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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 and 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 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. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at the speed of light. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum 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. He also 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 electrical currents and magnetized materials. In everyday life, the effects of magnetic fields are often 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. Magnetic fields 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 varies with location. As such, it is an example of a vector field.

- Differential form
- Integral form
- Vector potential
- Field lines
- Modification if magnetic monopoles exist
- History
- 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 the physics of wave propagation, a **plane wave** is a wave whose wavefronts are infinite parallel planes. Mathematically a plane wave takes the form

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 "infinite-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 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

An **electric field** is a vector field surrounding an electric charge that exerts force on other charges, attracting or repelling them. Mathematically the electric field is a vector field that associates to each point in space the force, called the Coulomb force, that would be experienced per unit of charge by an infinitesimal test charge at that point. The units of the electric field in the SI system are newtons per coulomb (N/C), or volts per meter (V/m). 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 positive 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 beyond the influence of the electric field charge 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 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 mathematics, **Poisson's equation** is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after the French mathematician, geometer, and physicist Siméon Denis Poisson.

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.

In physics, chemistry and biology, a **potential gradient** is the local rate of change of the potential with respect to displacement, i.e. spatial derivative, or gradient. This quantity frequently occurs in equations of physical processes because it leads to some form of flux.

"**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.

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**.

**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.

The **electromagnetic wave equation** is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field **E** or the magnetic field **B**, takes the form:

A **Gaussian surface** is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, the electric field, or magnetic field. It is an arbitrary closed surface *S* = ∂*V* used in conjunction with Gauss's law for the corresponding field by performing a surface integral, in order to calculate the total amount of the source quantity enclosed; e.g., amount of gravitational mass as the source of the gravitational field or amount of electric charge as the source of the electrostatic field, or vice versa: calculate the fields for the source distribution.

In electromagnetism and applications, an **inhomogeneous electromagnetic wave equation**, or **nonhomogeneous electromagnetic wave equation**, is one of a set of wave equations describing the propagation of electromagnetic waves generated by nonzero source charges and currents. The source terms in the wave equations makes the partial differential equations *inhomogeneous*, if the source terms are zero the equations reduce to the homogeneous electromagnetic wave equations. The equations follow from Maxwell's equations.

The **uniqueness theorem** for Poisson's equation states that, for a large class of boundary conditions, the equation may have many solutions, but the gradient of every solution is the same. In the case of electrostatics, this means that there is a unique electric field derived from a potential function satisfying Poisson's equation under the boundary conditions.

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.

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.

- 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).
*Handbook of Numerical Analysis*. p. 13. ISBN 978-0-444-51375-5. - ↑ Jackson, John David (1999).
*Classical Electrodynamics*(3rd ed.). Wiley. p. 180. ISBN 0-471-30932-X. - ↑ Moulin, F. (2001). "Magnetic monopoles and Lorentz force".
*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."

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