If the electric field associated with a radiation of...

If the electric field associated with a radiation of frequency $10 \mathrm{MH} z$ is $\mathrm{E}=10 \sin (\mathrm{kx}-\omega \mathrm{t}) \mathrm{mV} / \mathrm{m}$ then its energy density is $\mathrm{Jm}^{-3}\left(\varepsilon_{0}=8.85 \times 10^{-12} \mathrm{C}^{2} \mathrm{~N}^{-1} \mathrm{~m}^{-2}\right)$ (A) $4.425 \times 10^{-10}$ (B) $6.26 \times 10^{-14}$ (C) $8.85 \times 10^{-16}$ (D) $8.85 \times 10^{-14}$

If the electric field associated with a radiation of frequency $10 \mathrm{MH} z$ is $\mathrm{E}=10 \sin (\mathrm{kx}-\omega \mathrm{t}) \mathrm{mV} / \mathrm{m}$ then its energy density is
$\mathrm{Jm}^{-3}\left(\varepsilon_{0}=8.85 \times 10^{-12} \mathrm{C}^{2} \mathrm{~N}^{-1} \mathrm{~m}^{-2}\right)$
(A) $4.425 \times 10^{-10}$
(B) $6.26 \times 10^{-14}$
(C) $8.85 \times 10^{-16}$
(D) $8.85 \times 10^{-14}$

Key Concepts

Electromagnetic Energy Density

This concept refers to the energy stored per unit volume in an electromagnetic field. It is typically calculated by summing the contributions from the electric and magnetic fields, and in free space, the electric field part is given by one half of the product of the vacuum permittivity and the square of the electric field magnitude. This measure is fundamental in understanding how energy is distributed in fields propagated by electromagnetic waves.

Vacuum Permittivity

Vacuum permittivity, usually denoted as ??, is a physical constant that quantifies the ability of the classical vacuum to permit electric field lines. It plays a critical role in the equations of electromagnetism, particularly in Coulomb’s law and in the expression for electromagnetic energy density. Its value sets the scale for interactions between electric charges in free space.

Time Averaging of Sinusoidal Functions

For sinusoidally varying fields, such as those in electromagnetic radiation, the instantaneous quantities often vary with time. Since practical measurements are usually interested in average effects, the time average of the square of a sine or cosine function is used. This average value, typically 1/2 for a full cycle, is essential when calculating quantities like energy density which depend on the square of the field amplitude.

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