Question:

A beam of laser light with a wavelength of 600 nm and power 2 mW is incident on a diffraction grating. The grating consists of 4000 lines per centimeter and has a width of 4 cm.

a) Calculate the number of lines in the grating. b) Calculate the angle of diffraction for the first-order maximum. c) Determine the angular separation between the first-order maximum and the second-order maximum. d) If the laser beam is incident perpendicular to the grating, calculate the width of the central bright spot in the diffraction pattern on a screen placed 1 meter away from the grating. e) How does the width of the central bright spot change when the distance between the grating and the screen is increased?

Assume the speed of light in vacuum is 3.00 x 10^8 m/s.

Answer:

a) The number of lines in the grating can be calculated using the given information. The grating consists of 4000 lines per centimeter and has a width of 4 cm. Therefore, the total number of lines is:

Number of lines = 4000 lines/cm * 4 cm = 16,000 lines.

b) The angle of diffraction for the first-order maximum can be calculated using the equation:

d * sin(θ) = m * λ,

where d represents the grating spacing, θ is the angle of diffraction, m is the order of the maximum, and λ is the wavelength of the light.

The grating spacing, d, can be calculated by taking the reciprocal of the lines per centimeter and converting it to meters:

d = 1 cm / (4000 lines/cm) * (1 m / 100 cm) = 2.5 x 10^(-5) m.

Now, plug in the values into the equation to find θ:

2.5 x 10^(-5) m * sin(θ) = (1) * (600 nm) * (10^(-9) m/nm).

Simplifying,

sin(θ) = 2.4.

Since the sine function has a maximum value of 1, we can conclude that θ is approximately equal to 90°, or pi/2 radians.

c) The angular separation between the first-order maximum and the second-order maximum can be calculated using the formula:

Δθ = θ_m+1 - θ_m,

where Δθ is the angular separation, θ_m+1 is the angle for the (m+1)-th maximum, and θ_m is the angle for the m-th maximum.

For the first-order maximum, θ_1 is approximately equal to 90° as calculated in part (b). For the second-order maximum, θ_2, we can use the equation:

2 * d * sin(θ_2) = 2 * (2.5 x 10^(-5) m) * sin(θ_1).

Plugging in the values, we get:

2 * (2.5 x 10^(-5) m) * sin(θ_2) = 2 * (2.5 x 10^(-5) m) * sin(90°).

Simplifying,

2 * sin(θ_2) = 1.

This results in sin(θ_2) = 0.5, or θ_2 ≈ 30°.

Substituting the values into the formula for angular separation, we have:

Δθ = θ_2 - θ_1 = 30° - 90° = -60°.

Therefore, the angular separation between the first-order maximum and the second-order maximum is -60°.

d) When the laser beam is incident perpendicular to the grating, the central bright spot can be treated as a single-slit diffraction pattern. The width of the central bright spot on a screen can be calculated using the formula:

Δy = (λ * L) / w,

where Δy is the width of the central bright spot, λ is the wavelength of the light, L is the distance between the grating and the screen, and w is the width of the grating.

Plugging in the values,

Δy = (600 nm) * (10^(-9) m/nm) * (1 m) / (4 cm) = 1.5 x 10^(-5) m.

Therefore, the width of the central bright spot in the diffraction pattern on the screen is 1.5 x 10^(-5) m.

e) When the distance between the grating and the screen is increased, the width of the central bright spot (Δy) decreases. This is because the diffraction pattern becomes wider with increasing distance, causing the central peak to become narrower.

Note: In the analysis above, we made some approximations and assumptions for the sake of simplification. In practice, the phenomenon of diffraction is more complex and involves interference effects as well.