AP Physics 1 Exam Question:

In a double-slit experiment using electrons, a beam of electrons is incident on two slits separated by a distance of 0.1 mm. The distance between the screen and the slits is 1.0 m. The resulting interference pattern is observed on a screen and displayed in the diagram below:

a) Calculate the wavelength (λ) of the electrons used in the experiment.

b) If the distance between the first minimum and the central maximum on the screen is measured to be 2.0 cm, what is the distance between the first minimum and the second minimum?

Answer:

a) To calculate the wavelength (λ) of the electrons used in the experiment, we can use the equation:

sin(θ) = m * λ / d,

where:

• θ is the angle between the central maximum and the first minimum,
• m is the order of the minimum (in this case, m = 1),
• λ is the wavelength of the electrons, and
• d is the distance between the two slits.

From the diagram, we can see that the angle between the central maximum and the first minimum is 30 degrees (θ = 30°) since it forms an equilateral triangle. Let's convert this angle to radians:

θ_rad = (30°) * (π / 180°) = (π / 6) rad.

Now, we can rearrange the equation:

sin((π / 6)) = (1 * λ) / (0.1 mm).

Converting the distance between the slits to meters:

d = 0.1 mm = 0.1 * 10^(-3) m = 1 * 10^(-4) m.

Rearranging the equation and solving for λ:

λ = (sin((π / 6)) * d) / 1 λ = (sin((π / 6)) * (1 * 10^(-4)))

Let's calculate λ:

import math

d = 1 * 10**(-4)  # distance between slits in meters
angle_rad = math.pi / 6  # angle in radians

wavelength = math.sin(angle_rad) * d
wavelength

The calculated wavelength of the electrons used in the experiment is approximately 4.8 x 10^(-11) meters.

b) The distance between the first minimum and the central maximum is equal to half the distance between the slits (d) in this symmetric double-slit pattern. Therefore, the distance between the first minimum and the second minimum should be the same as the distance between the central maximum and the first minimum. From the given information, the distance between the first minimum and the central maximum is measured to be 2.0 cm.

Thus, the distance between the first minimum and the second minimum is also 2.0 cm.

Explanation:

In a double-slit experiment, interference occurs when waves from each slit interact with each other. The resulting interference pattern is caused by constructive and destructive interference of the waves.

For part a), we used the equation sin(θ) = m * λ / d, which relates the angle between the central maximum and the first minimum (θ), the order of the minimum (m), the wavelength of the electrons (λ), and the distance between the slits (d). By rearranging the equation and plugging in the given values, we can solve for the wavelength (λ) of the electrons.

For part b), we observed that the distance between the central maximum and the first minimum is equal to half the distance between the slits (d) in a symmetric pattern. Therefore, the distance between the first minimum and the second minimum is also the same.