JEE Main 2023 — Electromagnetic Waves Question with Solution

To find the energy contained in a volume of an electromagnetic wave, we need to calculate the energy density and then multiply it by the volume.

For an electromagnetic wave, the energy density $$u$$ is given by:

$$u=\frac{1}{2}{\epsilon }_0{E}^2$$

where $${\epsilon }_0$$ is the vacuum permittivity and $$E$$ is the electric field amplitude.

First, let's find the amplitude of the electric field. In the given equation:

$$\overrightarrow{\mathrm{E}}=20\sin \omega \left(\mathrm{t}-\frac{x}{\mathrm{c}}\right)\overrightarrow{\mathrm{j}}{\mathrm{NC}}^{-1}$$

The amplitude $$E$$ is $$20N/C$$.

Now, we can find the energy density:

$$u=\frac{1}{2}\cdot 8.85\times 10^{-12}{\mathrm{C}}^2/{\mathrm{Nm}}^2\cdot (20N/C{)}^2$$

$$u=\frac{1}{2}\cdot 8.85\times 10^{-12}\cdot 400$$

$$u=1.77\times 10^{-9}J/{\mathrm{m}}^3$$

The energy contained in a volume of $$5\times 10^{-4}{\mathrm{m}}^3$$ will be:

$$E_{\text{total}}=u\cdot V$$

$$E_{\text{total}}=1.77\times 10^{-9}J/{\mathrm{m}}^3\cdot 5\times 10^{-4}{\mathrm{m}}^3$$

$$E_{\text{total}}=8.85\times 10^{-13}\mathrm{J}$$