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<title>Magnetic diffusion</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Magnetic diffusion</span></span>

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<p><b>Magnetic diffusion</b> refers to the motion of <a href="Magnetic_field#Magnetic_field" title="Magnetic field">magnetic fields</a>, typically in the presence of a conducting solid or fluid such as a <a href="Plasma_(physics)" title="Plasma (physics)">plasma</a>. 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 <a href="Partial_differential_equation" title="Partial differential equation">partial differential equation</a> commonly used in physics. Understanding the phenomenon is essential to <a href="Magnetohydrodynamics" title="Magnetohydrodynamics">magnetohydrodynamics</a> and has important consequences in astrophysics, geophysics, and electrical engineering.

</p>

<div class="mw-heading mw-heading2"><h2 id="Equation">Equation</h2></div>

<p>The magnetic diffusion equation (also referred to as the <i><a href="Induction_equation" title="Induction equation">induction equation</a></i>) is

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}=\nabla \times \left[{\vec {v}}\times {\vec {B}}\right]+{\frac {1}{\mu _{0}\sigma }}\nabla ^{2}{\vec {B}}}</annotation>

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where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \mu _{0}}</annotation>

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</math></span><img src="./fe2fd9b8decb38a3cd158e7b6c0c6e2d987fefcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{0}}" loading="lazy"></span> is the <a href="Permeability_(electromagnetism)" title="Permeability (electromagnetism)">permeability</a> of free space and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math></span><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"></span> is the <a href="Electrical_resistivity_and_conductivity" title="Electrical resistivity and conductivity">electrical conductivity</a> of the material, which is assumed to be constant. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation>

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</math></span><img src="./85820588abd7333ef4d0c56539cb31c20e730753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}" loading="lazy"></span> denotes the (non-relativistic) velocity of the plasma. The first term on the right hand side accounts for effects from <a href="Electromagnetic_induction" title="Electromagnetic induction">induction</a> of the plasma, while the second accounts for <a href="Diffusion" title="Diffusion">diffusion</a>. 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 <a href="Magnetic_Reynolds_number" title="Magnetic Reynolds number">magnetic Reynolds number</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{m}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{m}}</annotation>

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</p><p>In the case of a non-uniform conductivity the magnetic diffusion equation is

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}=\nabla \times \left[{\vec {v}}\times {\vec {B}}\right]-{\frac {1}{\mu _{0}}}\nabla \times \left[{\frac {1}{\sigma }}\nabla \times {\vec {B}}\right]}</annotation>

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however, it becomes significantly harder to solve.

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<div class="mw-heading mw-heading2"><h2 id="Derivation">Derivation</h2></div>

<p>Starting from the generalized <a href="Ohm's_law#Magnetic_effects" title="Ohm's law">Ohm's law</a>:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {J}}=\sigma \left({\vec {E}}+{\vec {v}}\times {\vec {B}}\right)}</annotation>

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and the <a href="Maxwell's_equations#Formulation_in_SI_units_convention" title="Maxwell's equations">curl equations</a> for small <a href="Displacement_current" title="Displacement current">displacement currents</a> (i.e. low frequencies)

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<annotation encoding="application/x-tex">{\displaystyle \nabla \times {\vec {B}}=\mu _{0}{\vec {J}}+\epsilon _{0}\mu _{0}{\frac {\partial {\vec {E}}}{\partial t}}\approx \mu _{0}{\vec {J}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \nabla \times {\vec {E}}=-{\frac {\partial {\vec {B}}}{\partial t}}}</annotation>

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substitute <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {J}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\mu _{0}\sigma }}\nabla \times {\vec {B}}={\vec {E}}+{\vec {v}}\times {\vec {B}}\quad \Rightarrow \quad {\vec {E}}={\frac {1}{\mu _{0}\sigma }}\nabla \times {\vec {B}}-{\vec {v}}\times {\vec {B}}.}</annotation>

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Taking the curl of the above equation and substituting into Faraday's law,

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<annotation encoding="application/x-tex">{\displaystyle \nabla \times {\vec {E}}=\nabla \times \left({\frac {1}{\mu _{0}\sigma }}\nabla \times {\vec {B}}-{\vec {v}}\times {\vec {B}}\right)=-{\frac {\partial {\vec {B}}}{\partial t}}.}</annotation>

