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.
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.
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.
Method 1: Using the angle and distance to the eighth maximum
To determine the wavelength of light using the angle and distance to the eighth maximum, we can use the formula:
λ = (d * sinθ) / m
where λ is the wavelength, d is the distance between the slits, θ is the angle to the maximum, and m is the order of the maximum.
Using the given data:
d = 0.000250 m
θ = 1.12° = 0.01955 radians (1.12 * π / 180)
m = 8
Substituting these values into the formula:
λ = (0.000250 * sin(0.01955)) / 8
λ ≈ 5.936 × 10^-7 m
So, the wavelength of light being used to create the interference pattern is approximately 5.936 × 10^-7 m.
Method 2: Using the distance between the slits and the distance from the central maximum to the fifth minimum
To determine the wavelength of light using the distance between the slits and the distance from the central maximum to the fifth minimum, we can use the formula:
λ = (d * L) / x
where λ is the wavelength, d is the distance between the slits, L is the distance from the slits to the screen, and x is the distance from the central maximum to the fifth minimum.
Using the given data:
d = 0.000250 m
L = 302.0 cm = 3.02 m
x = 3.33 cm = 0.0333 m
Substituting these values into the formula:
λ = (0.000250 * 3.02) / 0.0333
λ ≈ 0.0226 m
So, the wavelength of light being used to create the interference pattern is approximately 0.0226 m.
Method 3: Using the distance between the slits and the distance from the central maximum to the fifth minimum (alternate method)
To determine the wavelength of light using the distance between the slits and the distance from the central maximum to the fifth minimum, we can use the formula:
λ = (m * λ * L) / x
where λ is the wavelength, m is the order of the minimum (in this case, -5 since it is a minimum), L is the distance from the slits to the screen, and x is the distance from the central maximum to the fifth minimum.
Using the given data:
m = -5
L = 302.0 cm = 3.02 m
x = 3.33 cm = 0.0333 m
Rearranging the formula:
λ = (m * λ * L) / x
(1 + m) * λ = (m * L) / x
λ = ((m * L) / x) / (1 + m)
Substituting the values into the formula:
λ = ((-5 * 3.02) / 0.0333) / (1 - 5)
λ ≈ 0.0226 m
So, the wavelength of light being used to create the interference pattern is approximately 0.0226 m.