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The frequency of optical light is $\approx {10}^{15}$ Hz and so the Poynting vector varies extremely quickly. So it is useful todetermine an time averaged quantity. So lets define the irradiance as $I={⟨S⟩}_{T}$ where the symbol ${⟨\phantom{\rule{thickmathspace}{0ex}}⟩}_{T}$ means find the average over a time $T\text{.}$ For $T$ we want to use an integer multiple of periods (such as 1). (What would you end up with otherwise? It wouldn't really make sense to me) This is a case whereit is easier to use a trig function for the wave. Let's consider the wave $\stackrel{⃗}{E}\left(\stackrel{⃗}{r},t\right)={\stackrel{⃗}{E}}_{0}{\mathrm{cos}}\left(\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}-\omega t\right)\text{.}$ Then $S={c}^{2}{\epsilon }_{0}{E}_{0}{B}_{0}{{\mathrm{cos}}}^{2}\left(\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}-\omega t\right)$ Now we need to find ${⟨S⟩}_{T}={c}^{2}{\epsilon }_{0}{E}_{0}{B}_{0}{⟨{{\mathrm{cos}}}^{2}\left(\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}-\omega t\right)⟩}_{T}$ ${⟨S⟩}_{T}={c}^{2}{\epsilon }_{0}{E}_{0}{B}_{0}\frac{1}{T}{\int }_{0}^{T}{{\mathrm{cos}}}^{2}\left(\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}-\omega t\right)dt$ we make a coordinate transformation $x=\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}-\omega t\text{.}$ Then $\begin{array}{c}{⟨S⟩}_{T}={c}^{2}{\epsilon }_{0}{E}_{0}{B}_{0}\frac{1}{T}{\int }_{0}^{T}{{\mathrm{cos}}}^{2}\left(\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}-\omega t\right)dt\\ ={c}^{2}{\epsilon }_{0}{E}_{0}{B}_{0}\frac{1}{T}{\int }_{\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}}^{\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}-\omega T}{{\mathrm{cos}}}^{2}xdx\frac{dt}{dx}\\ ={c}^{2}{\epsilon }_{0}{E}_{0}{B}_{0}\frac{-1}{\omega T}{\int }_{\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}}^{\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}-\omega T}{{\mathrm{cos}}}^{2}xdx\\ ={c}^{2}{\epsilon }_{0}{E}_{0}{B}_{0}\frac{-1}{\omega T}{\int }_{\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}}^{\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}-\omega T}\left[\frac{1+{\mathrm{cos}}2x}{2}\right]dx\\ ={c}^{2}{\epsilon }_{0}{E}_{0}{B}_{0}\frac{-1}{\omega T}{\left[\frac{x}{2}+\frac{{\mathrm{sin}}2x}{4}|}_{\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}}^{\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}-\omega T}\\ ={c}^{2}{\epsilon }_{0}{E}_{0}{B}_{0}\frac{-1}{\omega T}\frac{\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}-\omega T}{2}-\frac{\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}}{2}\phantom{\rule{thickmathspace}{0ex}}+\frac{{\mathrm{sin}}2\left(\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}-\omega T\right)}{4}-\frac{{\mathrm{sin}}2\left(\stackrel{⃗}{k}\cdot \stackrel{⃗}{r}\right)}{4}\\ ={c}^{2}{\epsilon }_{0}{E}_{0}{B}_{0}\frac{-1}{\omega T}\left[\frac{-\omega T}{2}\right]\\ ={c}^{2}{\epsilon }_{0}{E}_{0}{B}_{0}/2\end{array}$ Note that above I assume that I can pick $T$ to be an integer multiple of the period.

Now we have just gone through a rather involved derivation of something that should of been intuitively obvious to us. The time average of a harmonicfunction is zero. The time average of its square is 1/2. Small digression. For a harmonic function, a useful quantity is the Root MeanSquare: RMS. For example look at the power rating on a stereo, they have to specify whether it is peak, or RMS power that they are referring to.

Remember that $$ is the average power per unit area in the wave. So the Power in the wave is $A$ where A is the area crossed. The force of the wave when it hits something is Power/velocity or $F=\frac{A}{c}$

So the electromagnetic wave can exert a pressure (on a black object) $pressure=\frac{F}{A}=\frac{}{c}$ If it hits a reflective surface then it is $pressure=2\frac{}{c}$

Also we can write: $F=\frac{dp}{dt}=\frac{power}{c}=\frac{1}{c}\frac{dE}{dt}$ (here E is the energy). So the momentum in a unit volume of the EM field is theEnergy in unit volume of the EM field/c. The direction of the momentum is the direction of propagation of the wave. That is: $u=\frac{{B}^{2}}{{\mu }_{0}}=\frac{EB}{c{\mu }_{0}}=\frac{S}{c}$

and

$momentum/unit\phantom{\rule{thickmathspace}{0ex}}volume\equiv \stackrel{⃗}{g}=\frac{u}{c}\stackrel{̂}{k}=\frac{\stackrel{⃗}{S}}{{c}^{2}}$

A stationary charge or a uniformly moving charge can not produce an EM wave (or radiation). This is obvious when you consider a stationary charge. Youwould see a time independent $\stackrel{⃗}{E}$ field around it but no $\stackrel{⃗}{B}$ field. Thus there would not be a Poynting vector and no photons would be emitted.

What if you were driving by the charge at a constant speed. Then you would measure an $\stackrel{⃗}{E}$ and $\stackrel{⃗}{B}$ field but the irradience would integrate to zero. If you stopped moving with respect to the charge this can't make photons appear or disappear. The photonsdon't know what you are doing! If a charge moves nonuniformly though it will radiate.

Suppose you have an oscillating dipole $\stackrel{⃗}{\wp }={\stackrel{⃗}{\wp }}_{0}{\mathrm{cos}}\omega t$ When you get far from the dipole you get a wave with a fixed wavelength $E=\frac{{\wp }_{0}{k}^{2}{\mathrm{sin}}\theta }{4\pi {\epsilon }_{0}}\frac{{\mathrm{cos}}\left(kr-\omega t\right)}{r}$ Here $E$ is the electric field intensity, $\theta$ is with respect to the dipole moment (see figure 3.33 in Hecht Fourth Edition or figure 3.31 in the Third edition).The irradiance from this isgiven by $}_{T}=I\left(\theta \right)=\frac{{\wp }_{0}^{2}{\omega }^{4}}{32{\pi }^{2}{c}^{3}{\epsilon }_{0}}\frac{{{\mathrm{sin}}}^{2}\theta }{{r}^{2}}$ one can integrate over the angle (at any radius) and get the total energy radiated $\int }_{T}d\Omega =\frac{{\wp }_{0}^{2}{\omega }^{4}}{12\pi {c}^{3}{\epsilon }_{0}}$

how can chip be made from sand
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
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