91Ó°ÊÓ

To find the wavelength of light that gets deviated through a grating, we first need to understand what a diffraction grating is. It consists of multiple slits closely positioned together that break up light into various angles. These angles are related to the wavelength of light. In this context, you need to know two things:

• The angle at which the light deviates, which is given as \(13.5^{\circ}\), and
• The number of slits per centimeter, here it is 5000.

From the number of slits, we calculate the slit separation \(d\), which is the inverse of the number of slits. Thus, \(d = \frac{1}{5000}\, \text{cm} = 2 \times 10^{-4}\, \text{cm}\). Changing units for consistency, this becomes \(2 \times 10^{-6}\, \text{m}\). Now we use this distance to unravel the wavelength using the grating equation, aiming for clarity in combining physics and algebra.

The grating equation is a central concept when dealing with diffraction gratings. This equation is expressed as \(d \cdot \sin \theta = n \cdot \lambda\), where:

• \(d\) is the separation between slits,
• \(\theta\) is the angle of deviation,
• \(n\) is the order of diffraction, and
• \(\lambda\) is the wavelength of light.

In this scenario, for the first order of diffraction \((n=1)\), and with \(\theta = 13.5^{\circ}\), the task is to solve for \(\lambda\). That's done using the rearranged formula \(\lambda = \frac{d \cdot \sin \theta}{n}\). Substituting \(d = 2 \times 10^{-6}\, \text{m}\) and \(\theta = 13.5^{\circ}\), the wavelength becomes \(\lambda = 4.6689 \times 10^{-7}\, \text{m}\) or \(466.89\, \text{nm}\). This equation opens the doorway to understanding how light's wavelength relates to its angle of deflection.

The order of diffraction \((n)\) describes how many full wavelengths separate the light paths. Each order corresponds to an angle where light consistently diffracts through the grating. For the first order, we already calculated a deviation angle using the given conditions. Now, moving to the second order \((n=2)\), the angle will evidently change due to light needing to cover an additional full wavelength.

• We solve the grating equation again: \(d \cdot \sin \theta = n \cdot \lambda\).
• With \(\lambda = 4.6689 \times 10^{-7}\, \text{m}\) from the first calculation.
• Solving for \(\theta\) in second order, it is determined by \(\theta = \sin^{-1}\left( \frac{2 \cdot 4.6689 \times 10^{-7}}{2 \times 10^{-6}} \right)\).
• After computation, this yields approximately \(27.94^{\circ}\).

Understanding different orders of diffraction is crucial for exploring how light behaves when passed through a diffraction grating, particularly in scientific and technological applications where precise light measurements are needed.