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 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.

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.

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

Differential form

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

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

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

Integral form

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

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

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).

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.

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
${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$.

The vector field A is called the magnetic vector potential.

Note that there is more than one possible A 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):

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

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

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

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] ${\displaystyle \nabla \cdot \mathbf {B} =4\pi \rho _{\mathrm {m} }}$
SI units (weber convention) [8] ${\displaystyle \nabla \cdot \mathbf {B} =\rho _{\mathrm {m} }}$
SI units (ampere-meter convention) [9] ${\displaystyle \nabla \cdot \mathbf {B} =\mu _{0}\rho _{\mathrm {m} }}$

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.

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References

1. Chow, Tai L. (2006). Electromagnetic Theory: A modern perspective. Jones and Bartlett. p. 134. ISBN   0-7637-3827-1.
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:. 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:. 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."