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The energy passing through area A in time Δ t is

u × volume = u A c 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 Δ t .

S = Energy passing area A in time Δ t A Δ t = u c = ε 0 c E 2 = 1 μ 0 E B .

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 S , called the Poynting vector    , with the following definition:

S = 1 μ 0 E × B .

The cross-product of E and B points in the direction perpendicular to both vectors. To confirm that the direction of 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 + 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 S = 1 μ 0 E × 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 ( x , t ) = c ε 0 E 0 2 cos 2 ( k x ω t )

Because the frequency of visible light is very high, of the order of 10 14 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 avg = c ε 0 E 0 2 1 T 0 T cos 2 ( 2 π t T ) d t .

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 cos 2 ( ξ ) is the same as for sin 2 ( ξ ) , to obtain

cos 2 ξ = 1 2 [ cos 2 ξ + sin 2 ξ ] = 1 2 1 = 1 2 .

where the angle brackets stand for the time-averaging operation. The intensity of light moving at speed c in vacuum is then found to be

I = S avg = 1 2 c ε 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 = c B 0 2 2 μ 0

where B 0 is the magnetic field amplitude, which is the same as the maximum magnetic field strength. One more expression for I 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 = E 0 B 0 2 μ 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 = 2 I .

Questions & Answers

does your app come with video lessons?
Ahmed Reply
What is vector
Ajibola Reply
Vector is a quantity having a direction as well as magnitude
Damilare
tell me about charging and discharging of capacitors
Ahemen Reply
a big and a small metal spheres are connected by a wire, which of this has the maximum electric potential on the surface.
Bundi Reply
3 capacitors 2nf,3nf,4nf are connected in parallel... what is the equivalent capacitance...and what is the potential difference across each capacitor if the EMF is 500v
Prince Reply
four effect of heat on substances
Prince Reply
why we can find a electric mirror image only in a infinite conducting....why not in finite conducting plate..?
Rima Reply
because you can't fit the boundary conditions.
Jorge
what is the dimensions for VISCOUNSITY (U)
Branda
what is thermodynamics
Aniket Reply
the study of heat an other form of energy.
John
heat is internal kinetic energy of a body but it doesnt mean heat is energy contained in a body because heat means transfer of energy due to difference in temperature...and in thermo-dynamics we study cause, effect, application, laws, hypothesis and so on about above mentioned phenomenon in detail.
ing
It is abranch of physical chemistry which deals with the interconversion of all form of energy
Vishal
what is colamb,s law.?
Muhammad Reply
it is a low studied the force between 2 charges F=q.q`\r.r
Mostafa
what is the formula of del in cylindrical, polar media
Birengeso Reply
prove that the formula for the unknown resistor is Rx=R2 x R3 divided by R3,when Ig=0.
MAXWELL Reply
what is flux
Bundi Reply
Total number of field lines crossing the surface area
Kamru
Basically flux in general is amount of anything...In Electricity and Magnetism it is the total no..of electric field lines or Magnetic field lines passing normally through the suface
prince
what is temperature change
Celine
a bottle of soft drink was removed from refrigerator and after some time, it was observed that its temperature has increased by 15 degree Celsius, what is the temperature change in degree Fahrenheit and degree Celsius
Celine
process whereby the degree of hotness of a body (or medium) changes
Salim
Q=mcΔT
Salim
where The letter "Q" is the heat transferred in an exchange in calories, "m" is the mass of the substance being heated in grams, "c" is its specific heat capacity and the static value, and "ΔT" is its change in temperature in degrees Celsius to reflect the change in temperature.
Salim
what was the temperature of the soft drink when it was removed ?
Salim
15 degree Celsius
Celine
15 degree
Celine
ok I think is just conversion
Salim
15 degree Celsius to Fahrenheit
Salim
0 degree Celsius = 32 Fahrenheit
Salim
15 degree Celsius = (15×1.8)+32 =59 Fahrenheit
Salim
I dont understand
Celine
the question said you should convert 15 degree Celsius to Fahrenheit
Salim
To convert temperatures in degrees Celsius to Fahrenheit, multiply by 1.8 (or 9/5) and add 32.
Salim
what is d final ans for Fahrenheit and Celsius
Celine
it said what is temperature change in Fahrenheit and Celsius
Celine
the 15 is already in Celsius
Salim
So the final answer for Fahrenheit is 59
Salim
what is d final ans for Fahrenheit and Celsius
Celine
what are the effects of placing a dielectric between the plates of a capacitor
Bundi Reply
increase the capacitance.
Jorge
besides increasing the capacitance, is there any?
Bundi
mechanical stiffness and small size
Jorge
so as to increase the capacitance of a capacitor
Rahma
also to avoid diffusion of charges between the two plate since they are positive and negative.
Prince
why for an ideal gas internal energy is directly proportional to thermodynamics temperature?
Anne Reply
two charged particles are 8.45cm apart. They are moved and the force on each of them is found to have tripled. How far are they now?
Martin Reply
what is flux
Bundi
Bundi, flux is the number of electric field crossing a surface area
Mubanga
you right
Muhammad
martin,F/F`=(r`×r`)÷(r×r)
Mostafa
Practice Key Terms 1

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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