4.4.7 Double-Slit Equation

Interference Patterns

• Interference patterns depend on:
  • The coherent light incident on the slits
  • The distance between the slits and the screen where the interference pattern is observed
  • The separation between the slits

• In particular:
  • Successive bright fringes are further apart from each other if the wavelength λ of the incident light is longer

Difference in the interference pattern of red and blue light. Red light has a longer wavelength than blue light, so the fringes are more spaced out

• • Similarly, bright fringes are more spaced out if the screen is placed further away from the slits
  • If the separation between the slits is increased, instead successive bright fringes are closer to each other

Explaining the Interference Pattern

• Two slits are separated by a length d, and a screen is placed a distance D away

Light rays of wavelength λ incident on two slits a distance s apart interferes and form bright fringes on a screen placed a distance D away from the slits

• When light rays of wavelength λ are incident on the two slits:
  • A central bright fringe where the two waves have travelled the same distance to the screen and their path difference is zero 
  • Another bright fringe is formed either side of the central one where the path difference between the waves is  exactly one wavelength 
    • This is constructive interference 
  • Dark fringes (an absence of light) are formed where the path difference between the waves is some number of wavelengths plus half a wavelength 
    • This is destructive interference 

• Further bright fringes will be located at each position on the screen where the path difference is exactly equal to n number of wavelengths nλ
• For example: 2λ, 3λ, 4λ...
• Dark fringes are located in between the bright ones, where the path difference is exactly (n + )λ
• For example: 

Double-Slit Equation

• The separation s of successive fringes is given by:

• Where:
  • s = separation between successive fringes on the screen (m)
  • λ = wavelength of the waves incident on the slits (m)
  • D = distance between the screen and the slits (m)
  • d = separation between the slits (m)

• Note that s is the separation between two successive bright fringes or two successive dark ones
• The above equation shows that the separation between the fringes will increase if:
  • The wavelength of the incident light increases
  • The distance between the screen and the slits increases
  • The separation between the slits decreases

Dependence of the interference pattern on the separation between the slits. The further apart the slits, the closer together the bright fringes

Step 1: Write down the known quantities 

• • d = 0.40 mm = 4.0 x 10^–4 m
  • D = 0.50 m
  • s = 0.50 cm = 5.0 x 10^–3 m

Note that you must convert all lengths into metres (m)

Step 2: Write down the double-slit equation

Step 3: Rearrange the above equation to calculate the wavelength λ

Step 4: Substitute the numbers into the above equation 

λ = 4.0 × 10^–6 m = 4.0 μm

This corresponds to the infrared area of the electromagnetic spectrum