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## Main Question or Discussion Point

I reached that part and almost didn't understand anything.

I will write the whole text:

One of the common problems in electrostatics is the determination of electric field or potential due to a given surface distribution of charges.

Gauss law allows us to write a partial result directly. If a surface S with a unit normal ##\vec n ## has a surface charge density of ##\sigma(x)## and electric field ##\vec E_{1}## and ##\vec E_{2}## on either side of the surface, then gauss law tells us immediately that ##(\vec E_{1}-\vec E_{2})\cdot \vec n=\frac{\sigma}{\varepsilon_{0}} .....*##

This does not determine ##\vec E_{1}## and ##\vec E_{2}## if there are no other sources of field and the geometry and form ##\sigma## are especially simple.All that (*) says is that there is a discontinuity of ##\frac{\sigma}{\varepsilon_{0}}## in the normal component of electric field in crossing a surface with a surface charge density ##\sigma## the crossing being made from the "inner" to the "outer" side of the surface.

The tangential component of the electric field can be shown to be continuous across a boundary surface by using ##\oint \vec E \cdot \vec dl =0## for the line integral of ##\vec E## along a closed path. It is only necessarily to take a rectangular path with negligible ends and one side of either side of the boundary.

A general result for the potential (and hence, the field by diffrentiation) at any point in space not just at the surface, can be obtained from ##\phi(x)=\int {\frac{\sigma(x')}{|x-x'|}}\ d^3 x'## by replacing ##\rho d^3x## by##\sigma da## :$$\phi(x)=\int {\frac{\sigma(x')}{|x-x'|}}\ da',$$

My Questions are:

a) I didn't get that part:"If a surface S with a unit normal ##\vec n ## has a surface charge density of ##\sigma(x)## and electric field ##\vec E_{1}## and ##\vec E_{2}## on either side of the surface, then gauss law tells us immediately that ##(\vec E_{1}-\vec E_{2})\cdot \vec n=\frac{\sigma}{\varepsilon_{0}} .....*## ". E1 and E2 represent what? and where should I start to get this result.

b)what does he mean by this:"This does not determine ##\vec E_{1}## and ##\vec E_{2}## if there are no other sources of field and the geometry and form ##\sigma## are especially simple".

c)What did he mean by this, where is the discontinuity?:"All that (*) says is that there is a discontinuity of ##\frac{\sigma}{\varepsilon_{0}}## in the normal component of electric field in crossing a surface with a surface charge density ##\sigma## the crossing being made from the "inner" to the "outer" side of the surface".

d) can anybody explain this more deeply to me:"The tangential component of the electric field can be shown to be continuous across a boundary surface by using ##\oint \vec E \cdot \vec dl =0## for the line integral of ##\vec E## along a closed path. It is only necessarily to take a rectangular path with negligible ends and one side of either side of the boundary".

Thanks