# Resistive timescale

The resistive timescale is the typical time for the diffusion of a magnetic field into a resistive plasma. Based on Faraday's Law,

${\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}=-{\vec {\nabla }}\times {\vec {E}},}$

Ohm's Law,

${\displaystyle {\vec {E}}=\eta {\vec {j}},}$

where η is the resistivity (assumed homogeneous), and Ampère's Law,

${\displaystyle {\vec {\nabla }}\times {\vec {B}}=\mu _{0}{\vec {j}},}$

one immediately derives a diffusion type equation for the magnetic field:

${\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}=-{\frac {\eta }{\mu _{0}}}{\vec {\nabla }}\times {\vec {\nabla }}\times {\vec {B}}={\frac {\eta }{\mu _{0}}}\nabla ^{2}{\vec {B}},}$

since

${\displaystyle {\vec {\nabla }}\cdot {\vec {B}}=0.}$

From this, one can deduce the typical timescale

${\displaystyle \tau _{R}\simeq {\frac {\mu _{0}L^{2}}{\eta }}.}$

Here, L is the typical length scale of the problem, often taken equal to a, the minor radius of the toroidal plasma.