The electric field in a plane electromagnetic wave is given by
$$\vec{E} = 200 \cos[(0.5 \times 10^3 \text{ m}^{-1})x - (1.5 \times 10^{11} \text{ rad s}^{-1})t] \text{ V m}^{-1} \hat{j}$$.
If this wave falls normally on a perfectly reflecting surface having an area of 100 cm$$^2$$. If the radiation pressure exerted by the E.M. wave on the surface during a 10 min exposure is $$\frac{k}{10^9}$$ N m$$^{-2}$$. Find the value of $$k$$

Based on the provided solution steps for calculating radiation pressure, here is the formatted response:

To find the radiation pressure of an electromagnetic wave, we first relate the intensity of the wave to the electric field amplitude and then use the relationship between intensity and pressure for a perfectly reflecting surface.

1. Intensity of the Electromagnetic Wave ($$I$$)

The average intensity $$I$$ of an electromagnetic wave in a vacuum is given by:

$$I = \frac{1}{2} \varepsilon_0 E_0^2 c$$

Where:

• $$\varepsilon_0$$ = Permittivity of free space ($$8.85 \times 10^{-12} \text{ C}^2/\text{N}\cdot\text{m}^2$$)
• $$E_0$$ = Amplitude of the electric field
• $$c$$ = Speed of light

2. Radiation Pressure ($$P$$)

For a perfectly reflecting surface, the radiation pressure $$P$$ is twice the momentum density, or:

$$P = \frac{2I}{c}$$

Substituting the expression for intensity into the pressure formula:

$$P = \left( \frac{2}{c} \right) \left( \frac{1}{2} \varepsilon_0 E_0^2 c \right)$$

$$P = \varepsilon_0 E_0^2$$

3. Numerical Calculation

Given the electric field amplitude $$E_0 = 200 \text{ V/m}$$:

$$P = (8.85 \times 10^{-12}) \times (200)^2$$

$$P = 8.85 \times 10^{-12} \times 40,000$$

$$P = 8.85 \times 10^{-8} \times 4$$

$$P = 35.4 \times 10^{-8}$$

In fractional scientific notation:

$$\boxed{P = \frac{354}{10^9} \text{ N/m}^2}$$