Discontinuity in electric field

In summary, Jackson's book of electrodynamics discusses surface charge distributions and discontinuity in electric field. Gauss law allows us to determine the electric field or potential due to a given surface distribution of charges. The tangential component of the electric field is continuous across a boundary surface, while the normal component has a discontinuity at the surface. The tangential component can be shown to be continuous by using the line integral of the electric field along a closed path. If you are new to electromagnetism, it may be helpful to supplement Jackson's book with a more approachable textbook.
  • #1
amjad-sh
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I was recently reading jackson's book of electrodynamics regarding surface charge distributions and discontinuity in electric field.
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
 
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  • #2
amjad-sh said:
E1 and E2 represent what?
##\vec E_1## is the electric field on one side of the surface, and ##\vec E_2## is the electric field on the other side of the surface.
amjad-sh said:
where should I start to get this result.
Start by drawing a Gaussian surface as shown in the diagram here: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elesht.html
amjad-sh said:
b)what does he mean by this:"This does not determine ⃗E1\vec E_{1} and ⃗E2\vec E_{2} if there are no other sources of field [...]
If you know only the difference between two numbers, it does not tell you the numbers themselves. For example, if a - b = 5, then you might have a = 10 and b = 5; or a = 11 and b = 6; or a = 2 and b = -3; etc. In order to find a and b, you need more information or assumptions.
amjad-sh said:
where is the discontinuity?
The discontinuity is at the surface (where the charge density ##\sigma## is located). As you move across the surface, the normal component of ##\vec E## changes abruptly.
amjad-sh said:
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 ∮⃗E⋅⃗dl=0\oint \vec E \cdot \vec dl =0 for the line integral of ⃗E\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".
See for example the left part of the diagram at the top of this page: http://farside.ph.utexas.edu/teaching/em/lectures/node59.html

Jackson is a graduate-level (master's / PhD) textbook. Students who take a course using it have normally already completed an undergraduate-level course in electromagnetism using e.g. Griffiths, which covers material such as the above. If you're trying to learn electromagnetism from the beginning, or if it's been some time since you studied it at an intermediate undergraduate level, you might want to supplement Jackson with Griffiths or a similar textbook, which is more "approachable."
 
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Related to Discontinuity in electric field

1. What is discontinuity in an electric field?

Discontinuity in an electric field refers to a sudden change or disruption in the strength or direction of an electric field. This can occur at the boundary between two different materials or regions with different electrical properties.

2. How does discontinuity in an electric field affect the behavior of charged particles?

When a charged particle encounters a discontinuity in an electric field, its path and velocity may be altered. This can result in the particle being deflected, accelerated, or decelerated depending on the direction and strength of the electric field.

3. What causes a discontinuity in an electric field?

Discontinuities in electric fields can be caused by a variety of factors, including changes in the material properties, the presence of charged particles, or the geometry of the electric field. They can also occur naturally in the Earth's atmosphere due to variations in the Earth's magnetic field.

4. How is discontinuity in an electric field measured?

Discontinuity in an electric field can be measured using instruments such as electric field meters or voltmeters. These instruments measure the strength and direction of the electric field at a given point and can detect any sudden changes or disruptions.

5. What are some practical applications of studying discontinuity in electric fields?

Understanding discontinuities in electric fields is important for a variety of practical applications, such as designing electrical circuits and devices, studying atmospheric phenomena, and developing new technologies for controlling and manipulating charged particles.

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