Magnetic scalar potential

Magnetic scalar potential

The scalar potential is a useful quantity in describing the magnetic field, especially for permanent magnets.

Where there is no free current and no displacement current, $${\displaystyle \mathrm{\nabla }\times \mathbf{H}=\mathbf{0},}$$ so if this holds in simply connected domain we can define a magnetic scalar potential, ψ, as $${\displaystyle \mathbf{H}=-\mathrm{\nabla }\psi .}$$ The dimension of ψ in SI base units is ${\displaystyle \mathsf{A}}$, which can be expressed in SI units as amperes.

Using the definition of H: $${\displaystyle \mathrm{\nabla }\cdot \mathbf{B}={\mu }_0\mathrm{\nabla }\cdot \left(\mathbf{H}+\mathbf{M}\right)=0,}$$ it follows that $${\displaystyle {\mathrm{\nabla }}^2\psi =-\mathrm{\nabla }\cdot \mathbf{H}=\mathrm{\nabla }\cdot \mathbf{M}.}$$

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 $${\displaystyle {\rho }_m=-\mathrm{\nabla }\cdot \mathbf{M}}$$ is called the bound magnetic charge density. Magnetic charges $q_m=\int {\rho }_{m}dV$ 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 $${\displaystyle Q={\mu }_0{q}_m\psi ,}$$ and of a magnetic charge density distribution ρ_m in space $${\displaystyle Q={\mu }_0\int {\rho }_m\psi dV,}$$ where µ₀ is the vacuum permeability. This is analog to the energy ${\displaystyle Q=q{V}_E}$ of an electric charge q in an electric potential ${\displaystyle V_E}$.

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.