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, \nabla\times\mathbf{H} = \mathbf{0}, so if this holds in
simply connected domain we can define a , , as \mathbf{H} = -\nabla\psi. The dimension of in
SI base units is {{nowrap|\mathsf{A},}} which can be expressed in SI units as
amperes. Using the definition of : \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, acts as the source for magnetic field, much like 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 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 in a magnetic scalar potential is Q = \mu_0\,q_m\psi, and of a magnetic charge density distribution in space Q = \mu_0 \int \rho_m \psi \, dV , where is the
vacuum permeability. This is analog to the energy Q = q V_E of an electric charge 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. == See also ==