# 16.3 Energy carried by electromagnetic waves

 Page 1 / 5
By the end of this section, you will be able to:
• Express the time-averaged energy density of electromagnetic waves in terms of their electric and magnetic field amplitudes
• Calculate the Poynting vector and the energy intensity of electromagnetic waves
• Explain how the energy of an electromagnetic wave depends on its amplitude, whereas the energy of a photon is proportional to its frequency

Anyone who has used a microwave oven knows there is energy in electromagnetic waves. Sometimes this energy is obvious, such as in the warmth of the summer Sun. Other times, it is subtle, such as the unfelt energy of gamma rays, which can destroy living cells.

Electromagnetic waves bring energy into a system by virtue of their electric and magnetic fields. These fields can exert forces and move charges in the system and, thus, do work on them. However, there is energy in an electromagnetic wave itself, whether it is absorbed or not. Once created, the fields carry energy away from a source. If some energy is later absorbed, the field strengths are diminished and anything left travels on.

Clearly, the larger the strength of the electric and magnetic fields, the more work they can do and the greater the energy the electromagnetic wave carries. In electromagnetic waves, the amplitude is the maximum field strength of the electric and magnetic fields ( [link] ). The wave energy is determined by the wave amplitude.

For a plane wave traveling in the direction of the positive x -axis with the phase of the wave chosen so that the wave maximum is at the origin at $t=0$ , the electric and magnetic fields obey the equations

$\begin{array}{c}{E}_{y}\left(x,t\right)={E}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\left(kx-\omega t\right)\hfill \\ {B}_{z}\left(x,t\right)={B}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\left(kx-\omega t\right).\hfill \end{array}$

The energy in any part of the electromagnetic wave is the sum of the energies of the electric and magnetic fields. This energy per unit volume, or energy density     u , is the sum of the energy density from the electric field and the energy density from the magnetic field. Expressions for both field energy densities were discussed earlier ( ${u}_{E}$ in Capacitance and ${u}_{B}$ in Inductance ). Combining these the contributions, we obtain

$u\left(x,t\right)={u}_{E}+{u}_{B}=\frac{1}{2}{\epsilon }_{0}{E}^{2}+\frac{1}{2{\mu }_{0}}{B}^{2}.$

The expression $E=cB=\frac{1}{\sqrt{{\epsilon }_{0}{\mu }_{0}}}B$ then shows that the magnetic energy density ${u}_{B}$ and electric energy density ${u}_{E}$ are equal, despite the fact that changing electric fields generally produce only small magnetic fields. The equality of the electric and magnetic energy densities leads to

$u\left(x,t\right)={\epsilon }_{0}{E}^{2}=\frac{{B}^{2}}{{\mu }_{0}}.$

The energy density moves with the electric and magnetic fields in a similar manner to the waves themselves.

We can find the rate of transport of energy by considering a small time interval $\text{Δ}t$ . As shown in [link] , the energy contained in a cylinder of length $c\text{Δ}t$ and cross-sectional area A passes through the cross-sectional plane in the interval $\text{Δ}t.$

four effect of heat on substances
why we can find a electric mirror image only in a infinite conducting....why not in finite conducting plate..?
because you can't fit the boundary conditions.
Jorge
what is the dimensions for VISCOUNSITY (U)
Branda
what is thermodynamics
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.?
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
prove that the formula for the unknown resistor is Rx=R2 x R3 divided by R3,when Ig=0.
what is flux
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
increase the capacitance.
Jorge
besides increasing the capacitance, is there any?
Bundi
mechanical stiffness and small size
Jorge
why for an ideal gas internal energy is directly proportional to thermodynamics temperature?
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?
what is flux
Bundi
Bundi, flux is the number of electric field crossing a surface area
Mubanga
you right
martin,F/F=(r×r)÷(r×r)
Mostafa
determining dimensional correctness
determine dimensional correctness of,T=2π√L/g
PATRICK
somebody help me answer the question above
PATRICK
d=dQ+w
calculate the heat flow per square meter through a mineral roll insulation 5cm thick if the temperature on the two surfaces are 30degree Celsius and 20 degree Celsius respectively. thermal conduction of mineral roll is 0.04
what are the elementary compositions of a cell?
poles, chemical
prabir
when a current pass through a material does the velocity varies
no.
prabir
what is spin entropy ?and disorder in ferromagnetic material