A 3800-slit/cm grating produces a third-order fringe at a 35.0° angle. What wavelength of light is being used?

Verified step by step guidance

1. Identify the formula for diffraction grating: \( d \sin \theta = m \lambda \), where \( d \) is the distance between adjacent slits (grating spacing), \( \theta \) is the diffraction angle, \( m \) is the order of the fringe, and \( \lambda \) is the wavelength of light.
2. Convert the given grating value from slits per centimeter to grating spacing \( d \) in meters. Since there are 3800 slits per centimeter, \( d = \frac{1}{3800 \times 10^2} \) meters.
3. Substitute the known values into the formula: \( d \sin \theta = m \lambda \). Here, \( d \) is the grating spacing calculated in the previous step, \( \theta = 35.0^\circ \), and \( m = 3 \) (third-order fringe).
4. Rearrange the formula to solve for \( \lambda \): \( \lambda = \frac{d \sin \theta}{m} \).
5. Substitute the values of \( d \), \( \sin \theta \), and \( m \) into the rearranged formula to calculate the wavelength \( \lambda \). Ensure that \( \sin \theta \) is calculated using the angle in degrees.

Key Concepts

Diffraction Grating

A diffraction grating is an optical component with a periodic structure that disperses light into its component wavelengths. It consists of multiple slits, which cause constructive and destructive interference of light waves. The angle at which light is diffracted depends on the wavelength and the spacing of the slits, making it essential for analyzing light spectra.

Order of Diffraction

The order of diffraction refers to the integer (n) that indicates the number of wavelengths by which the path difference between light waves from adjacent slits differs. For example, the third-order fringe corresponds to n=3. Higher orders occur at larger angles and are associated with longer path differences, allowing for the observation of multiple wavelengths.

Wavelength Calculation

The wavelength of light can be calculated using the grating equation, which is given by d sin(θ) = nλ, where d is the distance between slits, θ is the angle of diffraction, n is the order of the fringe, and λ is the wavelength. This relationship allows us to determine the wavelength of light based on the angle at which a specific order of fringe appears.