# Can someone explain diamagnetism in plasma?

Diamagnetic plasma has an internal resistance to an outside magnetic field. I thought that a good analogy for this would be electrical resistivity.

In high school we learned that different materials would resist externally applied electric fields differently. You put the same voltage across a block of wood or plastic or metal - you get a different current - like ohms law.

What if we viewed plasma the same way? You put the same magnetic field across different plasmas and they resist conducting the magnetic field. They have a diagmagnetic constant (like solids do) and it is directly analogous to electrical resistivity. I drew it out:

Diamagnetism is never explained this way. People always talk about inducing magnetic fields inside plasmas, either by motion, organization or by externally applied fields.

My questions:

Anyone have a text which looks at plasma this way?

What is the function predicting this diamagnetic constant? I would guess:
diamagnetic constant = Function (plasma density, temperature, composition, ect...)

• I think you could treat this as the plasma pressure increases in a region forcing out the magnetic pressure to maintain a sort of pressure balance with the surrounding medium. Is that what you are looking for? Apr 1 '15 at 13:14
• No... That is not going to help. Plasma beta has been done to death. Let me work on this some more, do some research and I will return to this later. Apr 3 '15 at 18:13

If we assume a force-free situation, then we can write: $$\mathbf{J} \times \mathbf{B} = c_{o} \nabla \cdot \mathbb{P}$$ where $\mathbf{J}$ is the current density, $\mathbf{B}$ is the magnetic field vector, $c_{o}$ is some constant, and $\mathbb{P}$ is the pressure tensor. If we decompose vectors into those parallel and perpendicular to $\mathbf{b}$ = $\mathbf{B}$/B as: $$\mathbf{A}_{\parallel} = \mathbf{b} \left( \mathbf{b} \cdot \mathbf{A} \right) \\ \mathbf{A}_{\perp} = \left( \mathbb{I} - \mathbf{b} \mathbf{b} \right) \cdot \mathbf{A} \\ = \mathbf{b} \times \left( \mathbf{A} \times \mathbf{b} \right)$$ then we can define the diamagnetic current as: $$\mathbf{J}_{\perp} = \frac{c}{B} \mathbf{b} \times \left( \nabla \cdot \mathbb{P} \right)$$
This has an obvious simplification whereby one assumes a scalar pressure so that the term on the right-hand side goes to $\mathbf{b} \times \nabla P$. From this you can see that $\mathbf{J}_{\perp}$ will act to decrease $\mathbf{B}$ in regions of larger pressure. You can also show that $\nabla \cdot \mathbf{J}_{\perp} \neq 0$, which is related to the assumption of quasi-neutrality.