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Diffusion equation

The derivation of a useful equation for modelling the effect of eddy currents in a material starts with the differential, magnetostatic form of Ampère's Law, providing an expression for the magnetizing field H surrounding a current density J: ∇ × H = J. {\displaystyle \nabla \times \mathbf {H} =\mathbf {J}.}

Taking the curl on both sides of this equation and then using a common vector calculus identity for the curl of the curl results in ∇ (∇ ⋅ H) − ∇ 2 H = ∇ × J. {\displaystyle \nabla \left(\nabla \cdot \mathbf {H} \right)-\nabla ^{2}\mathbf {H} =\nabla \times \mathbf {J}.}

From Gauss's law for magnetism, ∇ ⋅ H = 0, so − ∇ 2 H = ∇ × J. {\displaystyle -\nabla ^{2}\mathbf {H} =\nabla \times \mathbf {J}.}

Using Ohm's law, J = σ E, which relates current density J to electric field E in terms of a material's conductivity σ, and assuming isotropic homogeneous conductivity, the equation can be written as − ∇ 2 H = σ ∇ × E. {\displaystyle -\nabla ^{2}\mathbf {H} =\sigma \nabla \times \mathbf {E}.}

Using the differential form of Faraday's law, ∇ × E = − ∂ B / ∂ t, this gives ∇ 2 H = σ ∂ B ∂ t. {\displaystyle \nabla ^{2}\mathbf {H} =\sigma {\frac {\partial \mathbf {B} }{\partial t}}.}

By definition, B = μ (H + M), where M is the magnetization of the material and μ is the vacuum permeability. The diffusion equation therefore is ∇ 2 H = μ 0 σ (∂ M ∂ t + ∂ H ∂ t). {\displaystyle \nabla ^{2}\mathbf {H} =\mu _{0}\sigma \left({\frac {\partial \mathbf {M} }{\partial t}}+{\frac {\partial \mathbf {H} }{\partial t}}\right).}