Question

A plane electromagnetic wave, has a frequency of $2.0\times {10}^{10}Hz$ and its energy density is $1.02\times {10}^{–8}\raisebox{1ex}{$J$}\!\left/ \!\raisebox{-1ex}{$m^3$}\right.$ in vacuum. The amplitude of the magnetic field of the wave is close

$($$\begin{array}{l}\frac{1}{4\pi {\epsilon }_0}=9\times {10}^9\frac{N{m}^2}{C^2}\end{array}$and speed of light $=3x{10}^{8}m{s}^{-1})$

• A. $160nT$
• B. $150nT$
• C. $180nT$
• D. $190nT$

Answer 1

The correct option is A

$160nT$

Step 1. Given Data:

Electromagnetic wave frequency $f=2.0\times {10}^{10}Hz$

Speed of light $C=3x{10}^8\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$

Energy density $U=1.02\times {10}^{–8}\raisebox{1ex}{$J$}\!\left/ \!\raisebox{-1ex}{$m^3$}\right.$

Assume,$\begin{array}{l}\frac{1}{4\pi {\epsilon }_0}=9\times {10}^9\frac{N{m}^2}{C^2}\end{array}$

Let $B$ is Magnetic Field.

Step 2. Energy Density and Magnetic Field,

We know that, Energy density,

$U=\begin{array}{l}\frac{B^2}{2{\mu }_0}\end{array}\dots \left(1\right)$ (where,${\mu }_0$ is the permeability of free space)

And, Relation between the permeability of free space ${\mu }_0$ and permittivity of free space ${ϵ}_0$ is,

$\begin{array}{l}\Rightarrow C=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}\end{array}$

$\begin{array}{l}\Rightarrow {\mu }_0=\frac{1}{C^2{\epsilon }_0}\end{array}\dots \left(2\right)$

Step 3. To calculate $B$:

From equation $\left(1\right)$ we can write,

$$\begin{gathered} U=\frac{B^2}{2{\mu }_0} \\ \Rightarrow B=\sqrt{U\times 2{\mu }_0}\dots \left(3\right) \end{gathered}$$

Using equation $\left(2\right)$ in equation $\left(3\right)$ we have,

$$\begin{gathered} \Rightarrow B=\sqrt{U\times 2\left(\frac{1}{C^2{\epsilon }_0}\right)}T \\ \begin{array}{l}\Rightarrow B=\sqrt{1.02\times {10}^{-8}\times 2\times \left(\frac{1}{9\times {10}^{16}}\times 4\pi \times 9\times {10}^9\right)}\end{array}T \\ \begin{array}{l}\Rightarrow B=\sqrt{1.02\times {10}^{-8}\times 2\times \left(4\pi \times {10}^{-7}\right)}\end{array}T \\ \Rightarrow B=\sqrt{1.02\times {10}^{-8}\times 2\times \left(4\times 3.14\times {10}^{-7}\right)}T \\ \Rightarrow B=\sqrt{25.6\times {10}^{-15}}T \\ \Rightarrow B=\sqrt{256\times {10}^{-16}}T \\ \Rightarrow B=16\times {10}^{-8}T \\ \Rightarrow B=160\times {10}^{-9}T \\ \Rightarrow B=160nT \end{gathered}$$

Therefore, $B=160nT$

Hence, the correct option is (A).