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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:

  1. 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 $\epsilon_r\approx1$.

  2. 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?

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1 Answer 1

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  1. On page 55 of Chen (1984) he shows that $$\epsilon_R=1+\frac{\mu_0\rho c^2}{B^2},$$ which for $n=10^{16}\mathrm{\,m}^{-3}$ and $B=0.1\mathrm{\,T}$ we have (for hydrogen) $$\frac{\mu_0\rho c^2}{B^2}=\frac{(4\pi\times10^{-7})(10^{16}) (1.67\times10^{-27})(9\times10^{16})}{(0.1)^2}=189.$$

  2. If the maths is correct then the questions have now been answered.

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