(a) What is the wavelength of light that is deviated in the first order through an angle of 13.5° by a transmission grating having 5000 slits/cm? (b) What is the second-order deviation of this wavelength? Assume normal incidence.

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Convert the grating's slit density into the slit spacing (d). The slit density is given as 5000 slits/cm. Use the formula:

d = \frac{1}{N}

, where N is the number of slits per unit length. Ensure the units are consistent (convert cm to meters).

Use the diffraction grating equation to find the wavelength of light for the first-order deviation. The equation is:

n λ = d sin θ

, where n is the order of diffraction (n=1 for first order), λ is the wavelength, d is the slit spacing, and θ is the angle of deviation (13.5°). Rearrange the equation to solve for λ:

λ = \frac{d}{n} sin θ

Substitute the known values into the equation from step 2. Use the slit spacing (d) calculated in step 1, n=1 for first order, and θ=13.5° (convert the angle to radians if necessary). This will give the wavelength λ in meters.

To find the second-order deviation for this wavelength, use the same diffraction grating equation:

n λ = d sin θ

. Here, n=2 for the second order, λ is the wavelength found in part (a), and d is the slit spacing. Rearrange the equation to solve for θ:

θ = arcsin \frac{n}{λ} d

Substitute the known values into the equation from step 4. Use n=2, the wavelength λ from part (a), and the slit spacing d. Calculate the angle θ for the second-order deviation. Ensure the angle is expressed in degrees if required.

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주요 개념

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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 or grooves, which cause constructive and destructive interference of light waves. The angle at which light is deviated depends on the wavelength and the spacing of the slits, allowing for the analysis of light spectra.

Order of Diffraction

The order of diffraction refers to the integer number that indicates the specific angle at which light is constructively interfered after passing through a grating. The first order corresponds to the first maximum of intensity observed, while higher orders (like second order) represent subsequent maxima. The angle of deviation increases with the order, affecting the wavelength calculation.

Wavelength Calculation

Wavelength calculation in the context of diffraction involves using the grating equation, which relates the angle of diffraction, the wavelength of light, and the grating spacing. The equation is given by d sin(θ) = mλ, where d is the distance between slits, θ is the angle of deviation, m is the order of diffraction, and λ is the wavelength. This relationship allows for determining the wavelength based on the observed angle and grating parameters.