# Resistive timescale

From FusionWiki

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

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

Ohm's Law,

- $ \vec E = \eta \vec j, $

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

- $ \vec \nabla \times \vec B = \mu_0 \vec j, $

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

- $ \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

- $ \vec \nabla \cdot \vec B = 0. $

From this, one can deduce the typical timescale

- $ \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.