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This expression can be simplified further by writing it in terms of the <i>i</i>-th component of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {B}}}">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ijk}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}-{\frac {\partial B_{i}}{\partial t}}&=\varepsilon _{ijk}\partial _{j}\left({\frac {1}{\mu _{0}\sigma }}\varepsilon _{klm}\partial _{l}B_{m}-\varepsilon _{klm}v_{l}B_{m}\right)\\&=\varepsilon _{kij}\varepsilon _{klm}\left({\frac {1}{\mu _{0}\sigma }}\partial _{j}\partial _{l}B_{m}-\left(v_{l}\partial _{j}B_{m}+B_{m}\partial _{j}v_{l}\right)\right)\end{aligned}}}</annotation>

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Using the identity<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{kij}\varepsilon _{klm}=\delta _{il}\delta _{jm}-\delta _{im}\delta _{jl}}">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{kij}\varepsilon _{klm}=\delta _{il}\delta _{jm}-\delta _{im}\delta _{jl}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \partial _{j}B_{j}=0}</annotation>

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</math></span><img src="./9565cbc0061f3c8a1ebd175478362cacd10871e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.079ex; height:2.843ex;" alt="{\displaystyle \partial _{j}B_{j}=0}" loading="lazy"></span>, the cross products can be eliminated:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}-{\frac {\partial B_{i}}{\partial t}}&={\frac {1}{\mu _{0}\sigma }}\left(\partial _{i}\partial _{j}B_{j}-\partial _{j}\partial _{j}B_{i}\right)-\left(v_{i}\partial _{j}B_{j}-v_{j}\partial _{j}B_{i}\right)-\left(B_{j}\partial _{j}v_{i}-B_{i}\partial _{j}v_{j}\right)\\&=-{\frac {1}{\mu _{0}\sigma }}\partial _{j}\partial _{j}B_{i}+v_{j}\partial _{j}B_{i}-\left(B_{j}\partial _{j}v_{i}-B_{i}\partial _{j}v_{j}\right)\end{aligned}}}</annotation>

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Written in vector form, the final expression is

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}+\left({\vec {v}}\cdot \nabla \right){\vec {B}}={\frac {D{\vec {B}}}{Dt}}=\left({\vec {B}}\cdot \nabla \right){\vec {v}}-{\vec {B}}\left(\nabla \cdot {\vec {v}}\right)+{\frac {1}{\mu _{0}\sigma }}\nabla ^{2}{\vec {B}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {D}{Dt}}={\frac {\partial }{\partial t}}+{\vec {v}}\cdot \nabla }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot {\vec {B}}=0}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}=\nabla \times [{\vec {v}}\times {\vec {B}}]+{\frac {1}{\mu _{0}\sigma }}\nabla ^{2}{\vec {B}}}</annotation>

</semantics>

</math></span></span>

In the case <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}=0}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=0}</annotation>

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</math></span><img src="./cb99f512669b606ab618c557ad45c433d42a7209.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.436ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}=0}" loading="lazy"></span>, this becomes a diffusion equation for the magnetic field,

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}={\frac {1}{\mu _{0}\sigma }}\nabla ^{2}{\vec {B}}=\eta \nabla ^{2}{\vec {B}}}</annotation>

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</math></span></span>

<mfrac>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \eta ={\frac {1}{\mu _{0}\sigma }}}</annotation>

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</math></span><img src="./11f56094897d51e2ad315196a1c1343fe9cb943c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.89ex; height:5.676ex;" alt="{\displaystyle \eta ={\frac {1}{\mu _{0}\sigma }}}" loading="lazy"></span> is the <a href="Magnetic_diffusivity" title="Magnetic diffusivity">magnetic diffusivity</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Limiting_Cases">Limiting Cases</h2></div>

<p>In some cases it is possible to neglect one of the terms in the magnetic diffusion equation. This is done by estimating the magnetic Reynolds number

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<annotation encoding="application/x-tex">{\displaystyle R_{m}={\frac {vL}{\eta }}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

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</math></span><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"></span> is the diffusivity, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math></span><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span> is the magnitude of the plasma's velocity and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> is a characteristic length of the plasma.

