A plane wavefront of wavelength 6000 angstorm is incident on a circular aperture of radius 0.09cm.

The first max intensity is obtained at a distance of ?

Thank You.

To determine the distance at which the first maximum intensity occurs after a plane wavefront passes through a circular aperture, we can use the principles of diffraction. Specifically, we will apply the formula related to the diffraction pattern produced by a circular aperture, which is described by the Airy pattern.

Understanding the Diffraction Pattern

When light passes through a circular aperture, it doesn't just travel in straight lines; instead, it spreads out and creates a pattern of bright and dark regions. The first maximum intensity occurs at a specific angle, which can be calculated using the following formula:

Formula for the First Maximum

The angle θ for the first maximum in the diffraction pattern can be approximated using:

• θ = 1.22 * (λ / D)

Where:

• λ is the wavelength of the light (in meters)
• D is the diameter of the aperture (in meters)

Given Values

In this case, we have:

• Wavelength (λ) = 6000 angstroms = 6000 x 10^-10 meters = 6 x 10^-7 meters
• Radius of the aperture (r) = 0.09 cm = 0.0009 meters
• Diameter (D) = 2 * r = 2 * 0.0009 = 0.0018 meters

Calculating the Angle

Now, substituting the values into the formula:

• θ = 1.22 * (6 x 10^-7 / 0.0018)

Calculating this gives:

• θ ≈ 1.22 * 3.33 x 10^-4 ≈ 4.07 x 10^-4 radians

Finding the Distance to the First Maximum

To find the distance (L) from the aperture to the screen where the first maximum occurs, we can use the small angle approximation, where:

• L = r / θ

Assuming we want to find the distance at a screen placed at a distance L from the aperture, we can rearrange this to:

• L = (0.0009) / (4.07 x 10^-4)

Calculating this gives:

• L ≈ 2.21 meters

Final Result

Thus, the first maximum intensity after the circular aperture occurs at a distance of approximately 2.21 meters from the aperture. This distance is where you would observe the first bright fringe in the diffraction pattern created by the circular aperture.