Magnetostatics is the study of
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 ...
s in systems where the
currents are
steady (not changing with time). It is the magnetic analogue of
electrostatics
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...
, where the
charges
Charge or charged may refer to:
Arts, entertainment, and media Films
* ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* '' Charge!!'', an album by The Aqu ...
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
Alternative or alternate may refer to:
Arts, entertainment and media
* Alternative (''Kamen Rider''), a character in the Japanese TV series ''Kamen Rider Ryuki''
* ''The Alternative'' (film), a 1978 Australian television film
* ''The Alternative ...
rapidly. Magnetostatics is widely used in applications of
micromagnetics
Micromagnetics is a field of physics dealing with the prediction of magnetic behaviors at sub-micrometer length scales. The length scales considered are large enough for the atomic structure of the material to be ignored (the continuum approxima ...
such as models of
magnetic storage devices as in
computer memory
In computing, memory is a device or system that is used to store information for immediate use in a computer or related computer hardware and digital electronic devices. The term ''memory'' is often synonymous with the term '' primary storag ...
.
Applications
Magnetostatics as a special case of Maxwell's equations
Starting from
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside 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 ...
and assuming that charges are either fixed or move as a steady current
, the equations separate into two equations for the
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
(see
electrostatics
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...
) and two for the
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 fields are independent of time and each other. The magnetostatic equations, in both differential and integral forms, are shown in the table below.
Where ∇ with the dot denotes
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
, and B is the
magnetic flux density, the first integral is over a surface
with oriented surface element
. Where ∇ with the cross denotes
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was ...
, J is the
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 a ...
and is the
magnetic field intensity
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 second integral is a line integral around a closed loop
with line element
. The current going through the loop is
.
The quality of this approximation may be guessed by comparing the above equations with the full version of
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside 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 ...
and considering the importance of the terms that have been removed. Of particular significance is the comparison of the
term against the
term. If the
term is substantially larger, then the smaller term may be ignored without significant loss of accuracy.
Re-introducing Faraday's law
A common technique is to solve a series of magnetostatic problems at incremental time steps and then use these solutions to approximate the term
. Plugging this result into
Faraday's Law finds a value for
(which had previously been ignored). This method is not a true solution of
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside 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 ...
but can provide a good approximation for slowly changing fields.
Solving for the magnetic field
Current sources
If all currents in a system are known (i.e., if a complete description of the current density
is available) then the magnetic field can be determined, at a position r, from the currents by the
Biot–Savart equation:
This technique works well for problems where the medium is a
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
or air or some similar material with a
relative permeability
In multiphase flow in porous media, the relative permeability of a phase is a dimensionless measure of the effective permeability of that phase. It is the ratio of the effective permeability of that phase to the absolute permeability. It can be ...
of 1. This includes
air-core inductors and
air-core transformer
A transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple circuits. A varying current in any coil of the transformer produces a varying magnetic flux in the transformer's ...
s. One advantage of this technique is that, if a coil has a complex geometry, it can be divided into sections and the integral evaluated for each section. Since this equation is primarily used to solve
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
problems, the contributions can be added. For a very difficult geometry,
numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equatio ...
may be used.
For problems where the dominant magnetic material is a highly permeable
magnetic core
A magnetic core is a piece of magnetic material with a high magnetic permeability used to confine and guide magnetic fields in electrical, electromechanical and magnetic devices such as electromagnets, transformers, electric motors, generators, ...
with relatively small air gaps, a
magnetic circuit
A magnetic circuit is made up of one or more closed loop paths containing a magnetic flux. The flux is usually generated by permanent magnets or electromagnets and confined to the path by magnetic cores consisting of ferromagnetic materials lik ...
approach is useful. When the air gaps are large in comparison to the
magnetic circuit
A magnetic circuit is made up of one or more closed loop paths containing a magnetic flux. The flux is usually generated by permanent magnets or electromagnets and confined to the path by magnetic cores consisting of ferromagnetic materials lik ...
length,
fringing becomes significant and usually requires a
finite element calculation. The
finite element calculation uses a modified form of the magnetostatic equations above in order to calculate
magnetic potential. The value of
can be found from the magnetic potential.
The magnetic field can be derived from the
vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field.
Formally, given a vector field v, a ''vecto ...
. Since the divergence of the magnetic flux density is always zero,
and the relation of the vector potential to current is:
Magnetization
Strongly magnetic materials (i.e.,
ferromagnetic
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
,
ferrimagnetic
A ferrimagnetic material is a material that has populations of atoms with opposing magnetic moments, as in antiferromagnetism, but these moments are unequal in magnitude so a spontaneous magnetization remains. This can for example occur when ...
or
paramagnetic
Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, ...
) have a
magnetization
In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or D ...
that is primarily due to
electron spin. In such materials the magnetization must be explicitly included using the relation
Except in the case of conductors, electric currents can be ignored. Then Ampère's law is simply
This has the general solution
where
is a scalar
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
.
Substituting this in Gauss's law gives
Thus, the divergence of the magnetization,
has a role analogous to the electric charge in electrostatics
and is often referred to as an effective charge density
.
The vector potential method can also be employed with an effective current density
See also
*
Darwin Lagrangian
The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order / between two charged particles in a vacuum and is given by
L = L_\text + L_\text,
where the free particle Lagrangian is
...
References
External links
*
{{Branches of physics
Electric and magnetic fields in matter
Potentials