Upon using Thomas Young’s double-slit experiment to obtain measurements, the following data were obtained. Use these data to determine the wavelength of light being used to create the interference pattern. Do this using three different methods.

Question

The angle to the eighth maximum is 1.12°.
The distance from the slits to the screen is 302.0 cm.
The distance from the central maximum to the fifth minimum is 3.33 cm.
The distance between the slits is 0.000250 m.

Answer 1

Method 1: Interference Fringe Spacing Formula
The interference fringe spacing formula for a double-slit experiment is given by:
d*sin(theta) = m*λ

where:
- d is the distance between the slits (0.000250 m)
- theta is the angle to the interference maximum (in radians)
- m is the order of the maximum (in this case, 8)
- λ is the wavelength of light

Using this formula, we can calculate the wavelength of light as follows:
theta = 1.12° = 1.12 * (π/180) radians = 0.0195 radians

d*sin(theta) = 0.000250 * sin(0.0195) = 4.56 × 10^-6 m

λ = (d*sin(theta))/m = (4.56 × 10^-6 m)/8 = 5.7 × 10^-7 m = 570 nm

Therefore, the wavelength of light being used to create the interference pattern is 570 nm.

Method 2: Distance Between Central Maximum and Fifth Minimum
The distance between the central maximum and the fifth minimum is directly related to the wavelength of light by the following formula:
y = (λ * L)/d

where:
- y is the distance between the central maximum and the fifth minimum (3.33 cm = 0.0333 m)
- L is the distance from the slits to the screen (302.0 cm = 3.02 m)
- d is the distance between the slits (0.000250 m)

Using this formula, we can solve for λ:
(λ * L)/d = 0.0333 m

λ = (0.0333 m * d)/L = (0.0333 m * 0.000250 m)/(3.02 m) = 2.75 × 10^-6 m = 2750 nm

Therefore, the wavelength of light being used to create the interference pattern is 2750 nm.

Method 3: Using the Distance Between Interference Maxima
The distance between interference maxima can be calculated using the following formula:
Δy = m * (λ * L)/d

where:
- Δy is the distance between interference maxima (in this case, the distance between the central maximum and the eight maximum)
- m is the order of the maximum (in this case, 8)
- L is the distance from the slits to the screen (3.02 m)
- d is the distance between the slits (0.000250 m)

Using this formula, we can solve for λ:
Δy = 1.12° = 1.12 * (π/180) radians = 0.0195 radians
Δy = (0.0195 * L)/(d*m) = (0.0195 * 3.02)/(0.000250 * 8) = 237 m

λ = (Δy * d)/(L * m) = (237 * 0.000250)/(3.02 * 8) = 6.22 × 10^-6 m = 6220 nm

Therefore, the wavelength of light being used to create the interference pattern is 6220 nm.