A plane electromagnetic wave propagating in x-direction is described by \( E_y = (200 \, \text{V/m}) \sin(1.5 \times 10^7 t - 0.05 x) \). The intensity of the wave is:
- 35.4 W/m²
- 53.1 W/m²
- 26.6 W/m²
- 106.2 W/m²
The Correct Option is B
Approach Solution - 1
To determine the intensity of the plane electromagnetic wave described by \( E_y = (200 \, \text{V/m}) \sin(1.5 \times 10^7 t - 0.05 x) \), we need to use the formula for the intensity of an electromagnetic wave:
The intensity \( I \) of an electromagnetic wave is given by the formula:
\(I = \frac{1}{2} \varepsilon_0 c E_0^2\)
Where:
- \(\varepsilon_0\) is the permittivity of free space, approximately \(8.85 \times 10^{-12} \, \text{F/m}\).
- \(c\) is the speed of light in a vacuum, approximately \(3 \times 10^8 \, \text{m/s}\).
- \(E_0\) is the amplitude of the electric field, given as \(200 \, \text{V/m}\) in the problem.
Substituting these values into the formula, we get:
\(I = \frac{1}{2} \times 8.85 \times 10^{-12} \, \text{F/m} \times 3 \times 10^8 \, \text{m/s} \times (200 \, \text{V/m})^2\)
Calculating step-by-step:
- First calculate \((200 \, \text{V/m})^2 = 40000 \, \text{V}^2/\text{m}^2\).
- Now compute \(8.85 \times 10^{-12} \times 3 \times 10^8 = 2.655 \times 10^{-3}\).
- Substitute these into the intensity equation:
\(I = \frac{1}{2} \times 2.655 \times 10^{-3} \times 40000\)
\(I = 1.3275 \times 10^{-3} \times 40000\)
\(I = 53.1 \, \text{W/m}^2\)
This matches the option 53.1 W/m², confirming it as the correct answer.
Approach Solution -2
The intensity \(I\) of an electromagnetic wave is given by:
\[ I = \frac{1}{2} \epsilon_0 c E_0^2 \]
where \(\epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2\), \(c = 3 \times 10^8 \, \text{m/s}\), and \(E_0 = 200 \, \text{V/m}\).
Substituting the values:
\[ I = \frac{1}{2} \times 8.85 \times 10^{-12} \times (3 \times 10^8) \times (200)^2 \] \[ I = 53.1 \, \text{W/m}^2 \]
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