Magnetic scalar potential

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.

In a simply connected domain where there is no free current,

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

hence we can define a magnetic scalar potential, ψ, as[1]

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

Using the definition of H:

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

it follows that

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

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

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

is called the bound magnetic charge.

If there is free current, one may subtract the contribution of free current per Biot–Savart law from total magnetic field and solve the remainder with the scalar potential method. To date there has not been any reproducible evidence for the existence of magnetic monopoles.[2]

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

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

• Vanderlinde, Jack (2005). Classical Electromagnetic Theory. Bibcode:2005cet..book.....V. doi:10.1007/1-4020-2700-1. ISBN 1-4020-2699-4.CS1 maint: ref=harv (link)