A laser beam of `10mW` power and wavelength `700nm` has aperture of `3mm`. If it is focused by a lens of focal length `5cm`, the intensity of image is

Options

A. (a) `1.56xx10^7W//m^2`
B. (b) `3xx10^6W//m^2`
C. (c) `3xx10^7W//m^2`
D. (d) `4xx10^7W//m^2`

Explanation

To find the intensity of the image formed by a laser beam focused by a lens, we will follow these steps:

### Step 1: Understand the Given Data
- Power of the laser beam (P) = 10 mW = \(10 \times 10^{-3}\) W
- Wavelength of the laser beam (λ) = 700 nm = \(700 \times 10^{-9}\) m
- Aperture diameter (D) = 3 mm = \(3 \times 10^{-3}\) m
- Focal length of the lens (f) = 5 cm = \(5 \times 10^{-2}\) m

### Step 2: Calculate the Radius of the Aperture
The radius (α) of the aperture is given by:
\[ \alpha = \frac{D}{2} = \frac{3 \times 10^{-3}}{2} = 1.5 \times 10^{-3} \text{ m} \]

### Step 3: Calculate the Angle θ using the Diffraction Formula
The angle θ for a circular aperture can be calculated using the formula:
\[ \theta = \frac{1.22 \lambda}{\alpha} \]
Substituting the values:
\[ \theta = \frac{1.22 \times 700 \times 10^{-9}}{1.5 \times 10^{-3}} = \frac{854 \times 10^{-9}}{1.5 \times 10^{-3}} = 5.693 \times 10^{-4} \text{ radians} \]

### Step 4: Calculate the Radius of the Image (r)
The radius of the image (r) formed by the lens can be calculated using:
\[ r = f \cdot \theta \]
Substituting the values:
\[ r = 5 \times 10^{-2} \cdot 5.693 \times 10^{-4} = 2.8465 \times 10^{-5} \text{ m} \]

### Step 5: Calculate the Area of the Image
The area (A) of the image is given by:
\[ A = \pi r^2 \]
Substituting the value of r:
\[ A = \pi (2.8465 \times 10^{-5})^2 \approx \pi (8.094 \times 10^{-10}) \approx 2.544 \times 10^{-9} \text{ m}^2 \]

### Step 6: Calculate the Intensity (I)
The intensity (I) is given by:
\[ I = \frac{P}{A} \]
Substituting the values:
\[ I = \frac{10 \times 10^{-3}}{2.544 \times 10^{-9}} \approx 3.93 \times 10^6 \text{ W/m}^2 \]

### Final Answer
The intensity of the image is approximately \(3.93 \times 10^6 \text{ W/m}^2\).

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