fraunhoffer diffraction pattern is obtained with slit of width0.3mm and a light wave length 5860a° .calculate the angles at which the dark band the next bright band are formed

To determine the angles at which the dark and bright bands are formed in a Fraunhofer diffraction pattern, we can use the principles of single-slit diffraction. The key to solving this problem lies in understanding the relationship between the slit width, the wavelength of light, and the angles of the diffraction pattern.

Understanding the Basics of Single-Slit Diffraction

In single-slit diffraction, light passing through a narrow slit spreads out and creates a pattern of bright and dark bands on a screen. The dark bands, or minima, occur at specific angles where destructive interference takes place, while the bright bands, or maxima, occur at angles where constructive interference happens.

Key Formulas

The positions of the dark bands in a single-slit diffraction pattern can be calculated using the formula:

• sin(θ) = mλ / a

Where:

• θ = angle of the dark band
• m = order of the dark band (m = 1, 2, 3, ...)
• λ = wavelength of light
• a = width of the slit

Given Values

From your question, we have:

• Slit width (a) = 0.3 mm = 0.3 × 10^-3 m
• Wavelength (λ) = 5860 Å = 5860 × 10^-10 m

Calculating the Angles for Dark Bands

To find the angles for the first dark band (m = 1), we can substitute the values into the formula:

sin(θ) = (1 × 5860 × 10^-10 m) / (0.3 × 10^-3 m)

Calculating this gives:

sin(θ) = 1.95 × 10^-6

Now, we can find θ:

θ = arcsin(1.95 × 10^-6)

Since the value is very small, we can approximate:

θ ≈ 1.95 × 10^-6 radians

To convert this to degrees, we multiply by (180/π):

θ ≈ 1.95 × 10^-6 × (180/π) ≈ 0.0001124°

Finding the Next Bright Band

The bright bands occur between the dark bands. The first bright band is located approximately halfway between the first and second dark bands. To find the angle for the second dark band (m = 2), we repeat the calculation:

sin(θ) = (2 × 5860 × 10^-10 m) / (0.3 × 10^-3 m)

sin(θ) = 3.90 × 10^-6

Calculating θ gives:

θ = arcsin(3.90 × 10^-6)

θ ≈ 3.90 × 10^-6 radians

Converting to degrees:

θ ≈ 3.90 × 10^-6 × (180/π) ≈ 0.000223°

Summary of Results

In summary, the angles for the dark and bright bands are:

• First dark band (m = 1): θ ≈ 0.0001124°
• Next bright band (between m = 1 and m = 2): θ ≈ 0.000223°

This analysis illustrates how diffraction patterns arise from the wave nature of light, and how we can use mathematical relationships to predict the behavior of light as it interacts with obstacles like slits.