Question

In an electromagnetic wave, the amplitude of electric field is $1V{m}^{-1}$ . the frequency wave is $5\times {10}^{14}Hz$. The wave is propagating along the z−axis. The average energy density of an electric field. In joule per meter, will be
A. $1.1\times {10}^{-11}$
B. $2.2\times {10}^{-12}$
C. $3.3\times {10}^{-13}$
D. $4.4\times {10}^{-14}$

Answer 1

Hint:To solve 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:
$U={\displaystyle \frac{{\epsilon }_0{E}^2}{2}}$
Where, $U$ is the energy density, $E$ is the electric field and ${\epsilon }_0=8.8\times {10}^{-12}{C}^{2}N{m}^{-2}$ is the permittivity of free space.

Complete step by step answer:
According to the question,
$$\begin{gathered} U_{avg}={\displaystyle \frac{{\epsilon }_0{E}^2}{2}} \\ \Rightarrow U_{avg}={\displaystyle \frac{1}{2}}{\epsilon }_0{\left({\displaystyle \frac{{E_0}^2}{\sqrt{2}}}\right)}^2 \\ \Rightarrow U_{avg}={\displaystyle \frac{1}{2}}{\epsilon }_0{\displaystyle \frac{{E_0}^2}{2}} \\ \Rightarrow U_{avg}={\displaystyle \frac{{\epsilon }_0{(E_0)}^2}{4}} \end{gathered}$$
Here,
${E_0}^2=1^2{V}^2{m}^{-2}$
Now putting all the values we will get the desired result.
$$\begin{gathered} U_{avg}={\displaystyle \frac{8.85\times {10}^{-12}\times 1^2}{4}} \\ \therefore U_{avg}=2.21\times {10}^{-12}J{m}^{-3} \end{gathered}$$
Therefore, we get the required energy density, which is given by the above equation

Hence, the correct option is B.

Additional information: 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.