# 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 $\text{Δ}t.$

what is unit
is electric field directly proportional to the squared of a distance
The bullet 2.00cm long is fired at 420/s and passes straight through a 10.0 cm thick board existing at 280 m/s.What is the average acceleration of the bullet through the board?
FAUSTINA
an unstretched spring is 12cm long .A load of 5N stretched it to 15cm .how long will it be under a load of 15N?
Morapeli
hi
Africa
hi
Benjamin
Benjamin how are u are u a freshman in the university
Africa
like 100 level
Africa
yes sir
Benjamin
l need a physics tutor
Benjamin
I think the best tutor is God and organic tutor in YouTube that guy is good
Africa
me too 100level
Africa
wow nice
Benjamin
from Nigeria and u
Africa
I am from Nigeria and u wow nice that something I use to always say
Africa
am from Ghana
Benjamin
ok
Africa
studying what
Africa
Compare the electric flux through the surface of a cube of side length a that has a charge q at its center to the flux through a spherical surface of radius a with a charge q at its center.
please I want to know how to solve increase in length
Ujah
Why a charged capacitor has potential difference but not emf
what is the dimension symbol of temperature?
what is the dimension symbol of temperature?
Keren
what's the meaning of enthalpy in terms of latent heat, internal energy, phase change
how to convert Kelvin to centigrade
what is the s, p, d, f in this table
Sangeetha
s, p, d, f in this table
Sangeetha
what kind of table this
Vengata
what are waves
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities
Discuss how would orient a planar surface of area A in a uniform electric field of magnitude E0 to obtain (a) the maximum flux and (b) the minimum flux through the area.
I'm just doing the first 3 with this message. but thankyou for the time your obviously intending to support us with. viva la accumulation
Marcel
Find the net capacitance of the combination of series and parallel capacitors shown belo
what is ohm?
calculate ideal gas pressure of 0.300mol,v=2L T=40°c
what is principle of superposition
what are questions that are likely to come out during exam