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From divergence theorem/Gauss's law: $$\oint_S \vec{E} . d\vec{a}=\frac{1}{\epsilon_0}\sigma A.$$ From what I understand, Gauss's law is where the electric field discontinuity (at a boundary) is derived from (mathematically):

$$\vec{E}_{above} - \vec{E}_{below} =\frac{\sigma}{\epsilon_0} \hat{n}.$$ However, what I don't understand, is what causes this discontinuity physically.

It has been confusing me for several months and the explanations that my teachers have given me, haven't resolved my problem. If anyone could clear this up for me that'd be great. Thanks.

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I'm not sure I understand what's bothering you. Let me try this:

In the case of a sheet of charge in a vacuum, it's the charges that cause the discontinuity. If the sheet is positively charged, then the field vectors point away from the sheet on both sides. This is exactly what a field discontinuity is: different fields on either side of a boundary. It's described by the second equation that you wrote.

In the case of a boundary between two media, the situation is almost the same. The only difference is that the sheet of charge is generated by the polarization of the media by an applied field rather than by a fixed static charge distribution.

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  • $\begingroup$ what is confusing me, is why the discontinuity is always $\frac{1}{\epsilon_0}\sigma A$ specifically $\endgroup$ – George Dec 24 '17 at 16:04
  • $\begingroup$ I think you mean $\sigma/\epsilon_0$ ... no $A$. It is sensible that the discontinuity depends on $\sigma$. That it is linear in $\sigma$ is due to the fact that electric fields add linearly. The $\epsilon_0$ is there only to make the units come out right. In certain systems of units other than the S.I. system, there is no $\epsilon_0$ $\endgroup$ – garyp Dec 24 '17 at 16:19
  • $\begingroup$ ok that makes sense, may I ask, is this discontinuity applicable only for sheets or cold we consider the surface a sphere in the same way $\endgroup$ – George Dec 24 '17 at 17:07
  • $\begingroup$ You can consider any surface that way. $\endgroup$ – garyp Dec 24 '17 at 17:15

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