I am reading this paper by Rosenbluth and Hinton (1996). In the second paragraph of the introduction, they say two things I don't understand:
The first, is that they claim that the "plasma dielectric constant is typically very large". Do you know a good source for what the dielectric constant, $\epsilon_r$, in fusion plasma typically is? I didn't know it is large, I thought it was okay to approximate the dielectric constant as $\epsilon_r\approx1$.
The second, is that they state that the large dielectric constant means that the net radial current can't be very large? Do you know why this is? I think the following bit of maths might explain why this is. From Gauss's law we know that $$\mathbf{\nabla\cdot E}=\frac{\rho}{\epsilon_0\epsilon_r}.$$ From the Ampère–Maxwell equation we know that $$\mathbf{\nabla\times B}=\mu_0\mathbf{J}+\mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}.$$ Taking the divergence and substituting for $\mathbf{\nabla\cdot E}$ gives $$\frac{\partial}{\partial t}\left(\frac{\rho}{\epsilon_r}\right)+\mathbf{\nabla\cdot J}=0.$$ Hence if $\epsilon_r\rightarrow \infty$ then $\mathbf{\nabla\cdot J}\rightarrow0$. Let $r$ denote the coordinate along the minor radius of the tokamak. If we assume poloidal and toroidal invariance then $$\mathbf{\nabla \cdot J} = \frac{1}{r}\frac{\partial}{\partial r}\left(rJ_r\right)=0,$$ $$\implies rJ_r = C.$$ where $C$ is an integration constant. Since $J_r$ is finite at $r=0$, we know that $C=0$ and hence $J_r=0$. Is this maths correct?
