MHD Equations: Deriving and Exploring Consequences

[Logarythmic]

Homework Statement

Two of the MHD equations can be written as

$$\vec{E} + \vec{v} \times \vec{B} = \eta \vec{J}$$

$$\vec{\nabla} \times \vec{B} = \mu_0 \vec{J}$$

where [itex]\eta[/tex] is the resistivity of the plasma.

Derive an expression for the magnetic field at a very high resistivity and describe the corresponding consequences for the behaviour of the magnetic field lines.

The Attempt at a Solution

I need a starter. I have no clue.. Use the maxwell equations?

 

[Jhero]

Hello! I would suggest starting by using the Maxwell equations to derive an expression for the magnetic field at high resistivity. The Maxwell equations describe the relationship between electric and magnetic fields, and they are fundamental to understanding MHD (magnetohydrodynamics) equations.

To begin, let's rewrite the MHD equations in terms of the Maxwell equations:

1. \vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_0} (Gauss's law)

2. \vec{\nabla} \cdot \vec{B} = 0 (Gauss's law for magnetism)

3. \vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} (Faraday's law)

4. \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} (Ampere's law)

Now, let's focus on the second MHD equation, \vec{\nabla} \times \vec{B} = \mu_0 \vec{J}. We can rearrange this equation to solve for \vec{B}:

\vec{B} = \frac{1}{\mu_0} \vec{\nabla} \times \vec{J}

Next, let's consider the high resistivity case, where \eta is very large. This means that the resistivity of the plasma is high, and therefore the current density \vec{J} is small. Plugging this into our equation for \vec{B}, we get:

\vec{B} = \frac{1}{\mu_0} \vec{\nabla} \times \vec{J} \approx \frac{1}{\mu_0} \vec{\nabla} \times 0 = 0

This tells us that at high resistivity, the magnetic field is approximately zero. This has significant consequences for the behaviour of magnetic field lines. In the absence of a strong magnetic field, the plasma will not be confined and will be free to move and expand. This can lead to instabilities and disruptions in the plasma, which can have negative effects on any plasma-based systems or experiments.

I hope this helps get you started on your solution. Remember to always use the fundamental equations and consider