Resistive timescale - Revision history

Admin at 15:28, 11 March 2024

2024-03-11T15:28:53Z

← Older revision		Revision as of 17:28, 11 March 2024
Line 4:		Line 4:
	Ohm's Law,		Ohm's Law,
	:<math>\vec E = \eta \vec j,</math>		:<math>\vec E = \eta \vec j,</math>
	where ''η'' is the resistivity (assumed homogeneous), and Ampère's Law,		where ''η'' is the [[:Wikipedia:Spitzer_resistivity\|resistivity]] (assumed homogeneous), and Ampère's Law,
	:<math> \vec \nabla \times \vec B = \mu_0 \vec j,</math>		:<math> \vec \nabla \times \vec B = \mu_0 \vec j,</math>
	one immediately derives a diffusion type equation for the magnetic field:		one immediately derives a diffusion type equation for the magnetic field:

Admin at 10:31, 12 October 2011

2011-10-12T10:31:33Z

← Older revision		Revision as of 12:31, 12 October 2011
Line 1:		Line 1:
	The resistive timescale is the typical time for the diffusion of a magnetic field into a resistive plasma.		The resistive timescale is the typical time for the diffusion of a magnetic field into a resistive plasma.
	Based on Faraday's Law,		Based on Faraday's Law,
	:<math>\frac{d\vec B}{dt} = \vec \nabla \times \vec E,</math>		:<math>\frac{\partial \vec B}{\partial t} = -\vec \nabla \times \vec E,</math>
	Ohm's Law,		Ohm's Law,
	:<math>\vec E = \eta \vec j,</math>		:<math>\vec E = \eta \vec j,</math>
Line 7:		Line 7:
	:<math> \vec \nabla \times \vec B = \mu_0 \vec j,</math>		:<math> \vec \nabla \times \vec B = \mu_0 \vec j,</math>
	one immediately derives a diffusion type equation for the magnetic field:		one immediately derives a diffusion type equation for the magnetic field:
	:<math>\frac{d\vec B}{dt} = \frac{\eta}{\mu_0} \vec \nabla \times \vec \nabla \times \vec B = - \frac{\eta}{\mu_0} \nabla^2 \vec B,</math>		:<math>\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,</math>
	since		since
	: <math>\vec \nabla \cdot \vec B = 0.</math>		: <math>\vec \nabla \cdot \vec B = 0.</math>

Admin at 10:25, 12 October 2011

2011-10-12T10:25:11Z

← Older revision		Revision as of 12:25, 12 October 2011
Line 3:		Line 3:
	:<math>\frac{d\vec B}{dt} = \vec \nabla \times \vec E,</math>		:<math>\frac{d\vec B}{dt} = \vec \nabla \times \vec E,</math>
	Ohm's Law,		Ohm's Law,
	:<math>\vec E = \eta \vec j</math>,		:<math>\vec E = \eta \vec j,</math>
	where ''η'' is the resistivity, and Ampère's Law,		where ''η'' is the resistivity (assumed homogeneous), and Ampère's Law,
	:<math> \vec \nabla \times \vec B = \mu_0 \vec j</math>,		:<math> \vec \nabla \times \vec B = \mu_0 \vec j,</math>
	one immediately derives a diffusion type equation for the magnetic field:		one immediately derives a diffusion type equation for the magnetic field:
	:<math>\frac{d\vec B}{dt} = \frac{\eta}{\mu_0} \vec \nabla \times \vec \nabla \times \vec B</math>,		:<math>\frac{d\vec B}{dt} = \frac{\eta}{\mu_0} \vec \nabla \times \vec \nabla \times \vec B = - \frac{\eta}{\mu_0} \nabla^2 \vec B,</math>
	~~from which~~ one can deduce the typical timescale		since
	:<math> \tau_R \simeq \frac{\mu_0 L^2}{\eta}</math>.		: <math>\vec \nabla \cdot \vec B = 0.</math>
			From this, one can deduce the typical timescale
			:<math> \tau_R \simeq \frac{\mu_0 L^2}{\eta}.</math>
	Here, ''L'' is the typical length scale of the problem, often taken equal to ''a'', the [[Toroidal coordinates\|minor radius]] of the toroidal plasma.		Here, ''L'' is the typical length scale of the problem, often taken equal to ''a'', the [[Toroidal coordinates\|minor radius]] of the toroidal plasma.

			== See also ==

			* [[:Wikipedia:Lundquist number\|Lundquist number]]

Admin: Created page with 'The resistive timescale is the typical time for the diffusion of a magnetic field into a resistive plasma. Based on Faraday's Law, : $\frac{d\vec B}{dt} = \vec \nabla \times …'$

2011-10-12T10:06:28Z

Created page with 'The resistive timescale is the typical time for the diffusion of a magnetic field into a resistive plasma. Based on Faraday's Law, :<math>\frac{d\vec B}{dt} = \vec \nabla \times …'

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The resistive timescale is the typical time for the diffusion of a magnetic field into a resistive plasma.
Based on Faraday's Law,
:<math>\frac{d\vec B}{dt} = \vec \nabla \times \vec E,</math>
Ohm's Law,
:<math>\vec E = \eta \vec j</math>,
where ''η'' is the resistivity, and Ampère's Law,
:<math> \vec \nabla \times \vec B = \mu_0 \vec j</math>,
one immediately derives a diffusion type equation for the magnetic field:
:<math>\frac{d\vec B}{dt} = \frac{\eta}{\mu_0} \vec \nabla \times \vec \nabla \times \vec B</math>,
from which one can deduce the typical timescale
:<math> \tau_R \simeq \frac{\mu_0 L^2}{\eta}</math>.
Here, ''L'' is the typical length scale of the problem, often taken equal to ''a'', the [[Toroidal coordinates|minor radius]] of the toroidal plasma.

Resistive timescale - Revision history

Admin at 15:28, 11 March 2024

Admin at 10:31, 12 October 2011

Admin at 10:25, 12 October 2011

Admin: Created page with 'The resistive timescale is the typical time for the diffusion of a magnetic field into a resistive plasma. Based on Faraday's Law, :\frac{d\vec B}{dt} = \vec \nabla \times …'

Admin: Created page with 'The resistive timescale is the typical time for the diffusion of a magnetic field into a resistive plasma. Based on Faraday's Law, : $\frac{d\vec B}{dt} = \vec \nabla \times …'$