How to derive the induction equation for spatial varying conductivity?

Question

How to derive the induction equation from Maxwell equations (without displacement current) and Ohm’s law for Spatial varying conductivity?

Answer 1

First start with two of Maxwell's equations, Ampere's and Faraday's law: $$ \begin{align} \nabla \times \mathbf{E} & = -\partial_{t} \ \mathbf{B} \tag{0a} \\ \nabla \times \mathbf{B} & = \mu_{o} \ \mathbf{j} \tag{0b} \end{align} $$

Next, we need to make some approximations regarding the generalized Ohm's law, which is given by: $$ \mathbf{E} + \mathbf{v} \times \mathbf{B} \approx \frac{ \mathbf{j} \times \mathbf{B} }{ n \ e } - \frac{ \nabla}{ n \ e } \cdot \left( \mathcal{P}_{e} + \frac{ m_{e} }{ m_{i} } \mathcal{P}_{i} \right) + \eta \ \mathbf{j} + \frac{ m_{e} }{ n \ e^{2} } \frac{ d \mathbf{j} }{ d t } \tag{1} $$ where $\mathbf{j}$ is the total current density, $n$ is the total number density (assuming quasi-neutrality, i.e., $n_{e} = n_{i}$), $e$ is the fundamental charge, $\mathcal{P}_{s}$ is the pressure tensor of species $s$, $m_{s}$ is the mass of species $s$ ($s$ can be $e$ for electron or $i$ for ion), and $\eta$ is the scalar electrical resistivity (see also https://physics.stackexchange.com/a/438272/59023 or https://physics.stackexchange.com/a/363523/59023 or https://physics.stackexchange.com/a/261223/59023 for more on Ohm's law and conductivities).

Generally, one can assume that the $\tfrac{ d \mathbf{j} }{ d t }$ is small for most settings and $\tfrac{ m_{e} }{ m_{i} }$ ~ 1/1836 which is small too so the ion pressure contribution can be dropped. If we assume an isotropic electron pressure, then the electron pressure term can be neglected since we will take the curl of Ohm's law in the next step. So taking the curl of Equation 1 gives: $$ \nabla \times \mathbf{E} \approx - \nabla \times \left( \mathbf{v} \times \mathbf{B} \right) + \frac{ 1 }{ n \ e } \nabla \times \left( \mathbf{j} \times \mathbf{B} \right) + \nabla \times \left( \eta \ \mathbf{j} \right) \tag{2} $$

In the force-free limit, the $\mathbf{j} \times \mathbf{B}$ $\rightarrow$ 0 and we can use Equation 0b to rewrite the curl of the $\eta \ \mathbf{j}$ term to give: $$ \nabla \times \mathbf{E} \approx - \nabla \times \left( \mathbf{v} \times \mathbf{B} \right) + \nabla \ \eta \times \mathbf{j} - \frac{ \eta }{ \mu_{o} } \ \nabla^{2} \mathbf{B} \tag{3} $$

Finally we now insert this into Faraday's law (Equation 0a) to get: $$ \partial_{t} \ \mathbf{B} \approx \nabla \times \left( \mathbf{v} \times \mathbf{B} \right) + \frac{ \eta }{ \mu_{o} } \ \nabla^{2} \mathbf{B} - \nabla \ \eta \times \mathbf{j} \tag{4} $$

A more formal derivation would actually start with a resistivity tensor, not a scalar. The conductivity is just the inverse of the resistivity, which is simple for scalars and can be complicated for tensors with imaginary parts and possibly non-invertable forms.