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[edit]
Magnetic scalar potential of flat cylinder magnets encoded as color from positive (fuchsia) through zero (yellow) to negative (aqua).
The scalar potential is a useful quantity in describing the magnetic field,
especially for permanent magnets.
Where there is no free current,
so if this holds in simply connected domain we can define a magnetic scalar
potential, ψ, as [1]
The dimensions of ψ in SI base units are .
Using the definition of H:
, it follows that
Here, ∇ ⋅ M acts as the source for magnetic field, much like ∇ ⋅ P 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 monopoles, but only within dipoles and in magnets
with a total magnetic charge sum of zero. The energy of a localized magnetic
charge q in a magnetic scalar potential is
m
,
and of a magnetic charge density distribution ρ in space
m
,
where µ is the vacuum permeability. This is analog to the energy
0 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.
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: .
Together with the electric potential φ, the magnetic vector potential can be used to
specify the electric field E as well. Therefore, many equations of electromagnetism
can be written either in terms of the fields E and B, or equivalently in terms of the
potentials φ and A. In more advanced theories such as quantum mechanics, most
equations use potentials rather than fields.
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[edit]
Magnetic scalar potential of flat cylinder magnets encoded as color from positive (fuchsia) through zero (yellow) to negative (aqua).
The scalar potential is a useful quantity in describing the magnetic field,
especially for permanent magnets.
Where there is no free current,
so if this holds in simply connected domain we can define a magnetic scalar
potential, ψ, as [1]
The dimensions of ψ in SI base units are .
Using the definition of H:
, it follows that
Here, ∇ ⋅ M acts as the source for magnetic field, much like ∇ ⋅ P 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 monopoles, but only within dipoles and in magnets
with a total magnetic charge sum of zero. The energy of a localized magnetic
charge q in a magnetic scalar potential is
m
,
and of a magnetic charge density distribution ρ in space
m
,
where µ is the vacuum permeability. This is analog to the energy
0 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.
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: .
Together with the electric potential φ, the magnetic vector potential can be used to
specify the electric field E as well. Therefore, many equations of electromagnetism
can be written either in terms of the fields E and B, or equivalently in terms of the
potentials φ and A. In more advanced theories such as quantum mechanics, most
equations use potentials rather than fields.
