A radiation of wavelength 200nm is propagating in the form of a parallel surface. The power of the beam is 5mW and its cross-sectional area is 1.0`mm^(2)`. Find the pressure exerted by radiation on the metallic surface if the radiation is completely reflected.

Options

A. `3.33xx10^-5`
B. `4.33xx10^-5`
C. `5xx10^-2`
D. None of the above

Explanation

To solve the problem of finding the pressure exerted by radiation on a metallic surface when the radiation is completely reflected, we can follow these steps:

### Step 1: Calculate the Energy of One Photon
The energy \( E \) of a photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)),
- \( c \) is the speed of light (\( 3.00 \times 10^{8} \, \text{m/s} \)),
- \( \lambda \) is the wavelength of the radiation (200 nm = \( 200 \times 10^{-9} \, \text{m} \)).

Substituting the values:
\[ E = \frac{(6.626 \times 10^{-34} \, \text{J s})(3.00 \times 10^{8} \, \text{m/s})}{200 \times 10^{-9} \, \text{m}} = \frac{1.9878 \times 10^{-25} \, \text{J m}}{200 \times 10^{-9} \, \text{m}} = 9.939 \times 10^{-19} \, \text{J} \]

### Step 2: Calculate the Number of Photons Incident per Second
The number of photons \( n \) incident on the surface per second can be calculated using the formula:
\[ n = \frac{P}{E} \]
where \( P \) is the power of the beam (5 mW = \( 5 \times 10^{-3} \, \text{W} \)).

Substituting the values:
\[ n = \frac{5 \times 10^{-3} \, \text{W}}{9.939 \times 10^{-19} \, \text{J}} \approx 5.03 \times 10^{15} \, \text{photons/s} \]

### Step 3: Calculate the Change in Momentum per Photon
The momentum \( p \) of a photon is given by:
\[ p = \frac{h}{\lambda} \]
Substituting the values:
\[ p = \frac{6.626 \times 10^{-34} \, \text{J s}}{200 \times 10^{-9} \, \text{m}} = 3.313 \times 10^{-24} \, \text{kg m/s} \]

Since the radiation is completely reflected, the change in momentum \( \Delta p \) for each photon is:
\[ \Delta p = 2p = 2 \times 3.313 \times 10^{-24} \, \text{kg m/s} = 6.626 \times 10^{-24} \, \text{kg m/s} \]

### Step 4: Calculate the Total Change in Momentum per Second
The total change in momentum per second (which is the force \( F \)) is given by:
\[ F = n \cdot \Delta p \]
Substituting the values:
\[ F = (5.03 \times 10^{15} \, \text{photons/s}) \cdot (6.626 \times 10^{-24} \, \text{kg m/s}) \approx 3.33 \times 10^{-8} \, \text{N} \]

### Step 5: Calculate the Pressure Exerted by the Radiation
Pressure \( P \) is defined as force per unit area:
\[ P = \frac{F}{A} \]
where \( A \) is the cross-sectional area (1.0 mm² = \( 1.0 \times 10^{-6} \, \text{m}^2 \)).

Substituting the values:
\[ P = \frac{3.33 \times 10^{-8} \, \text{N}}{1.0 \times 10^{-6} \, \text{m}^2} = 0.0333 \, \text{N/m}^2 = 3.33 \times 10^{-2} \, \text{Pa} \]

### Final Answer
The pressure exerted by the radiation on the metallic surface is approximately:
\[ P \approx 3.33 \times 10^{-2} \, \text{Pa} \]