</p>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle (R_{m})}</annotation>

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</math></span><img src="./4e2d800c6ed413b3067cd21644ec0d537c311e9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.248ex; height:2.843ex;" alt="{\displaystyle (R_{m})}" loading="lazy"></span></th>

<th>Physical Condition</th>

<th>Dominating Term</th>

<th>Magnetic Diffusion Equation</th>

<th>Examples

</th></tr>

<tr>

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<annotation encoding="application/x-tex">{\displaystyle \gg 1}</annotation>

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</math></span><img src="./518b42113f45ccf4550daf3ffa4e403ffbe6ebd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.131ex; height:2.176ex;" alt="{\displaystyle \gg 1}" loading="lazy"></span>  </td>

<td>Large electrical conductivity, large length scales or high plasma velocity.</td>

<td>The inductive term dominates in this case. The motion of magnetic fields is determined by the flow of the plasma. This is the case for most naturally occurring plasmas in the universe.  </td>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}\approx \nabla \times [{\vec {v}}\times {\vec {B}}]}</annotation>

</semantics>

</math></span><img src="./44643f3303f0137feaabdae4cec7a960f4d69bd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.866ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}\approx \nabla \times [{\vec {v}}\times {\vec {B}}]}" loading="lazy"></span></td>

<msub>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle (R_{m}\approx 10^{6})}</annotation>

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</math></span><img src="./c1b83b7c14d86b38f952284fb9f048773953caa0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.726ex; height:3.176ex;" alt="{\displaystyle (R_{m}\approx 10^{6})}" loading="lazy"></span> or the <a href="Dynamo_theory" title="Dynamo theory">core of the earth</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (R_{m}\approx 10^{3})}">

<msub>

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</msub>

<msup>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle (R_{m}\approx 10^{3})}</annotation>

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</math></span><img src="./37a63704abcbcefd37ec27fe7ee6ef50ec9b8e9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.726ex; height:3.176ex;" alt="{\displaystyle (R_{m}\approx 10^{3})}" loading="lazy"></span>

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ll 1}</annotation>

</semantics>

</math></span><img src="./08b246b342d27748c9d86028d8a5a37e127ddffb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.131ex; height:2.176ex;" alt="{\displaystyle \ll 1}" loading="lazy"></span></td>

<td>Small electrical conductivity, small length scales or low plasma velocity.</td>

<td>The diffusive term dominates in this case. The motion of the magnetic field obeys the typical (nonconducting) fluid diffusion equation.</td>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}\approx {\frac {1}{\mu _{0}\sigma }}\nabla ^{2}{\vec {B}}}</annotation>

</semantics>

</math></span><img src="./c7ed524efee61c358b8fcc864a6cf86464c17fbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.393ex; height:6.509ex;" alt="{\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}\approx {\frac {1}{\mu _{0}\sigma }}\nabla ^{2}{\vec {B}}}" loading="lazy"></span></td>

<td>Solar flares or created in laboratories using mercury or other <a href="Liquid_metal" title="Liquid metal">liquid metals</a>.

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Relation_to_Skin_Effect">Relation to Skin Effect</h3></div>

<p>At low frequencies, the <a href="Skin_effect#Formulat" title="Skin effect">skin depth</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>

</semantics>

</math></span><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"></span> for the penetration of an AC electromagnetic field into a conductor is:

<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle \delta ={\sqrt {\frac {2}{\mu \sigma \omega }}}}</annotation>

</semantics>

</math></span></span>

Comparing with the formula for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

</semantics>

</math></span><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"></span>, the skin depth is the diffusion length of the field over one period of oscillation:

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<annotation encoding="application/x-tex">{\displaystyle \delta ={\sqrt {\frac {2\eta }{\omega }}}={\sqrt {\frac {\eta T}{\pi }}}}</annotation>

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</math></span></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="Examples_and_Visualization">Examples and Visualization</h2></div>

<p>For the limit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{m}\gg 1}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle R_{m}\gg 1}</annotation>

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</math></span><img src="./db875b833d9bcde367276c6f806841891c99bf68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.216ex; height:2.509ex;" alt="{\displaystyle R_{m}\gg 1}" loading="lazy"></span>, the magnetic field lines become "<a href="Alfv%C3%A9n's_theorem" title="Alfvén's theorem">frozen in</a>" to the motion of the conducting fluid. A simple example illustrating this behavior has a sinusoidally-varying shear flow

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<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=v_{0}\sin(ky){\hat {x}}}</annotation>

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</math></span></span>

with a uniform initial magnetic field <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {B}}\left({\vec {r}},0\right)=B_{0}{\hat {y}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {B}}\left({\vec {r}},0\right)=B_{0}{\hat {y}}}</annotation>

