# 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,

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

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.