Final answer:
The maximum number of dark fringes produced by light passing through a single slit and reaching a screen can be found by dividing the slit width by the wavelength of the light. Substituting the given values, the setup can produce a maximum of 5 dark fringes on the screen.
Step-by-step explanation:
To calculate the maximum number of dark fringes (fringes) that can be produced on the screen in a single-slit diffraction pattern, we can use the condition for dark fringes given by the formula:
d sin(θ) = nλ, where d is the slit width, θ is the diffraction angle, λ is the wavelength, and n is the order number of the dark fringe.
However, in this scenario, the maximum number of dark fringes occurs just before the light reaches an angle of 90° from the center. Mathematically:
d sin(90°) = nmaxλ
By rearranging this equation and substituting the values in, we get:
nmax = d / λ
Where:
- d = 3.1 × 10-6 m (slit width)
- λ = 545 × 10-9 m (wavelength)
Substituting the given values and calculating we have:
nmax = (3.1 × 10-6) / (545 × 10-9)
nmax ≈ 5.688 full dark fringes
We can only have whole dark fringes, so we round down to the nearest whole number:
nfringes = 5
Therefore, the maximum number of dark fringes or lines this setup can produce on the screen is 5.