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</math></span><img src="./bc5578dc8641311269db02b5c82c2dbc4e170cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.599ex; height:3.343ex;" alt="{\displaystyle {\vec {B}}\left({\vec {r}},0\right)=B_{0}{\hat {y}}}" loading="lazy"></span>. The equation for this limit, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}=\nabla \times [{\vec {v}}\times {\vec {B}}]}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}=\nabla \times [{\vec {v}}\times {\vec {B}}]}</annotation>

</semantics>

</math></span><img src="./ca7f9b2e75b59ed2959c5387b157373fcdbff2d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.866ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}=\nabla \times [{\vec {v}}\times {\vec {B}}]}" loading="lazy"></span>, has the solution<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {B}}\left({\vec {r}},t\right)=B_{0}kv_{0}t\cos(ky){\hat {x}}+B_{0}{\hat {y}}}</annotation>

</semantics>

</math></span></span>

As can be seen in the figure to the right, the fluid drags the magnetic field lines so that they obtain the sinusoidal character of the flow field.

</p><p>For the limit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{m}\ll 1}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle R_{m}\ll 1}</annotation>

</semantics>

</math></span><img src="./2713ab9b5bbe4ad89c1e42607f4c6da9d5839b79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.216ex; height:2.509ex;" alt="{\displaystyle R_{m}\ll 1}" loading="lazy"></span>, the magnetic diffusion equation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}={\frac {1}{\mu _{0}\sigma }}\nabla ^{2}{\vec {B}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}={\frac {1}{\mu _{0}\sigma }}\nabla ^{2}{\vec {B}}}</annotation>

</semantics>

</math></span><img src="./d6cb8245ae44ac4f187cf67f027edcc8c6cfe724.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.393ex; height:6.509ex;" alt="{\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}={\frac {1}{\mu _{0}\sigma }}\nabla ^{2}{\vec {B}}}" loading="lazy"></span> is just a vector-valued form of the <a href="Heat_equation" title="Heat equation">heat equation</a>. For a localized initial magnetic field (e.g. Gaussian distribution) within a conducting material, the maxima and minima will asymptotically decay to a value consistent with <a href="Laplace's_equation" title="Laplace's equation">Laplace's equation</a> for the given boundary conditions.

</p>

<div class="mw-heading mw-heading2"><h2 id="Diffusion_Times_for_Stationary_Conductors">Diffusion Times for Stationary Conductors</h2></div>

<p>For stationary conductors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (R_{m}=0)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle (R_{m}=0)}</annotation>

</semantics>

</math></span><img src="./d39fa5012e3e48a8f82d7ba7765b2715c9362682.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.509ex; height:2.843ex;" alt="{\displaystyle (R_{m}=0)}" loading="lazy"></span> with simple geometries a time constant called magnetic diffusion time can be derived.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Different one-dimensional equations apply for conducting slabs and conducting cylinders with constant magnetic permeability. Also, different diffusion time equations can be derived for nonlinear saturable materials such as steel.

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFHoltHaskell1965" class="citation book cs1">Holt, E. H.; Haskell, R. E. (1965). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/foundationsofpla0000holt"><i>Foundations of Plasma Dynamics</i></a></span>. New York: Macmillan. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/foundationsofpla0000holt/page/429">429-431</a>.</cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFChen2016" class="citation book cs1">Chen, Francis F. (2016). <i>Introduction to Plasma Physics and Controlled Fusion</i> (3rd ed.). Heidelberg: Springer. pp. <span class="nowrap">192–</span>194. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-22308-7</bdi>.</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFLandauLifshitz2013" class="citation book cs1">Landau, L. D.; Lifshitz, E. M. (2013). <i>The Classical Theory of Fields</i> (4th revised ed.). New York: Elsevier. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781483293288</bdi>.</cite></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFLongcope2002" class="citation web cs1">Longcope, Dana (2002). <a rel="nofollow" class="external text" href="http://solar.physics.montana.edu/dana/mhd_notes.pdf">"Notes on Magnetohydrodynamics"</a> <span class="cs1-format">(PDF)</span>. <i>Montana State University - Department of Physics</i><span class="reference-accessdate">. Retrieved <span class="nowrap">30 April</span> 2019</span>.</cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFBrauer2014" class="citation book cs1">Brauer, J. R. (2014). <i>Magnetic Actuators and Sensors</i> (2nd ed.). Hoboken NJ: Wiley IEEE Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-118-50525-0</bdi>.</cite></span>

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