An electromagnetic wave of frequency $$1 \times 10^{14}$$ hertz is propagating along z-axis. The amplitude of electric field is 4 V/m. If $$\varepsilon_0 = 8.8 \times 10^{-12}$$ C$$^2$$/N-m$$^2$$, then average energy density of electric field will be:

We are given an electromagnetic wave propagating along the z-axis with a frequency of $$1 \times 10^{14}$$ Hz. The amplitude of the electric field is $$E_0 = 4$$ V/m, and the permittivity of free space is $$\varepsilon_0 = 8.8 \times 10^{-12}$$ C$$^2$$/N-m$$^2$$. We need to find the average energy density of the electric field.

The instantaneous energy density due to the electric field in an electromagnetic wave is given by $$u_E = \frac{1}{2} \varepsilon_0 E^2$$, where $$E$$ is the instantaneous electric field. Since the electric field oscillates sinusoidally, we must find the average over one complete cycle. The electric field varies as $$E = E_0 \cos(kz - \omega t)$$, so $$E^2 = E_0^2 \cos^2(kz - \omega t)$$. The average value of $$\cos^2(\theta)$$ over a full cycle is $$\frac{1}{2}$$. Therefore, the average energy density for the electric field component is:

$$$ \langle u_E \rangle = \frac{1}{2} \varepsilon_0 \langle E^2 \rangle = \frac{1}{2} \varepsilon_0 \times \frac{1}{2} E_0^2 = \frac{1}{4} \varepsilon_0 E_0^2 $$$

Now, substitute the given values: $$\varepsilon_0 = 8.8 \times 10^{-12}$$ C$$^2$$/N-m$$^2$$ and $$E_0 = 4$$ V/m.

First, compute $$E_0^2$$:

$$$ E_0^2 = (4)^2 = 16 $$$

Now plug into the formula:

$$$ \langle u_E \rangle = \frac{1}{4} \times (8.8 \times 10^{-12}) \times 16 $$$

Simplify the expression step by step. First, multiply $$\frac{1}{4}$$ and 16:

$$$ \frac{1}{4} \times 16 = 4 $$$

So the expression becomes:

$$$ \langle u_E \rangle = 4 \times (8.8 \times 10^{-12}) $$$

Now multiply 4 and 8.8:

$$$ 4 \times 8.8 = 35.2 $$$

Therefore:

$$$ \langle u_E \rangle = 35.2 \times 10^{-12} \text{ J/m}^3 $$$

Note that the frequency of the wave ($$1 \times 10^{14}$$ Hz) is not required for this calculation, as the average energy density of the electric field depends only on the amplitude $$E_0$$ and $$\varepsilon_0$$.

Comparing with the options:

A. $$35.2 \times 10^{-10}$$ J/m$$^3$$

B. $$35.2 \times 10^{-11}$$ J/m$$^3$$

C. $$35.2 \times 10^{-12}$$ J/m$$^3$$

D. $$35.2 \times 10^{-13}$$ J/m$$^3$$

Our result matches option C.

Hence, the correct answer is Option C.