Magnetic scalar potential, ''ψ'', is a quantity in
classical electromagnetism
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fie ...
analogous to
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
. 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
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 ...
. One important use of ''ψ'' is to determine the magnetic field due to
permanent magnets
A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, steel, nickel, ...
when their
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 ...
is known. The potential is valid in any region with zero
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 ...
, 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
In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
is a useful quantity in describing the magnetic field, especially for
permanent magnet
A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, steel, nickel ...
s.
Where there is no free current,
:
so if this holds in
simply connected domain we can define a ''magnetic scalar potential'', ''ψ'', as
:
The dimensions of ''ψ'' in
SI base units are
.
Using the definition of H:
:
it follows that
:
Here, acts as the source for magnetic field, much like acts as the source for electric field. So analogously to
bound electric charge, the quantity
:
is called the ''bound magnetic charge'' density. Magnetic charges
never occur isolated as
magnetic monopole
In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magneti ...
s, but only within dipoles and in magnets with a total magnetic charge sum of zero. The energy of a localized magnetic charge ''q
m'' in a magnetic scalar potential is
:
,
and of a magnetic charge density distribution ''ρ
m'' in space
:
,
where ''µ
0'' is the
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, ...
. This is analog to the energy
of an electric charge ''q'' in an electric potential
.
If there is free current, one may subtract the contributions of free current per
Biot–Savart law from total magnetic field and solve the remainder with the scalar potential method.
See also
*
Magnetic vector potential
In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic ...
Notes
References
*
*
*{{Cite book
, isbn = 1-4020-2699-4
, last = Vanderlinde
, first = Jack
, title = Classical Electromagnetic Theory
, year = 2005
, doi = 10.1007/1-4020-2700-1
, bibcode = 2005cet..book.....V
, url = http://cds.cern.ch/record/1250088
Potentials
Magnetism