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.
The dimension of ψ in
SI base units is , which can be expressed in SI units as
amperes.
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 qm in a magnetic scalar potential is
and of a magnetic charge density distribution ρm in space
where µ0 is the
vacuum permeability. 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.