In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of $2.0 \times 10^{10} \mathrm{~Hz}$ and amplitude $48 \mathrm{~V} \mathrm{~m}^{-1}$. Show that the average energy density of the $E$ field equals the average energy density of the B field. $\left[\mathrm{c}=3 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1}\right]$
In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of $2.0 \times 10^{10} \mathrm{~Hz}$ and amplitude $48 \mathrm{~V} \mathrm{~m}^{-1}$. Show that the average energy density of the $E$ field equals the average energy density of the B field. $\left[\mathrm{c}=3 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1}\right]$

Frequency of the electromagnetic wave is given as $\mathrm{v}=2 \times 10^{10} \mathrm{~Hz}$

Electric field amplitude is given as $E_{0}=48 \vee \mathrm{m}^{-1}$

Speed of light is known as $c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$

Energy density of the electric field is represented as,

$U_{E}=\frac{1}{2} \epsilon_{0} E^{2}$

And, energy density of the magnetic field is given as:

$U_{B}=\frac{1}{2 \mu_{0}} B^{2}$

Where,

$\epsilon_{0}$ is the permittivity of free space

$\mu_{0}=$ is the permeability of free space

$E=c B$…(1)

Now,

$c=\frac{1}{\sqrt{\epsilon_{0} \mu_{0}}} . \ldots(2)$

On putting equation (2) in equation (1), we get

$E=\frac{1}{\sqrt{\epsilon_{0} \mu_{0}}} B$

On squaring on both sides, we get

$E^{2}=\frac{1}{\epsilon_{0} \mu_{0}} B^{2}$

$\epsilon_{0} E^{2}=\frac{B^{2}}{\mu_{0}}$

$\frac{1}{2} \epsilon_{0} E^{2}=\frac{1}{2} \frac{B^{2}}{\mu_{0}}$

$\Rightarrow U_{E}=U_{B}$