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Main Question or Discussion Point
In Griffith's section about electrostatic boundary conditions, he says that given a surface with charge density [itex] \sigma [/itex], and take a waferthin Gaussian pillbox extending over the top and bottom of the surface, Gauss's law states that: [tex] \oint_{S} \mathbf{E} \cdot d \mathbf{a} = \frac{1}{\epsilon_{0}} Q_{enc} = \frac{1}{\epsilon_{0}} \sigma A [/tex] Now, in the limit that the thickness of the pillbox goes to zero, we have: [tex] E_{above}^{\perp}  E_{below}^{\perp} = \frac{1}{\epsilon_{0}} \sigma [/tex] The image he gives is attached in this post. He says for consistency to let upward be the positive direction for both, but I don't understand why he has E pointing up above and also up below the surface. I would think E is pointing up above the surface and down below the surface so that when we take [tex] \oint_{S} \mathbf{E} \cdot d \mathbf{a} [/tex] we would actually get [tex] E_{above}^{\perp} + E_{below}^{\perp} = \frac{1}{\epsilon_{0}} \sigma [/tex] getting a plus instead of minus since [itex] \mathbf{E}_{below} [/itex] and [itex] d \mathbf{a} [/itex] both point down canceling the negatives. What am I thinking wrong?
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