Question

When laser light of wavelength 633.2 nm passes through a diffraction grating, the first bright spots occur at ±18.6º from the central maximum.

A. What is the line density (in lines/cm) of this grating?
B. How many additionals pares of bright spots are there beyond the first bright spots?
C. At what angles do the second bright spots occur?
D. At what angles do the third bright spots occur?

Answer 1

The line density of a diffraction grating can be calculated using the formula d ⋅ sin(θ) = m ⋅ λ, where the inverse of d gives the line density in lines/cm. To find additional bright spots and their respective angles, we use the same equation with higher order numbers until the sine of the angle exceeds 1, indicating no further spots are visible.

Step-by-step explanation:

When laser light of wavelength 633.2 nm passes through a diffraction grating, the first bright spots occur at ±18.6° from the central maximum, we can use the equation for diffraction grating maxima, which is d ⋅ sin(θ) = m ⋅ λ, where d is the distance between grating lines, θ is the angle of the bright spot, m is the order number, and λ is the wavelength of the light. To calculate the line density, which is the inverse of d and typically expressed as lines per centimeter, we can rearrange this equation and use m = 1 for the first order maximum.

A. To find the line density (N), N = ⅛ / d, we first calculate d using d = m ⋅ λ / sin(θ) and then take its inverse. For θ = 18.6° and λ = 633.2 nm, the line density N comes out to be the inverse of “d” calculated value.

B. To determine the number of additional pairs of bright spots beyond the first, we need to calculate at which order number (m) the sine of the angle exceeds 1, since the sine function cannot exceed 1 (sine values are between -1 and 1). This represents the number of visible pairs of bright spots.

C. & D. To find the angles for the second and third bright spots, we use m = 2 and m = 3 in the formula given above and solve for θ, keeping in mind that as θ approaches 90°, no more bright spots will be observable beyond that angle.

Answer 2

The line density of the diffraction grating is 0.00158 lines/cm. There are additional pairs of bright spots beyond the first bright spots at an angle of 37.2º. The second bright spots occur at an angle of 2.001º and the third bright spots occur at an angle of 3.002º.

Step-by-step explanation:

To calculate the line density of the diffraction grating, we can use the formula:
line density (lines/cm) = 1 / wavelength (nm)
Plugging in the given wavelength of 633.2 nm, we get:
line density = 1 / 633.2 = 0.00158 lines/cm

For part B, the additional pairs of bright spots beyond the first bright spots can be calculated by finding the difference in angles between each pair of bright spots. Let's assume the distance between each pair is Δθ. The formula becomes:
Δθ = 2 × θ
Δθ = 2 × 18.6º = 37.2º

For part C, the angles for the second bright spots can be calculated using the formula:
θ = n × λ / d
Plugging in the values n = 2 (second order) and λ = 633.2 nm, we need to find d. Using the line density formula from earlier, we can rewrite it as:
d = 1 / line density
Plugging in the line density of 0.00158 lines/cm, we get d = 632.91 cm. Substituting the values into the formula, we get θ = 2 × 633.2 / 632.91 = 2.001º.

For part D, we can use the same formula as in part C, but with n = 3 (third order). Plugging in the values, we get θ = 3 × 633.2 / 632.91 = 3.002º.