In a plane EM wave, the electric field oscillates sinusoidally at a frequency of \( 5 \times 10^{10} \, \text{Hz} \) and an amplitude of \( 50 \, \text{Vm}^{-1} \). The total average energy density of the electromagnetic field of the wave is: \([ \text{Use } \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2 / \text{Nm}^2 ]\)
- \( 1.106 \times 10^{-8} \, \text{Jm}^{-3} \)
- \( 4.425 \times 10^{-8} \, \text{Jm}^{-3} \)
- \( 2.212 \times 10^{-8} \, \text{Jm}^{-3} \)
- \( 2.212 \times 10^{-10} \, \text{Jm}^{-3} \)
The Correct Option is A
Approach Solution - 1
To find the total average energy density of an electromagnetic wave, we must consider both the electric field and the magnetic field components. For a plane electromagnetic wave, the energy density can be given by:
\(u = \frac{1}{2} \epsilon_0 E^2 + \frac{1}{2} \mu_0 H^2\)
However, in a vacuum, the energy density contributed by the electric field is equal to that contributed by the magnetic field. Thus, the total average energy density can be defined using only the electric field:
\(u = \epsilon_0 E^2\)
Given:
- \(E = 50 \, \text{Vm}^{-1}\) (Amplitude of the electric field)
- \(\epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2 / \text{Nm}^2\) (Permittivity of free space)
Substituting the values, the average energy density is calculated as follows:
\(u = \frac{1}{2} \times 8.85 \times 10^{-12} \times (50)^2\)
We'll proceed with the calculations:
\(u = \frac{1}{2} \times 8.85 \times 10^{-12} \times 2500\)
\(u = \frac{1}{2} \times 2.2125 \times 10^{-8}\)
Finally:
\(u = 1.106 \times 10^{-8} \, \text{Jm}^{-3}\)
This matches the given correct answer. Therefore, the total average energy density is:
- \(1.106 \times 10^{-8} \, \text{Jm}^{-3}\)
Hence, the correct option is:
\(< 1.106 \times 10^{-8} \, \text{Jm}^{-3} \)
Approach Solution -2
The average energy density of the electric field is given by:
\[ U_E = \frac{1}{2} \epsilon_0 E^2 \]
Substituting the given values:
\[ U_E = \frac{1}{2} \times 8.85 \times 10^{-12} \times (50)^2 \]
Calculating:
\[ U_E = 1.106 \times 10^{-8} \, \text{Jm}^{-3} \]
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