Magnetostatics

Magnetostatics as a special case of Maxwell's equations Starting from Maxwell's equations and assuming that charges are either fixed or move as a steady current \mathbf{J}, the equations separate into two equations for the electric field (see electrostatics) and two for the magnetic field. The fields are independent of time and each other. The magnetostatic equations, in both differential and integral forms, are shown in the table below. Where ∇ with the dot denotes divergence, and B is the magnetic flux density, the first integral is over a surface S with oriented surface element d\mathbf{S}. Where ∇ with the cross denotes curl, J is the current density and is the magnetic field intensity, the second integral is a line integral around a closed loop C with line element \mathbf{l}. The current going through the loop is I_\text{enc}. The quality of this approximation may be guessed by comparing the above equations with the full version of Maxwell's equations and considering the importance of the terms that have been removed. Of particular significance is the comparison of the \mathbf{J} term against the \partial \mathbf{D} / \partial t term. If the \mathbf{J} term is substantially larger, then the smaller term may be ignored without significant loss of accuracy. Re-introducing Faraday's law A common technique is to solve a series of magnetostatic problems at incremental time steps and then use these solutions to approximate the term \partial \mathbf{B} / \partial t. Plugging this result into Faraday's Law finds a value for \mathbf{E} (which had previously been ignored). This method is not a true solution of Maxwell's equations but can provide a good approximation for slowly changing fields. ==Solving for the magnetic field==

Current sources If all currents in a system are known (i.e., if a complete description of the current density \mathbf{J}(\mathbf{r}) is available) then the magnetic field can be determined, at a position r, from the currents by the Biot–Savart equation: \mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int{\frac{\mathbf{J}(\mathbf{r}') \times \left(\mathbf{r} - \mathbf{r}'\right)} \mathrm{d}^3\mathbf{r}'}. Magnetization Strongly magnetic materials (i.e., ferromagnetic, ferrimagnetic or paramagnetic) have a magnetization that is primarily due to electron spin. In such materials the magnetization must be explicitly included using the relation \mathbf{B} = \mu_0(\mathbf{M}+\mathbf{H}). Except in the case of conductors, electric currents can be ignored. Then Ampère's law is simply \nabla\times\mathbf{H} = 0. This has the general solution \mathbf{H} = -\nabla \Phi_M, where \Phi_M is a scalar potential. and is often referred to as an effective charge density \rho_M. The vector potential method can also be employed with an effective current density \mathbf{J_M} = \nabla \times \mathbf{M}. ==See also==