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The "Introduction to Plasma Physics C17 Lecture Notes", which I'm reading online, set out the following scenario in section 1.1.2:

Consider the electric field due to a 1-D line of charge (see Fig. 1.2). Applying Gauss' theorem to the pillbox shown, we find $$ \int E.ds = 2AE = \rho A dx/\epsilon_0$$

Here is the diagram:

enter image description here

I'm struggling to reconcile this with my understanding of Gauss' law. Looking at the above scenario I would say that the total charge $Q$ is equal to the charge density times the height, $A$ (area in 1 dimension). The total flux is the $E$ times the area at each side, so $2A$. Equating these via Gauss' law gives:

$$\Phi_E = 2AE = Q/\epsilon_0 = \rho A/\epsilon_0$$

This is not the same as the result in the lecture notes ($\rho A dx / \epsilon_0$). Why is the $dx$ relevant here? I have a suspicion that it's to do with my (mis)understanding of the geometry of a "Pillbox", but I'm not sure.

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  • $\begingroup$ Note that your charge density appears to be per unit length, whilst the author's is per unit area. $\endgroup$
    – jacob1729
    Commented Jun 7, 2020 at 13:53
  • $\begingroup$ @jacob1729 in the lecture notes the author says $\rho$ is the charge per unit length of the line. $\endgroup$
    – quant
    Commented Jun 7, 2020 at 13:56
  • $\begingroup$ If $\rho$ is charge per unit length then the author's expression $\rho A dx$ is not a charge. They must be interpreting $\rho$ as charge per unit volume (area). $\endgroup$
    – jacob1729
    Commented Jun 7, 2020 at 13:58

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