|
Force-free Magnetic Field
A force-free magnetic field is a magnetic field in which the Lorentz force is equal to zero and the magnetic pressure greatly exceeds the plasma pressure such that non-magnetic forces can be neglected. For a force-free field, the electric current density is either zero or parallel to the magnetic field. Definition When a magnetic field is approximated as force-free, all non-magnetic forces are neglected and the Lorentz force vanishes. For non-magnetic forces to be neglected, it is assumed that the ratio of the plasma pressure to the magnetic pressure—the plasma ''β''—is much less than one, i.e., \beta \ll 1. With this assumption, magnetic pressure dominates over plasma pressure such that the latter can be ignored. It is also assumed that the magnetic pressure dominates over other non-magnetic forces, such as gravity, so that these forces can similarly be ignored. In SI units, the Lorentz force condition for a static magnetic field \mathbf can be expressed as :\mathbf \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetic Scalar Potential
Magnetic scalar potential, ''ψ'', is a quantity in classical electromagnetism analogous to electric potential. It is used to specify the magnetic H-field in cases when there are no free currents, in a manner analogous to using the electric potential to determine the electric field in electrostatics. One important use of ''ψ'' is to determine the magnetic field due to permanent magnets when their magnetization is known. The potential is valid in any region with zero current density, thus if currents are confined to wires or surfaces, piecemeal solutions can be stitched together to provide a description of the magnetic field at all points in space. Magnetic scalar potential The scalar potential is a useful quantity in describing the magnetic field, especially for permanent magnets. Where there is no free current, :\nabla\times\mathbf = 0, so if this holds in simply connected domain we can define a ''magnetic scalar potential'', ''ψ'', as :\mathbf = -\nabla\psi. The di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetic Helicity
In plasma physics, magnetic helicity is a measure of the linkage, twist, and writhe of a magnetic field. In ideal magnetohydrodynamics, magnetic helicity is conserved. When a magnetic field contains magnetic helicity, it tends to form large-scale structures from small-scale ones. This process can be referred as an inverse transfer in Fourier space. This second property makes magnetic helicity special: three-dimensional turbulent flows tend to "destroy" structure, in the sense that large-scale vortices break-up in smaller and smaller ones (a process called "direct energy cascade", described by Lewis Fry Richardson and Andrey Nikolaevich Kolmogorov). At the smallest scales, the vortices are dissipated in heat through viscous effects. Through a sort of "inverse cascade of magnetic helicity", the opposite happens: small helical structures (with a non-zero magnetic helicity) lead to the formation of large-scale magnetic fields. This is for example visible in the heliospheric curren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chandrasekhar–Kendall Function
Chandrasekhar–Kendall functions are the axisymmetric eigenfunctions of the curl operator derived by Subrahmanyan Chandrasekhar and P. C. Kendall in 1957 while attempting to solve the force-free magnetic fields. The functions were independently derived by both, and the two decided to publish their findings in the same paper. If the force-free magnetic field equation is written as \nabla\times\mathbf=\lambda\mathbf, where \mathbf is the magnetic field and \lambda is the force-free parameter, with the assumption of divergence free field, \nabla\cdot\mathbf=0, then the most general solution for the axisymmetric case is :\mathbf = \frac\nabla\times(\nabla\times\psi\mathbf) + \nabla \times \psi \mathbf where \mathbf is a unit vector and the scalar function \psi satisfies the Helmholtz equation, i.e., :\nabla^2\psi + \lambda^2\psi=0. The same equation also appears in Beltrami flows from fluid dynamics where, the vorticity vector is parallel to the velocity vector, i.e., \nabla\times\math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Woltjer's Theorem
In plasma physics, Woltjer's theorem states that force-free magnetic fields in a closed system with constant force-free parameter \alpha represent the state with lowest magnetic energy in the system and that the magnetic helicity is invariant under this condition. It is named after Lodewijk Woltjer who derived it in 1958.Kholodenko, A. L. (2013). Applications of contact geometry and topology in physics. World Scientific. The force-free field strength \mathbf equation is :\nabla \times \mathbf = \alpha \mathbf. The helicity \mathcal invariant is given by :\frac =0. where \mathcal is related to \mathbf=\nabla\times \mathbf through the vector potential \mathbf as below :\mathcal = \int_V \mathbf\cdot\mathbf\ dV = \int_V \mathbf \cdot (\nabla \times \mathbf) \ dV. See also *Chandrasekhar–Kendall function *Hydrodynamical helicity :''This page is about helicity in fluid dynamics. For helicity of magnetic fields, see magnetic helicity. For helicity in particle physics, see helicit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solar Corona
