If a diffraction grating produces its third-order bright band at an angle of 78.4° for light of wavelength 681 nm, find (a) the number of slits per centimeter for the grating and (b) the angular location of the first-order and second-order bright bands. (c) Will there be a fourth-order bright band? Explain.

Verified step by step guidance

1. Step 1: Begin by recalling the diffraction grating equation: \( d \sin \theta = m \lambda \), where \( d \) is the distance between adjacent slits (grating spacing), \( \theta \) is the diffraction angle, \( m \) is the order of the bright band, and \( \lambda \) is the wavelength of light. Rearrange the equation to solve for \( d \): \( d = \frac{m \lambda}{\sin \theta} \).
2. Step 2: Substitute the given values for the third-order bright band (\( m = 3 \)), wavelength (\( \lambda = 681 \, \text{nm} = 681 \times 10^{-9} \, \text{m} \)), and angle (\( \theta = 78.4^\circ \)) into the equation \( d = \frac{m \lambda}{\sin \theta} \). This will give the grating spacing \( d \).
3. Step 3: To find the number of slits per centimeter, recall that the number of slits per unit length is the reciprocal of the grating spacing \( d \). Convert \( d \) from meters to centimeters and calculate \( \text{slits per cm} = \frac{1}{d} \).
4. Step 4: For the angular location of the first-order (\( m = 1 \)) and second-order (\( m = 2 \)) bright bands, use the diffraction grating equation \( \sin \theta = \frac{m \lambda}{d} \). Substitute \( m = 1 \) and \( m = 2 \) along with the previously calculated \( d \) and solve for \( \theta \) in each case.
5. Step 5: To determine if a fourth-order bright band exists, check if \( \sin \theta \) for \( m = 4 \) satisfies \( \sin \theta \leq 1 \). Use \( \sin \theta = \frac{m \lambda}{d} \) with \( m = 4 \) and compare the result to 1. If \( \sin \theta > 1 \), the fourth-order bright band does not exist.

Key Concepts

Diffraction Grating

A diffraction grating is an optical component with a periodic structure that disperses light into its constituent wavelengths. It works on the principle of interference, where light waves overlap and combine, producing bright and dark bands at specific angles. The number of slits and their spacing determine the angles at which these bands appear, making it essential for analyzing light behavior.

Order of Diffraction

The order of diffraction refers to the integer multiples of the wavelength that correspond to the angles at which constructive interference occurs. For a given wavelength and grating, the first-order, second-order, and higher orders represent the angles where bright bands appear. The order is calculated using the grating equation, which relates the angle, wavelength, and slit spacing.

Grating Equation

The grating equation, given by d sin(θ) = mλ, relates the angle of diffraction (θ), the wavelength of light (λ), the slit spacing (d), and the order of diffraction (m). This equation is fundamental for determining the number of slits per unit length and predicting the angles for various orders of bright bands. It helps in understanding the limits of diffraction and the conditions under which higher orders can be observed.