If a diffraction grating produces a third-order bright spot for red light (of wavelength 700 nm) at 65.0° from the central maximum, at what angle will the second-order bright spot be for violet light (of wavelength 400 nm)?

Verified step by step guidance

1. Step 1: Recall the diffraction grating equation: $n\; \backslash (\backslash lambda\backslash )\; =\; d\; \backslash (\backslash sin\backslash )\; \backslash (\backslash theta\backslash )$, where $n$ is the order of the bright spot, $\backslash (\backslash lambda\backslash )$ is the wavelength of light, $d$ is the spacing between adjacent slits in the grating, and $\backslash (\backslash theta\backslash )$ is the diffraction angle.
2. Step 2: Use the given information for the red light to calculate the grating spacing $d$. For the third-order bright spot ($n\; =\; 3$) of red light ($\backslash (\backslash lambda\backslash )\; =\; 700\; \backslash (\backslash text\{\; nm\}\backslash )\; =\; 700\; \backslash (\backslash times\backslash )\; 10^\{-9\}\; \backslash (\backslash text\{\; m\}\backslash )$) at an angle of $\backslash (\backslash theta\backslash )\; =\; 65.0^\backslash (\backslash circ\backslash )$, rearrange the equation to solve for $d$: $d\; =\; \backslash (\backslash frac\{n\; \backslash lambda\}\{\backslash sin\; \backslash theta\}\backslash )$.
3. Step 3: Once $d$ is determined, use it to find the angle for the second-order bright spot ($n\; =\; 2$) of violet light ($\backslash (\backslash lambda\backslash )\; =\; 400\; \backslash (\backslash text\{\; nm\}\backslash )\; =\; 400\; \backslash (\backslash times\backslash )\; 10^\{-9\}\; \backslash (\backslash text\{\; m\}\backslash )$). Rearrange the diffraction equation to solve for $\backslash (\backslash theta\backslash )$: $\backslash (\backslash sin\backslash )\; \backslash (\backslash theta\backslash )\; =\; \backslash (\backslash frac\{n\; \backslash lambda\}\{d\}\backslash )$.
4. Step 4: Substitute the values of $n\; =\; 2$, $\backslash (\backslash lambda\backslash )\; =\; 400\; \backslash (\backslash times\backslash )\; 10^\{-9\}\; \backslash (\backslash text\{\; m\}\backslash )$, and the previously calculated $d$ into the equation $\backslash (\backslash sin\backslash )\; \backslash (\backslash theta\backslash )\; =\; \backslash (\backslash frac\{n\; \backslash lambda\}\{d\}\backslash )$. Compute $\backslash (\backslash sin\backslash )\; \backslash (\backslash theta\backslash )$.
5. Step 5: Use the inverse sine function to find the angle $\backslash (\backslash theta\backslash )$: $\backslash (\backslash theta\backslash )\; =\; \backslash (\backslash arcsin\backslash )(\backslash (\backslash sin\backslash )\; \backslash (\backslash theta\backslash ))$. This will give the angle for the second-order bright spot of violet light.

Key Concepts

Diffraction Grating

A diffraction grating is an optical component with a periodic structure that disperses light into its component wavelengths. It works on the principle of interference, where light waves overlap and combine, creating bright and dark spots at specific angles. The angle at which these spots appear depends on the wavelength of the light and the spacing of the grating lines.

Order of Diffraction

The order of diffraction refers to the integer (m) that indicates the number of wavelengths by which the path difference between light waves contributes to constructive interference. For example, the first-order bright spot corresponds to m=1, the second-order to m=2, and so on. Each order appears at a specific angle determined by the wavelength of light and the grating spacing.

Wavelength and Angle Relationship

The relationship between wavelength and angle in diffraction can be described by the grating equation: d sin(θ) = mλ, where d is the grating spacing, θ is the angle of the bright spot, m is the order of diffraction, and λ is the wavelength. This equation allows us to calculate the angle for different wavelengths and orders, illustrating how varying wavelengths produce different diffraction patterns.