Question:
A double-slit experiment is conducted using light of wavelength 600 nm. The separation between the slits is 0.1 mm and the distance between the screen and the slits is 1.5 m.
a) Calculate the distance between the central bright fringe and the first-order bright fringe on the screen.
b) If a single-slit experiment is performed using the same setup, calculate the width of the central maximum on the screen.
Answer:
a) The distance between the central bright fringe (m = 0) and the first-order bright fringe (m = 1) on the screen can be calculated using the equation for the position of bright fringes in a double-slit interference pattern:
d * sin(θ) = m * λ
where: d is the slit separation, θ is the angle with respect to the central axis, m is the order of the bright fringe, and λ is the wavelength of light.
To calculate the distance between the central bright fringe and the first-order bright fringe, we need to calculate the angle θ for m = 1.
Given: λ = 600 nm = 600 × 10^(-9) m d = 0.1 mm = 0.1 × 10^(-3) m
We can use the equation:
d * sin(θ) = m * λ
Substituting the values:
0.1 × 10^(-3) m * sin(θ) = 1 * 600 × 10^(-9) m
Simplifying, we find:
sin(θ) = 6
As this value for sin(θ) is not possible (sin(θ) can take values only between -1 and 1), we conclude that there is no first-order bright fringe on the screen. Therefore, the distance between the central bright fringe and the first-order bright fringe is undefined.
b) To calculate the width of the central maximum on the screen in a single-slit experiment, we can use the equation for the angular width of the central maximum:
θ ≈ λ / (2 * s)
where: λ is the wavelength of light, and s is the slit width.
In a double-slit setup, the slit separation is analogous to the slit width in a single-slit experiment. Therefore, we can use the same value for s, which is given as 0.1 mm.
Given: λ = 600 nm = 600 × 10^(-9) m s = 0.1 mm = 0.1 × 10^(-3) m
Substituting these values into the equation:
θ ≈ 600 × 10^(-9) m / (2 * 0.1 × 10^(-3) m)
Simplifying, we find:
θ ≈ 3
Therefore, the angular width of the central maximum on the screen is approximately 3 radians.