Gauss's law for magnetism

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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 Study of the fundamental properties of matter and energy

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.

Maxwells equations set of partial differential equations that describe how electric and magnetic fields are generated and altered by each other and by charges and currents

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.

Magnetic field spatial distribution of vectors allowing the calculation of the magnetic force on a test particle

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.

Contents

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.

Magnetic monopole hypothetical particle with one magnetic pole

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.

Differential form

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

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

Integral form

Definition of a closed surface.
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. SurfacesWithAndWithoutBoundary.svg
Definition of a closed surface.
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.

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

OiintLaTeX.svg

where S is any closed surface (see image right), and dA 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).

Infinitesimal extremely small quantity in calculus; thing so small that there is no way to measure them

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.

Surface integral generalization of a multiple integral to (possibly)-curved surfaces

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 measure of the flow of something through a surface, in some cases per surface area

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.

Vector potential

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

There exists a vector field A such that
.

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.

Field lines

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.

Modification if magnetic monopoles exist

If magnetic monopoles were discovered, then Gauss's law for magnetism would state the divergence of B would be proportional to the magnetic charge densityρm, analogous to Gauss's law for electric field. For zero net magnetic charge density (ρm = 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.

UnitsEquation
cgs units [7]
SI units (weber convention) [8]
SI units (ampere-meter convention) [9]

where μ0 is the vacuum permeability.

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

History

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.

Numerical computation

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.

See also

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References

  1. 1 2 Chow, Tai L. (2006). Electromagnetic Theory: A modern perspective. Jones and Bartlett. p. 134. ISBN   0-7637-3827-1.
  2. 1 2 Jackson, John David (1999). Classical Electrodynamics (3rd ed.). Wiley. p. 237. ISBN   0-471-30932-X.
  3. Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. p. 321. ISBN   0-13-805326-X.
  4. 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.
  5. Schilders, W. H. A.; et al. (2005). Handbook of Numerical Analysis. p. 13. ISBN   978-0-444-51375-5.
  6. Jackson, John David (1999). Classical Electrodynamics (3rd ed.). Wiley. p. 180. ISBN   0-471-30932-X.
  7. Moulin, F. (2001). "Magnetic monopoles and Lorentz force". Il Nuovo Cimento B . 116 (8): 869–877. arXiv: math-ph/0203043 . Bibcode:2001NCimB.116..869M.
  8. Jackson, John David (1999). Classical Electrodynamics (3rd ed.). Wiley. p. 273, eq. 6.150.
  9. 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.
  10. 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."
  11. Brackbill, J.U; Barnes, D.C (May 1980). "The Effect of Nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations". Journal of Computational Physics. 35 (3): 426–430. doi:https://doi.org/10.1016/0021-9991(80)90079-0 Check |doi= value (help).
  12. 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.
  13. 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.
  14. Jardin, Stephen (2010). Computational Methods in Plasma Physics (1st ed.). Boca Raton: CRC Press. ISBN   9780429075537.
  15. 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.
  16. 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).