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:

$$u = \frac{1}{2} \varepsilon_0 E^2$$

where $$\varepsilon_0$$ 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 $$20 \mathrm{~N/C}$$.

Now, we can find the energy density:

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

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

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

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

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

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

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