law should be replaced by Jefimenko's equations. The law is valid in the magnetostatic approximation, and is consistent with both Ampère's circuital law and Gauss's law for magnetism. It is named after Jean-Baptiste Biot and Félix Savart, who discovered this relationship in 1820.

In classical electromagnetism, **Ampère's circuital law** relates the integrated magnetic field around a closed loop to the electric current passing through the loop. James Clerk Maxwell derived it using hydrodynamics in his 1861 paper "On Physical Lines of Force" and it is now one of the Maxwell equations, which form the basis of classical electromagnetism.

A **continuity equation** in physics is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.

"**A Dynamical Theory of the Electromagnetic Field**" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. In the paper, Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and deduces that light is an electromagnetic wave.

In electrodynamics, **Poynting's theorem** is a statement of conservation of energy for the electromagnetic field, in the form of a partial differential equation, due to the British physicist John Henry Poynting. Poynting's theorem is analogous to the work-energy theorem in classical mechanics, and mathematically similar to the continuity equation, because it relates the energy stored in the electromagnetic field to the work done on a charge distribution, through energy flux.

The term **magnetic potential** can be used for either of two quantities in classical electromagnetism: the *magnetic vector potential*, or simply *vector potential*, **A**; and the *magnetic scalar potential**ψ*. Both quantities can be used in certain circumstances to calculate the magnetic field **B**.

A **classical field theory** is a physical theory that predicts how one or more physical fields interact with matter through **field equations**. The term 'classical field theory' is commonly reserved for describing those physical theories that describe electromagnetism and gravitation, two of the fundamental forces of nature. Theories that incorporate quantum mechanics are called quantum field theories.

**Magnetostatics** is the study of magnetic fields in systems where the currents are steady. It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast magnetic switching events that occur on time scales of nanoseconds or less. Magnetostatics is even a good approximation when the currents are not static — as long as the currents do not alternate rapidly. Magnetostatics is widely used in applications of micromagnetics such as models of magnetic storage devices as in computer memory. Magnetostatic focussing can be achieved either by a permanent magnet or by passing current through a coil of wire whose axis coincides with the beam axis.

There are various **mathematical descriptions of the electromagnetic field** that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

**Computational magnetohydrodynamics (CMHD)** is a rapidly developing branch of magnetohydrodynamics that uses numerical methods and algorithms to solve and analyze problems that involve electrically conducting fluids. Most of the methods used in CMHD are borrowed from the well established techniques employed in Computational fluid dynamics. The complexity mainly arises due to the presence of a magnetic field and its coupling with the fluid. One of the important issues is to numerically maintain the condition, from Maxwell's equations, to avoid the presence of unrealistic effects, namely magnetic monopoles, in the solutions.

In physics, **Gauss's law for gravity**, also known as **Gauss's flux theorem for gravity**, is a law of physics that is essentially equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. Although Gauss's law for gravity is equivalent to Newton's law, there are many situations where Gauss's law for gravity offers a more convenient and simple way to do a calculation than Newton's law.

**Magnetohydrodynamic turbulence** concerns the chaotic regimes of magnetofluid flow at high Reynolds number. Magnetohydrodynamics (MHD) deals with what is a quasi-neutral fluid with very high conductivity. The fluid approximation implies that the focus is on macro length-and-time scales which are much larger than the collision length and collision time respectively.

- 1 2 Chow, Tai L. (2006).
*Electromagnetic Theory: A modern perspective*. Jones and Bartlett. p. 134. ISBN 0-7637-3827-1. - 1 2 Jackson, John David (1999).
*Classical Electrodynamics*(3rd ed.). Wiley. p. 237. ISBN 0-471-30932-X. - ↑ Griffiths, David J. (1998).
*Introduction to Electrodynamics*(3rd ed.). Prentice Hall. p. 321. ISBN 0-13-805326-X. - ↑ Joannopoulos, John D.; Johnson, Steve G.; Winn, Joshua N.; Meade, Robert D. (2008).
*Photonic Crystals: Molding the Flow of Light*(2nd ed.). Princeton University Press. p. 9. ISBN 978-0-691-12456-8. - ↑ Schilders, W. H. A.; et al. (2005).
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*Il Nuovo Cimento B*.**116**(8): 869–877. arXiv: math-ph/0203043 . Bibcode:2001NCimB.116..869M. - ↑ Jackson, John David (1999).
*Classical Electrodynamics*(3rd ed.). Wiley. p. 273, eq. 6.150. - ↑ See for example equation 4 in Nowakowski, M.; Kelkar, N. G. (2005). "Faraday's law in the presence of magnetic monopoles".
*Europhysics Letters*.**71**(3): 346. arXiv: physics/0508099 . Bibcode:2005EL.....71..346N. doi:10.1209/epl/i2004-10545-2. - ↑ Magnetic Monopoles, report from Particle data group, updated August 2015 by D. Milstead and E.J. Weinberg. "To date there have been no confirmed observations of exotic particles possessing magnetic charge."
- ↑ Brackbill, J.U; Barnes, D.C (May 1980). "The Effect of Nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations".
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value (help). - ↑ Tóth, Gábor (1 July 2000). "The ∇·B=0 Constraint in Shock-Capturing Magnetohydrodynamics Codes".
*Journal of Computational Physics*.**161**(2): 605–652. doi:10.1006/jcph.2000.6519. ISSN 0021-9991. - ↑ Hernquist, Lars; Vogelsberger, Mark; Mocz, Philip (21 July 2014). "A constrained transport scheme for MHD on unstructured static and moving meshes".
*Monthly Notices of the Royal Astronomical Society*.**442**(1): 43–55. doi:10.1093/mnras/stu865. ISSN 0035-8711. - ↑ Jardin, Stephen (2010).
*Computational Methods in Plasma Physics*(1st ed.). Boca Raton: CRC Press. ISBN 9780429075537. - ↑ Hu, Kaibo; Ma, Yicong; Xu, Jinchao (1 February 2017). "Stable finite element methods preserving ∇·B=0 exactly for MHD models".
*Numerische Mathematik*.**135**(2): 371–396. doi:10.1007/s00211-016-0803-4. ISSN 0945-3245. - ↑ Ma, Yicong; Hu, Kaibo; Hu, Xiaozhe; Xu, Jinchao (July 2016). "Robust preconditioners for incompressible MHD models".
*Journal of Computational Physics*.**316**: 721–746. doi:https://doi.org/10.1016/j.jcp.2016.04.019 Check`|doi=`

value (help).

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