Question

An electromagnetic wave of frequency $1 \times {10^{14}}Hz$ is propagating along the z-axis. The amplitude of the electric field is 4 V/m. If ${\varepsilon _0} = 8.8 \times {10^{ - 12}}{C^2}/N - {m^2}$, then the average energy density of the electric field will be?

Answer 1

In this question, we will use the relation between energy density and electric field. Further, substituting the given values will give us the required answer. We will also discuss some of the basics of an electromagnetic wave.
Formula used-
$e = \dfrac{{{\varepsilon _0}{E^2}}}{2}$

Complete answer:
As we know that the energy density is defined as the amount of energy stored in a given system or region of space per unit volume. Energy density can also be used as energy per unit mass.
Here, energy density will be given as:
$e = \dfrac{{{\varepsilon _0}{E^2}}}{2}$
Here, e/2 is energy density and E is electric field experienced.
Now, substituting the given values in the above equation we get:
\eqalign{ & e = 8.8 \times {10^{ - 12}} \times 8 \cr & \Rightarrow e = 70.4 \times {10^{ - 12}}\dfrac{J}{{{m^3}}} \cr & \therefore \dfrac{e}{2} = 35.2 \times {10^{ - 12}}\dfrac{J}{{{m^3}}} \cr}
Therefore, we get the required energy density, which is given by the above equation.

Additional information:
Waves involve the transfer of energy without the transfer of the matter. So, it can be said that waves can be described as a disturbance that travels through a medium, transporting energy from one location to another location without transfer of matter.
Further, the frequency is defined as the number of waves that pass a fixed point in unit time. It can also be defined as the number of cycles or vibrations undergone during one unit of time.
Two waves are said to be coherent if they are moving with the same frequency and have constant phase difference.
The summation or adding or subtraction of all the waves travelling in a particular medium, gives us the superposition of waves. If the direction or amplitude of the waves are opposite then the superposition of waves is calculated by subtracting the waves, whereas if the two waves are travelling in the same direction or have the same amplitude the resultant is given by adding up the two or more waves.
The S.I unit of frequency is Hertz or Hz and the unit of wavelength is meter or m. Furthermore we also know the S.I unit of time which is given by second or s.
Phase of a wave specifies the location of a point within a wave cycle of a repetitive waveform. Generally, the phase differences between two or more sound waves are important. When two sound waves combine, like- the difference between the phases of the two waves is important in determining the resulting waveform.

Note:
We have to observe that energy and energy density, both are different measures. The phase of the wave can be positive or negative depending on its direction of propagation. A sine wave starts from zero, whereas the cosine wave starts from one. A wave which has the same amplitude but opposite orientation will cancel out each other and thereby give zero output.