Question

The electric field in a plane electromagnetic wave is given by
\(\vec{E} = 200 \cos \left[ \left( \frac{0.5 \times 10^3}{\text{m}} \right) x - (1.5 \times 10^{11} \frac{\text{rad}}{\text{s}} \times t) \right] \frac{\text{V}}{\text{m}} \hat{j}\).
If this wave falls normally on a perfectly reflecting surface having an area of \(100 \text{ cm}^2\). If the radiation pressure exerted by the E.M. wave on the surface during a 10 minute exposure is \(\frac{x}{10^9} \frac{\text{N}}{\text{m}^2}\). Find the value of \(x\).

Hint:

Radiation Pressure is Intensity divided by speed of light (\(I/c\)) for absorption and \(2I/c\) for reflection. Notice that the pressure \(P = \epsilon_0 E_0^2\) doesn't depend on wavelength or frequency.

Answer 1

Correct Answer: 354

Step 1: Understanding the Concept:
Electromagnetic waves carry momentum. When they strike a surface, they exert pressure. For a perfectly reflecting surface, the radiation pressure is twice that of a perfectly absorbing surface.
Step 2: Key Formula or Approach:
1. Average intensity \(I = \frac{1}{2} \epsilon_0 c E_0^2\).
2. Radiation pressure for perfect reflection: \(P = \frac{2I}{c} = \epsilon_0 E_0^2\).
Step 3: Detailed Explanation:
Given: \(E_0 = 200 \text{ V/m}\).
Permittivity of free space \(\epsilon_0 \approx 8.854 \times 10^{-12} \text{ C}^2/\text{N}\cdot\text{m}^2\).

Calculate Radiation Pressure \(P\):
\[ P = \epsilon_0 E_0^2 \]
\[ P = (8.854 \times 10^{-12}) \times (200)^2 \]
\[ P = 8.854 \times 10^{-12} \times 40000 \]
\[ P = 354.16 \times 10^{-9} \text{ N/m}^2 \]

Given that \(P = \frac{x}{10^9} \text{ N/m}^2\):
\[ \frac{x}{10^9} = 354.16 \times 10^{-9} \]
\[ x \approx 354 \]
Step 4: Final Answer:
The value of \(x\) is 354.