Magnetic field 2D

Consider magnetostatics — the stationary field of a rectangular permanent magnet, uniformly magnetized ($\mathbf{M}$​${} = \text{const}$). There are no free currents, so the field is potential: $\nabla \times \mathbf{H}$​${} = 0$, hence $\mathbf{H}$​${} = -\nabla \varphi$. Substituting this into $\nabla \cdot \mathbf{B}$​${} = 0$ with $\mathbf{B}$​${} = \mu_0 (\mathbf{H} + \mathbf{M})$, we obtain a single elliptic equation for the scalar potential (with the Laplace operator) and a Dirichlet condition on the outer boundary — the field decays far away

On the left is the Laplace operator ($\nabla^2$​${} = \Delta$). The right-hand side is the source $\sigma$​${} = \nabla \cdot \mathbf{M}$: inside a uniform magnet it vanishes, while on the pole faces it gives a bound surface charge (essentially the normal “flux” of the magnetization $\mathbf{M} \cdot \mathbf{n}$), which depends on which face we are on

Equation (7.3$\displaystyle \frac{\displaystyle \partial^2 \varphi}{\displaystyle \partial x^2}$​$\displaystyle {} + \frac{\displaystyle \partial^2 \varphi}{\displaystyle \partial y^2}$​$\displaystyle {} = \sigma(x, y),$$\displaystyle (x, y)$ $\displaystyle {} \in \Omega,$$\displaystyle \varphi\big|_{\partial \Omega}$ $\displaystyle {} = 0.$) is a Poisson equation, i.e. an elliptic boundary value problem; for elliptic equations and reducing transient problems to them see Appendix N. We solve it numerically by the finite element method on a triangular mesh: the weak form is $\int_\Omega \nabla \varphi \cdot \nabla v \, d\Omega$​${} = \int_{\text{magnet}} \mathbf{M} \cdot \nabla v \, d\Omega$.