To solve this problem, we need to understand the relationship between the wavelength of light, the distance between fringes in an interference pattern, and the distance from the slits to the screen. The key here is to use the formula for fringe spacing in a double-slit interference setup, which is given by:
Fringe Spacing Formula
The fringe spacing (β) can be calculated using the formula:
β = λL / d
Where:
- λ = wavelength of light
- L = distance from the slits to the screen
- d = distance between the slits
In this case, we can derive the fringe spacing for both light sources and set up a relationship between them. Let's denote:
- λ₁ = 5890 Å (the wavelength of sodium light)
- n₁ = 20 (number of fringes for sodium light)
- x₁ = 2.30 cm (distance occupied by fringes for sodium light)
- λ₂ = ? (the wavelength of the new light source)
- n₂ = 30 (number of fringes for the new light source)
- x₂ = 2.80 cm (distance occupied by fringes for the new light source)
Calculating Fringe Spacing
The fringe spacing for the sodium light can be calculated as:
β₁ = x₁ / n₁ = 2.30 cm / 20 = 0.115 cm
For the new light source, the fringe spacing is:
β₂ = x₂ / n₂ = 2.80 cm / 30 = 0.0933 cm
Setting Up the Relationship
Since the distance from the slits to the screen (L) and the distance between the slits (d) remain unchanged, we can set up a ratio of the fringe spacings:
β₁ / β₂ = λ₁ / λ₂
Substituting the values we have:
0.115 cm / 0.0933 cm = 5890 Å / λ₂
Solving for the New Wavelength
Now we can rearrange this equation to find λ₂:
λ₂ = 5890 Å * (0.0933 cm / 0.115 cm)
Calculating this gives:
λ₂ = 5890 Å * 0.811 = 4785.79 Å
Final Result
Thus, the wavelength of the light from the new monochromatic source is approximately 4786 Å.
This calculation illustrates how interference patterns can provide insights into the properties of light, such as wavelength, by analyzing the spacing and number of fringes produced in a controlled experiment.
