The wavelength range of the visible spectrum is approximately 380–750 nm. White light falls at normal incidence on a diffraction grating that has 350 slits/mm. Find the angular width of the visible spectrum in the first order.

Verified step by step guidance

1. Step 1: Start by identifying 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 diffraction order, and $\lambda$ is the wavelength of light.
2. Step 2: Calculate the grating spacing $d$ from the given number of slits per millimeter. Since there are 350 slits/mm, the spacing is $d\; =\; \backslash frac\{1\}\{350\; \backslash times\; 10^3\}$ meters.
3. Step 3: For the first-order diffraction ($m\; =\; 1$), use the grating equation to calculate the diffraction angle for the shortest wavelength ($\lambda \; =\; 380\; \backslash times\; 10^\{-9\}$ m). Rearrange the equation to solve for $\theta$: $\theta \; =\; \backslash arcsin\backslash left(\backslash frac\{m\lambda \}\{d\}\backslash right)$.
4. Step 4: Repeat the calculation for the longest wavelength ($\lambda \; =\; 750\; \backslash times\; 10^\{-9\}$ m) to find the corresponding diffraction angle.
5. Step 5: The angular width of the visible spectrum in the first order is the difference between the two angles calculated in Steps 3 and 4: $\Delta \theta \; =\; \theta \_\{750\}\; -\; \theta \_\{380\}$.

Key Concepts

Diffraction Grating

A diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams. The angle at which light is diffracted depends on the wavelength and the spacing of the slits. The number of slits per millimeter determines the grating's resolving power, allowing for the separation of different wavelengths of light.

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. The first order (m=1) is the first angle where light is maximally intensified after passing through the grating. Higher orders (m=2, 3, etc.) correspond to angles where additional maxima appear, but they are less intense and can overlap.

Angular Width of the Spectrum

The angular width of the visible spectrum is the range of angles over which the visible wavelengths are diffracted by the grating. It can be calculated by determining the angles for the longest and shortest wavelengths in the visible spectrum. The difference between these angles gives the total angular width, which is crucial for understanding how much of the spectrum can be observed at a given order.