Magnetic scalar potential

Magnetic scalar potential, ψ, is a quantity in

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

permanent magnets

.

Where there is no free current,

\nabla \times \mathbf {H} =\mathbf {0} ,

so if this holds in

simply connected domain we can define a magnetic scalar potential,

ψ

, as^[1]

\mathbf {H} =-\nabla \psi .

The dimension of

ψ

in

SI base units

is

{\mathsf {A}}

, which can be expressed in SI units as amperes.

Using the definition of $H$ :

\nabla \cdot \mathbf {B} =\mu _{0}\nabla \cdot \left(\mathbf {H} +\mathbf {M} \right)=0,

it follows that

\nabla ^{2}\psi =-\nabla \cdot \mathbf {H} =\nabla \cdot \mathbf {M} .

Here, $\nabla \cdot M$ acts as the source for magnetic field, much like $\nabla \cdot P$ acts as the source for electric field. So analogously to

bound electric charge

, the quantity

\rho _{m}=-\nabla \cdot \mathbf {M}

is called the bound magnetic charge density. Magnetic charges

{\textstyle q_{m}=\int \rho _{m}\,dV}

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 m

in a magnetic scalar potential is

Q=\mu _{0}\,q_{m}\psi ,

and of a magnetic charge density distribution

ρ m

in space

Q=\mu _{0}\int \rho _{m}\psi \,dV,

where

µ 0

is the vacuum permeability. This is analog to the energy

Q=qV_{E}

of an electric charge

q

in an electric potential

V_{E}

.

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.

Notes

^ Vanderlinde 2005, pp. 194–199

References

Duffin, W.J. (1980). Electricity and Magnetism, Fourth Edition. McGraw-Hill.
ISBN 007084111X
.

Jackson, John David (1999), Classical Electrodynamics (3rd ed.),
ISBN 0-471-30932-X

Vanderlinde, Jack (2005). Classical Electromagnetic Theory.
ISBN 1-4020-2699-4
.