A corona ( coronas or coronae) is the outermost layer of a star's atmosphere. It consists of plasma. The Sun's corona lies above the chromosphere and extends millions of kilometres into outer space. It is most easily seen during a total solar eclipse, but it is also observable with a coronagraph. Spectroscopic measurements indicate strong ionization in the corona and a plasma temperature in excess of , much hotter than the surface of the Sun, known as the photosphere. The word ''corona'' is , in turn derived . History In 1724, French-Italian astronomer Giacomo F. Maraldi recognized that the aura visible during a solar eclipse belongs to the Sun, not to the Moon. In 1809, Spanish astronomer José Joaquín de Ferrer coined the term 'corona'. Based in his own observations of the 1806 solar eclipse at Kinderhook (New York), de Ferrer also proposed that the corona was part of the Sun and not of the Moon. English astronomer Norman Lockyer identified the first element unknown on E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromosphere
A chromosphere ("sphere of color") is the second layer of a star's atmosphere, located above the photosphere and below the solar transition region and corona. The term usually refers to the Sun's chromosphere, but not exclusively. In the Sun's atmosphere, the chromosphere is roughly in height, or slightly more than 1% of the Sun's radius at maximum thickness. It possesses a homogeneous layer at the boundary with the photosphere. Hair-like jets of plasma, called spicules, rise from this homogeneous region and through the chromosphere, extending up to into the corona above. The chromosphere has a characteristic red color due to electromagnetic emissions in the ''H''α spectral line. Information about the chromosphere is primarily obtained by analysis of its emitted electromagnetic radiation. Chromospheres have also been observed on stars other than the Sun. On large stars, chromospheres sometimes make up a significant proportion of the entire star. For example, the chro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmholtz Equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplacian"), is the eigenvalue, and is the (eigen)function. When the equation is applied to waves, is known as the wave number. The Helmholtz equation has a variety of applications in physics, including the wave equation and the diffusion equation, and it has uses in other sciences. Motivation and uses The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis. For example, consider the wave equation \left(\nabla^2-\frac\frac\right) u(\mathbf,t)=0. Separation of variables begins by assumi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace's Equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nabla \cdot \nabla = \nabla^2 is the Laplace operator,The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, \Delta x = x_1 - x_2. Its use to represent the Laplacian should not be confused with this use. \nabla \cdot is the divergence operator (also symbolized "div"), \nabla is the gradient operator (also symbolized "grad"), and f (x, y, z) is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function, h(x, y, z), we have \Delta f = h. This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuum Permeability
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, conventionally written as ''μ''0 (pronounced "mu nought" or "mu zero"). Its purpose is to quantify the strength of the magnetic field emitted by an electric current. Expressed in terms of SI base units, it has the unit kg⋅m⋅s−2·A−2. Since the redefinition of SI units in 2019 (when the values of ''e'' and ''h'' were fixed as defined quantities), ''μ''0 is an experimentally determined constant, its value being proportional to the dimensionless fine-structure constant, which is known to a relative uncertainty of about , with no other dependencies with experimental uncertainty. Its value in SI units as recommended by CODATA 2018 (published in May 2019) is: From 1948 to 2019, ''μ''0 had a defined value (per the former defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetic Field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field. A permanent magnet's magnetic field pulls on ferromagnetic materials such as iron, and attracts or repels other magnets. In addition, a nonuniform magnetic field exerts minuscule forces on "nonmagnetic" materials by three other magnetic effects: paramagnetism, diamagnetism, and antiferromagnetism, although these forces are usually so small they can only be detected by laboratory equipment. Magnetic fields surround magnetized materials, and are created by electric currents such as those used in electromagnets, and by electric fields varying in time. Since both strength and direction of a magnetic field may vary with location, it is described mathematically by a function assigning a vector to each point of space, cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Current Density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional area at a given point in space, its direction being that of the motion of the positive charges at this point. In SI base units, the electric current density is measured in amperes per square metre. Definition Assume that ''A'' (SI unit: m2) is a small surface centred at a given point ''M'' and orthogonal to the motion of the charges at ''M''. If ''I'' (SI unit: A) is the electric current flowing through ''A'', then electric current density ''j'' at ''M'' is given by the limit: :j = \lim_ \frac = \left.\frac \_, with surface ''A'' remaining centered at ''M'' and orthogonal to the motion of the charges during the limit process. The current density vector j is the vector whose magnitude is the electric current density, and whose dire ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.jpg "Click for more on -> Force-free Magnetic Field")
