# 6.2 Electromagnetism at an interface

 Page 1 / 1
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 $\stackrel{⃗}{E}$ and $\stackrel{⃗}{B}$ fields: First lets look at the $\stackrel{⃗}{E}$ field using Gauss' law. Recall $\oint \epsilon \stackrel{⃗}{E}\cdot d\stackrel{⃗}{s}=\int \rho dV$

Consider the diagram, the field on the incident side is $\stackrel{⃗}{{E}_{i}}+\stackrel{⃗}{{E}_{r}}$ . On the transmission side, the field is $\stackrel{⃗}{{E}_{t}}$ . 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; $\oint \epsilon \stackrel{⃗}{E}\cdot d\stackrel{⃗}{s}=\oint \left[{\epsilon }_{i}\left({E}_{i\perp }+{E}_{r\perp }\right)-{\epsilon }_{t}{E}_{t\perp }\right]ds=0.$ I have set $\int \rho dV=0$ since we will only consider cases without free charges. So we have ${\epsilon }_{i}{E}_{i\perp }+{\epsilon }_{i}{E}_{r\perp }={\epsilon }_{t}{E}_{t\perp }$ if ${\stackrel{̂}{u}}_{n}$ is a unit vector normal to the surface this can be written ${\epsilon }_{i}{\stackrel{̂}{u}}_{n}\cdot {\stackrel{⃗}{E}}_{i}+{\epsilon }_{i}{\stackrel{̂}{u}}_{n}\cdot {\stackrel{⃗}{E}}_{r}={\epsilon }_{t}{\stackrel{̂}{u}}_{n}\cdot {\stackrel{⃗}{E}}_{t}$

Similarly Gauss' law of Magnetism $\oint \stackrel{⃗}{B}\cdot d\stackrel{⃗}{s}=0$ gives ${B}_{i\perp }+{B}_{r\perp }={B}_{t\perp }$ or ${\stackrel{̂}{u}}_{n}\cdot {\stackrel{⃗}{B}}_{i}+{\stackrel{̂}{u}}_{n}\cdot {\stackrel{⃗}{B}}_{r}={\stackrel{̂}{u}}_{n}\cdot {\stackrel{⃗}{B}}_{t}$

Amperes law can also be applied to an interface.Then

$\int \frac{\stackrel{⃗}{B}}{\mu }\cdot d\stackrel{⃗}{l}=\int \stackrel{⃗}{j}\cdot d\stackrel{⃗}{s}+\frac{d}{dt}\int \epsilon \stackrel{⃗}{E}\cdot d\stackrel{⃗}{s}$ (note that in this case $d\stackrel{⃗}{s}$ 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 $\int \epsilon \stackrel{⃗}{E}\cdot d\stackrel{⃗}{s}=0$ . Thus we get at asurface $\frac{{B}_{i\parallel }+{B}_{r\parallel }}{{\mu }_{i}}=\frac{{B}_{t\parallel }}{{\mu }_{t}}$ or $\frac{{\stackrel{̂}{u}}_{n}×{\stackrel{⃗}{B}}_{i}}{{\mu }_{i}}+\frac{{\stackrel{̂}{u}}_{n}×{\stackrel{⃗}{B}}_{r}}{{\mu }_{r}}=\frac{{\stackrel{̂}{u}}_{n}×{\stackrel{⃗}{B}}_{t}}{{\mu }_{t}}$

Similarly we can use Faraday's law $\int \stackrel{⃗}{E}\cdot d\stackrel{⃗}{l}=-\frac{d}{dt}\int \stackrel{⃗}{B}\cdot d\stackrel{⃗}{s}$ and play the same game with the edges to get ${E}_{i\parallel }+{E}_{r\parallel }={E}_{t\parallel }$ or ${\stackrel{̂}{u}}_{n}×{\stackrel{⃗}{E}}_{i}+{\stackrel{̂}{u}}_{n}×{\stackrel{⃗}{E}}_{r}={\stackrel{̂}{u}}_{n}×{\stackrel{⃗}{E}}_{t}$ (notice $\epsilon$ does not appear)

In summary we have derived what happens to the $\stackrel{⃗}{E}$ and $\stackrel{⃗}{B}$ fields at the interface between two media: ${\epsilon }_{i}{\stackrel{̂}{u}}_{n}\cdot {\stackrel{⃗}{E}}_{i}+{\epsilon }_{i}{\stackrel{̂}{u}}_{n}\cdot {\stackrel{⃗}{E}}_{r}={\epsilon }_{t}{\stackrel{̂}{u}}_{n}\cdot {\stackrel{⃗}{E}}_{t}$

${\stackrel{̂}{u}}_{n}\cdot {\stackrel{⃗}{B}}_{i}+{\stackrel{̂}{u}}_{n}\cdot {\stackrel{⃗}{B}}_{r}={\stackrel{̂}{u}}_{n}\cdot {\stackrel{⃗}{B}}_{t}$

$\frac{{\stackrel{̂}{u}}_{n}×{\stackrel{⃗}{B}}_{i}}{{\mu }_{i}}+\frac{{\stackrel{̂}{u}}_{n}×{\stackrel{⃗}{B}}_{r}}{{\mu }_{i}}=\frac{{\stackrel{̂}{u}}_{n}×{\stackrel{⃗}{B}}_{t}}{{\mu }_{t}}$

${\stackrel{̂}{u}}_{n}×{\stackrel{⃗}{E}}_{i}+{\stackrel{̂}{u}}_{n}×{\stackrel{⃗}{E}}_{r}={\stackrel{̂}{u}}_{n}×{\stackrel{⃗}{E}}_{t}$

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers!