Magnetic diffusion refers to the motion of magnetic fields, typically in the presence of a conducting solid or fluid such as a plasma. The motion of magnetic fields is described by the magnetic diffusion equation and is due primarily to induction and diffusion of magnetic fields through the material. The magnetic diffusion equation is a partial differential equation commonly used in physics. Understanding the phenomenon is essential to magnetohydrodynamics and has important consequences in astrophysics, geophysics, and electrical engineering.
Equation The magnetic diffusion equation (also referred to as the induction equation) is
∂ B ∂ t = ∇ × ( v × B ) + 1 μ 0 σ ∇ 2 B {\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times \left(\mathbf {v} \times \mathbf {B} \right)+{\frac {1}{\mu _{0}\sigma }}\nabla ^{2}\mathbf {B} }
where μ 0 {\displaystyle \mu _{0}} is the permeability of free space and σ {\displaystyle \sigma } is the electrical conductivity of the material, which is assumed to be constant. v {\displaystyle \mathbf {v} } denotes the (non-relativistic) velocity of the plasma. The first term on the right hand side accounts for effects from induction of the plasma, while the second accounts for diffusion. The latter acts as a dissipation term, resulting in a loss of magnetic field energy to heat. The relative importance of the two terms is characterized by the magnetic Reynolds number, R m {\displaystyle R_{m}} . In the case of a non-uniform conductivity the magnetic diffusion equation is
∂ B ∂ t = ∇ × ( v × B ) − 1 μ 0 ∇ × ( 1 σ ∇ × B ) {\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times \left(\mathbf {v} \times \mathbf {B} \right)-{\frac {1}{\mu _{0}}}\nabla \times \left({\frac {1}{\sigma }}\nabla \times \mathbf {B} \right)}
however, it becomes significantly harder to solve.
Derivation Starting from the generalized Ohm's law:
J = σ ( E + v × B ) {\displaystyle \mathbf {J} =\sigma \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)}
and the curl equations for small displacement currents (i.e. low frequencies)
∇ × B = μ 0 J + 1 c 2 ∂ E ∂ t ≈ μ 0 J ∇ × E = − ∂ B ∂ t {\displaystyle {\begin{aligned}\nabla \times \mathbf {B} &=\mu _{0}\mathbf {J} +{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}\approx \mu _{0}\mathbf {J} \\\nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}}\end{aligned}}}
substitute J {\displaystyle \mathbf {J} } into the Ampere–Maxwell law to get
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