A convex lens of diameter 8.0 cm is used to focus a parallel beam of light of wavelength 620 nm. If the light be focused at a distance of 20 cm from the lens, what would be the radius of the central bright spot formed ?

Explanation

To find the radius of the central bright spot formed by a convex lens when focusing a parallel beam of light, we can use the formula for the radius of the central bright spot in diffraction patterns:

\[ r = \frac{1.22 \lambda D}{d} \]

Where:
- \( r \) is the radius of the central bright spot,
- \( \lambda \) is the wavelength of the light,
- \( D \) is the distance from the lens to the screen where the spot is formed,
- \( d \) is the diameter of the lens.

### Step-by-Step Solution:

1. **Identify the given values:**
- Diameter of the lens, \( d = 8.0 \, \text{cm} = 8 \times 10^{-2} \, \text{m} \)
- Wavelength of the light, \( \lambda = 620 \, \text{nm} = 620 \times 10^{-9} \, \text{m} \)
- Distance from the lens to the screen, \( D = 20 \, \text{cm} = 20 \times 10^{-2} \, \text{m} \)

2. **Substitute the values into the formula:**
\[ r = \frac{1.22 \times (620 \times 10^{-9}) \times (20 \times 10^{-2})}{8 \times 10^{-2}} \]

3. **Calculate the numerator:**
- Calculate \( 1.22 \times 620 \times 10^{-9} \):
\[ 1.22 \times 620 = 756.4 \times 10^{-9} \]
- Now multiply by \( 20 \times 10^{-2} \):
\[ 756.4 \times 10^{-9} \times 20 \times 10^{-2} = 15128 \times 10^{-11} = 1.5128 \times 10^{-7} \, \text{m} \]

4. **Calculate the denominator:**
- The denominator is \( 8 \times 10^{-2} \).

5. **Divide the numerator by the denominator:**
\[ r = \frac{1.5128 \times 10^{-7}}{8 \times 10^{-2}} = 1.89 \times 10^{-6} \, \text{m} \]

6. **Final result for the radius of the central bright spot:**
The radius of the central bright spot is approximately:
\[ r \approx 1.89 \, \mu m \]