The electric field in an electromagnetic wave is given as $$\vec{E} = 20\sin(\omega t - \frac{x}{c})\hat{j}$$ N C$$^{-1}$$, where $$\omega$$ and $$c$$ are angular frequency and velocity of electromagnetic wave respectively. The energy contained in a volume of $$5 \times 10^{-4}$$ m$$^3$$ will be
(Given $$\varepsilon_0 = 8.85 \times 10^{-12}$$ C$$^2$$ N$$^{-1}$$ m$$^{-2}$$)

We have an electromagnetic wave with electric field $$\vec{E} = 20\sin\left(\omega t - \frac{x}{c}\right)\hat{j}$$ N C$$^{-1}$$, so the amplitude is $$E_0 = 20$$ N/C.

The average energy density of an electromagnetic wave is $$u_{avg} = \varepsilon_0 E_0^2 / 2$$ (since the electric and magnetic contributions are equal, each giving $$\varepsilon_0 E_0^2 / 4$$). Substituting:

$$u_{avg} = \frac{8.85 \times 10^{-12} \times (20)^2}{2} = \frac{8.85 \times 10^{-12} \times 400}{2} = 1770 \times 10^{-12} \text{ J/m}^3$$

Now, the total energy in the given volume is:

$$U = u_{avg} \times V = 1770 \times 10^{-12} \times 5 \times 10^{-4} = 8.85 \times 10^{-13} \text{ J}$$

So, the answer is $$8.85 \times 10^{-13}$$ J.