Calculating Wavelength Using Slit Separation and Fringe Degrees Khan Academy

Introduction

When light passes through two closely spaced slits, it creates an interference pattern with bright and dark fringes. The position of these fringes depends on the wavelength of the light and the separation between the slits. By measuring the angle of a fringe and knowing the slit separation, we can determine the wavelength.

This technique is used in spectroscopy to analyze the composition of light sources. The calculation involves converting the fringe angle to radians and using the slit separation to find the wavelength.

Formula

The wavelength (λ) can be calculated using the following formula:

Where:

• λ = wavelength of light (in meters)
• d = separation between the two slits (in meters)
• θ = angle of the fringe (in degrees)
• m = order of the fringe (integer, typically 1 for the first bright fringe)

Note: The angle θ must be converted from degrees to radians before calculation. The sine function is used because the path difference between the two slits creates constructive interference when the path difference equals an integer multiple of the wavelength.

Calculation Steps

1. Convert the fringe angle from degrees to radians: θ_radians = θ × (π / 180)
2. Calculate the sine of the angle: sinθ = sin(θ_radians)
3. Multiply the slit separation by the sine of the angle: d × sinθ
4. Divide the result by the fringe order: (d × sinθ) / m
5. The result is the wavelength in meters

Example Calculation

Let's calculate the wavelength for a light source with:

• Slit separation (d) = 0.0005 meters (500 micrometers)
• Fringe angle (θ) = 10 degrees
• Fringe order (m) = 1

Step 1: Convert angle to radians: 10 × (π / 180) ≈ 0.1745 radians

Step 2: Calculate sine: sin(0.1745) ≈ 0.1736

Step 3: Multiply by slit separation: 0.0005 × 0.1736 ≈ 0.0000868

Step 4: Divide by fringe order: 0.0000868 / 1 = 0.0000868 meters

The wavelength is approximately 86.8 nanometers (0.0000868 meters).

Interpreting Results

The calculated wavelength tells you the characteristic length of the light wave. Different wavelengths correspond to different colors of light in the visible spectrum. For example:

• Violet light has wavelengths around 400-450 nm
• Blue light is 450-495 nm
• Green is 495-570 nm
• Yellow is 570-590 nm
• Orange is 590-620 nm
• Red is 620-750 nm

If your calculated wavelength falls outside the visible spectrum, it indicates the light source is ultraviolet or infrared. The fringe order affects the result - higher-order fringes will give smaller wavelengths for the same angle and slit separation.