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We derive the properties of E and B fields at an interface between two media.

Em at an interface

We want to understand with Electromagnetism what happens at a surface. From Maxwell's equations we can understand what happens to the components of the E and B fields: First lets look at the E field using Gauss' law. Recall ε E d s = ρ V

Consider the diagram, the field on the incident side is E i + E r . On the transmission side, the field is E t . We can collapse the cylinder down so that it is a pancake with an infinitelysmall height. When we do this there are no field lines through the side of the cylinder. Thus there is only a flux through the top and the bottom of thecylinder and we have; ε E d s = [ ε i ( E i + E r ) ε t E t ] s = 0. I have set ρ V = 0 since we will only consider cases without free charges. So we have ε i E i + ε i E r = ε t E t if u ̂ n is a unit vector normal to the surface this can be written ε i u ̂ n E i + ε i u ̂ n E r = ε t u ̂ n E t

Similarly Gauss' law of Magnetism B d s = 0 gives B i + B r = B t or u ̂ n B i + u ̂ n B r = u ̂ n B t

Amperes law can also be applied to an interface.Then

B μ d l = j d s + t ε E d s (note that in this case d s is perpendicular to the page)

Now we will not consider cases with surface currents. Also we can shrink the vertical ends of the loop so that the area of the box is 0 so that ε E d s = 0 . Thus we get at asurface B i + B r μ i = B t μ t or u ̂ n × B i μ i + u ̂ n × B r μ r = u ̂ n × B t μ t

Similarly we can use Faraday's law E d l = t B d s and play the same game with the edges to get E i + E r = E t or u ̂ n × E i + u ̂ n × E r = u ̂ n × E t (notice ε does not appear)

In summary we have derived what happens to the E and B fields at the interface between two media: ε i u ̂ n E i + ε i u ̂ n E r = ε t u ̂ n E t

u ̂ n B i + u ̂ n B r = u ̂ n B t

u ̂ n × B i μ i + u ̂ n × B r μ i = u ̂ n × B t μ t

u ̂ n × E i + u ̂ n × E r = u ̂ n × E t

Questions & Answers

who was the first nanotechnologist
Lizzy Reply
technologist's thinker father is Richard Feynman but the literature first user scientist Nario Tagunichi.
Norio Taniguchi
I need help
anyone have book of Abdel Salam Hamdy Makhlouf book in pdf Fundamentals of Nanoparticles: Classifications, Synthesis
Naeem Reply
what happen with The nano material on The deep space.?
pedro Reply
It could change the whole space science.
the characteristics of nano materials can be studied by solving which equation?
sibaram Reply
plz answer fast
synthesis of nano materials by chemical reaction taking place in aqueous solvents under high temperature and pressure is call?
hydrothermal synthesis
how can chip be made from sand
Eke Reply
is this allso about nanoscale material
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where is the latest information on a no technology how can I find it
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
STM - Scanning Tunneling Microscope.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Waves and optics. OpenStax CNX. Nov 17, 2005 Download for free at http://cnx.org/content/col10279/1.33
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