To find the angular width of the central maximum in the diffraction pattern, we use the formula for the angular width of the central maximum caused by a circular aperture (lens):

$\theta = 1.22 \frac{\lambda}{D}$

Where:

• $\lambda = 500 \, \text{nm} = 500 \times 10^{-9} \, \text{m}$ (wavelength of light),
• $D = 120 \, \text{mm} = 0.120 \, \text{m}$ (diameter of the lens).

Step 1: Calculate the Angular Width of the Central Maximum

Substitute the values into the formula:

$\theta = 1.22 \frac{500 \times 10^{-9}}{0.120}$

$\theta = 1.22 \times \frac{500}{120} \times 10^{-9}$

$\theta = 1.22 \times 4.167 \times 10^{-6}$

$\theta \approx 5.08 \times 10^{-6} \, \text{radians}$

Step 2: Convert the Angular Width to Degrees

To convert from radians to degrees:

$\theta (\text{degrees}) = \theta (\text{radians}) \times \frac{180}{\pi}$

$\theta (\text{degrees}) = 5.08 \times 10^{-6} \times \frac{180}{3.1416}$

$\theta (\text{degrees}) \approx 2.91 \times 10^{-4} \, \text{degrees}$

Step 3: Total Angular Width

The total angular width of the central maximum is:

$2\theta$

So, the total angular width is:

$2 \times 5.08 \times 10^{-6} = 1.016 \times 10^{-5} \, \text{radians}$

Or in degrees:

$2 \times 2.91 \times 10^{-4} \approx 5.82 \times 10^{-4} \, \text{degrees}$

Final Answer:

• Angular width in radians: $1.016 \times 10^{-5} \, \text{radians}$,
• Angular width in degrees: $5.82 \times 10^{-4} \, \text{degrees}$.

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Would you like further clarification or numerical details?
Here are 5 related questions for further exploration:

1. How does the diameter of the lens affect the angular width of the central maximum?
2. What happens to the angular width if the wavelength of light is increased?
3. How is the focal length of the lens related to diffraction effects in this setup?
4. Could diffraction limit the resolution of an optical system, and how?
5. How would you calculate the position of the first diffraction minimum on a screen?

Tip: Diffraction effects become significant when the lens diameter is comparable to the wavelength of light. Always consider this in high-precision optical systems.