Question

The energy associated with electric field (${{U}_{E}}$) and with magnetic field is (${{U}_{B}}$) for an electromagnetic wave in free space. Then:
    $\begin{align}
  & A){{U}_{E}}=\dfrac{{{U}_{B}}}{2} \\
 & B){{U}_{E}}<{{U}_{B}} \\
 & C){{U}_{E}}={{U}_{B}} \\
 & D){{U}_{E}}>{{U}_{B}} \\
\end{align}$

Answer 1

Hint: To solve this question, we must understand the basic equations for energy associated with electric and magnetic fields. We will relate them by using the relation between their field intensities, which states electric field intensity is equal to the product of speed of light and intensity of the magnetic field. i.e. ${{E}_{0}}=C{{B}_{0}}$. Then we will use the relation between velocity of light, permittivity and permeability of vacuum which is given as \[C=\dfrac{1}{\sqrt{{{\mu }_{0}}{{\varepsilon }_{0}}}}\] in that equation. Then we deduce the relation between energies associated with both fields.

Complete step-by-step answer:
Energy associated with the electric field is ${{U}_{E}}$ (which is given).
Energy associated with the magnetic field is ${{U}_{B}}$ (which is also given).
So now the energy density of the magnetic field will be,
${{U}_{B}}=\dfrac{{{B}_{0}}^{2}}{2{{\mu }_{0}}}$
In the above formula, ${{B}_{0}}$ is the maximum value of a magnetic field.
Now the average density is given as,
${{U}_{E}}=\dfrac{{{\varepsilon }_{0}}{{E}^{2}}}{2}$
Thus we can say that, ${{E}_{0}}=C{{B}_{0}}$ and ${{C}^{2}}=\dfrac{1}{{{\mu }_{0}}{{\varepsilon }_{0}}}$ .
So now we can write following:
${{U}_{E}}=\dfrac{{{\varepsilon }_{0}}}{2}\times {{C}^{2}}{{B}^{2}}=\dfrac{{{\varepsilon }_{0}}}{2}\times \dfrac{1}{{{\mu }_{0}}{{\varepsilon }_{0}}}\times {{B}^{2}}=\dfrac{{{B}_{0}}^{2}}{2{{\mu }_{0}}}={{U}_{B}}$
Therefore, from the above equation we can say that ${{U}_{E}}={{U}_{B}}$.
Hence, the energy density of electric and magnetic fields is the same, thus the energy associated with the electric field and magnetic field for an electromagnetic wave in free space will be equal.

So, the correct answer is “Option C”.

Note: Free space here refers to a space that is fully vacuum, that is, a space devoid of any particles. In quantum physics, this term refers to the ground state of any electromagnetic field that is subject to fluctuations about a zero average-field condition. The concept of vacuum varies among various authors. Still the fundamental concept of vacuum remains the same. From quantum to classical physics, one thing remains constant always. This is the conservation of energy.