# 16.3 Energy carried by electromagnetic waves  (Page 2/5)

 Page 2 / 5

The energy passing through area A in time $\text{Δ}t$ is

$u\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{volume}=uAc\text{Δ}t.$

The energy per unit area per unit time passing through a plane perpendicular to the wave, called the energy flux and denoted by S , can be calculated by dividing the energy by the area A and the time interval $\text{Δ}t$ .

$S=\frac{\text{Energy passing area}\phantom{\rule{0.2em}{0ex}}A\phantom{\rule{0.2em}{0ex}}\text{in time}\phantom{\rule{0.2em}{0ex}}\text{Δ}t}{A\text{Δ}t}=uc={\epsilon }_{0}c{E}^{2}=\frac{1}{{\mu }_{0}}EB.$

More generally, the flux of energy through any surface also depends on the orientation of the surface. To take the direction into account, we introduce a vector $\stackrel{\to }{S}$ , called the Poynting vector    , with the following definition:

$\stackrel{\to }{S}=\frac{1}{{\mu }_{0}}\stackrel{\to }{E}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{B}.$

The cross-product of $\stackrel{\to }{E}$ and $\stackrel{\to }{B}$ points in the direction perpendicular to both vectors. To confirm that the direction of $\stackrel{\to }{S}$ is that of wave propagation, and not its negative, return to [link] . Note that Lenz’s and Faraday’s laws imply that when the magnetic field shown is increasing in time, the electric field is greater at x than at $x+\text{Δ}x$ . The electric field is decreasing with increasing x at the given time and location. The proportionality between electric and magnetic fields requires the electric field to increase in time along with the magnetic field. This is possible only if the wave is propagating to the right in the diagram, in which case, the relative orientations show that $\stackrel{\to }{S}=\frac{1}{{\mu }_{0}}\stackrel{\to }{E}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{B}$ is specifically in the direction of propagation of the electromagnetic wave.

The energy flux at any place also varies in time, as can be seen by substituting u from [link] into [link] .

$S\left(x,t\right)=c{\epsilon }_{0}{E}_{0}^{2}{\text{cos}}^{2}\left(kx-\omega t\right)$

Because the frequency of visible light is very high, of the order of ${10}^{14}\phantom{\rule{0.2em}{0ex}}\text{Hz,}$ the energy flux for visible light through any area is an extremely rapidly varying quantity. Most measuring devices, including our eyes, detect only an average over many cycles. The time average of the energy flux is the intensity I of the electromagnetic wave and is the power per unit area. It can be expressed by averaging the cosine function in [link] over one complete cycle, which is the same as time-averaging over many cycles (here, T is one period):

$I={S}_{\text{avg}}=c{\epsilon }_{0}{E}_{0}^{2}\frac{1}{T}\underset{0}{\overset{T}{\int }}{\text{cos}}^{2}\left(2\pi \frac{t}{T}\right)dt.$

We can either evaluate the integral, or else note that because the sine and cosine differ merely in phase, the average over a complete cycle for ${\text{cos}}^{2}\left(\xi \right)$ is the same as for ${\text{sin}}^{2}\left(\xi \right)$ , to obtain

$⟨{\text{cos}}^{2}\xi ⟩=\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\left[⟨{\text{cos}}^{2}\xi ⟩+⟨{\text{sin}}^{2}\xi ⟩\right]=\frac{1}{2}⟨1⟩=\frac{1}{2}.$

where the angle brackets $⟨\text{⋯}⟩$ stand for the time-averaging operation. The intensity of light moving at speed c in vacuum is then found to be

$I={S}_{\text{avg}}=\frac{1}{2}c{\epsilon }_{0}{E}_{0}^{2}$

in terms of the maximum electric field strength ${E}_{0},$ which is also the electric field amplitude. Algebraic manipulation produces the relationship

$I=\frac{c{B}_{0}^{2}}{2{\mu }_{0}}$

where ${B}_{0}$ is the magnetic field amplitude, which is the same as the maximum magnetic field strength. One more expression for ${I}_{\text{avg}}$ in terms of both electric and magnetic field strengths is useful. Substituting the fact that $c{B}_{0}={E}_{0},$ the previous expression becomes

$I=\frac{{E}_{0}{B}_{0}}{2{\mu }_{0}}.$

We can use whichever of the three preceding equations is most convenient, because the three equations are really just different versions of the same result: The energy in a wave is related to amplitude squared. Furthermore, because these equations are based on the assumption that the electromagnetic waves are sinusoidal, the peak intensity is twice the average intensity; that is, ${I}_{0}=2I.$

what is motion?
where the solving of questions of this topic?
According to Nernst's distribution law there are about two solvents in which solutes undergo equilibria. But i don't understand how can you know which of two solvents goes bottom and one top? I real want to understand b'coz some books do say why they prefer one to top/bottom.
I need chapter 25 last topic
What is physics?
Abdulaziz
physics is the study of matter and energy in space and time and how they related to each other
Manzoor
interaction of matter and eneegy....
Abdullah
thanks for correcting me bro
Manzoor
What is electrostatics bassically?
study of charge at rest
wamis
A branch in physics that deals with statics electricity
Akona
what is PN junction?
Manzoor
please I don't understand the solution of the first example as in d working
what's the question? Write it here.
SABYASACHI
a cold body of 100°C and a hot body is of 100°F . Transfer heat = ?
you are given two metal spheres mounted on portable insulating support. Find a way to give them equal and opposite charges. you may use a glass rod rubbed with silk but may not touch it to the spheres. Do the spheres have to be of equal size for your method to work?
what is emotion?
Abdulaziz
in the 2nd example, for chapter 8.2 on page 3/3, I don't understand where the value 48uC comes from, I just couldn't get that value in my calculator.
are you talking about the capacitance combination problem
sam
please write the problem or send a snap of th page....I don't have the book in my vicinity.
SABYASACHI
yes, the 2nd example called Network of Capacitors on page 3/3 of section 8.2.
Anita
12 V = (Q1/12uF)+(Q1/6uF). So, Q1 = 12x4 = 48 uC.
sam
ohhhh OK thanks so much!!!!!!!
Anita
hello guys,, I'm asking to know something about, How can i know which solvent goes down and which does up in determination of partion coefficient(Nernst's distribution law). Please Need help because i have seen many contradictions via few of text books even some videos on youtube they don't say
Elia
what is electromagnetic force. do electric and magnetic force happen differently
yes
yes
Pranay
why
Godson
how?
Godson
when electric charge exert force on another electric charge then this force is known as electrostatic force and when a magnet exert force on another magnet then this force is known as magnetic force and when force exerted on magnet due to varying electric field then this electromagnetic force
Ilyas
Yes
Akona
derived the electric potential due to disk of charge
how can we derived potential electric due to the disk
aron
how can you derived electric potential of a disk
aron
how can you derived electric potential due to disk
aron
where is response?
aron
what is difference between heat and temperature?
temperature is the measure of degree of hotness or coldness. on the other hand, heat is the form of energy, which causes temperature. So we can safely say, heat is the reason and temperature is its consequence.
SABYASACHI
Heat is the reason and temperature is the consequences
Angela
how many liquid metals do we have
do we have gasses as metals
Jeffery
who knows should please tell us
yes...gallium & cesium
Idris
Hg is liquid. No metal gasses at standard temp and pressure
Shane
I don't ever understand any of this formulae
which formula
How to determine a temperature scale
what is the formula for absolute error
Nyro