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Calculating the wavelength of light involves determining how far waves of light travel. The wavelength, denoted as \( \lambda \), is a vital measure which tells us the distance over which the wave's shape repeats. In the exercise, we calculated the wavelength using the grating equation and the values provided. The formula for the wavelength derivation is \( \lambda = \frac{a \sin \theta_{2}}{m} \). Here, \( a \) is the spacing between the grating lines, \( \theta_{2} \) is the angle of the second-order diffraction, and \( m \) is the order of diffraction. Substituting the known values and performing conversions led to the light's wavelength being 699 nanometers. This calculation illustrates how the wavelength of light is closely connected to both the physical properties of the grating and the geometry of the diffraction setup.

The grating equation is crucial in understanding how a diffraction grating works. It is given as \( m \lambda = a \sin \theta_m \). Here is a breakdown of each component:

• \(m\) - The order of diffraction, indicating which diffraction fringe (first, second, etc.) we observe.
• \(\lambda\) - The wavelength of the light being used.
• \(a\) - The spacing between lines on the grating. For a grating with 4000 lines per centimeter, this value is \(a = \frac{1}{4000} \text{ cm/line}\).
• \(\theta_m\) - The angle at which the light of order \(m\) is diffracted.

This equation allows us to relate the properties of the grating and the light together to find unknown quantities. By doing so, it demonstrates the underlying physics of how light behaves when passing through or reflecting off periodic structures like gratings.

The order of diffraction is indicated by the integer \( m \) in the grating equation. This number denotes which set of diffraction lines or fringes is being considered. For example, if \( m = 2 \), we are looking at the second set of bright lines that appear on either side of a central maximum. Each order represents a different path difference that photons can take, leading to a separate diffraction angle. Higher orders generally involve more wave interference, resulting in different angles and often less intensity. In this exercise, we used a second-order diffraction image to calculate the wavelength of the light.

The angle of diffraction, represented by \( \theta \), is the angle at which light emerges after passing through the grating. It is crucial for determining how light will spread and is directly calculated using the grating equation. In this problem, the second-order angle \( \theta_2 \) was provided as \(34.0^{\circ}\). This angle relates to the wavelength, as each wavelength diffracts at a particular angle. Knowing the angle lets us use trigonometric identities, like \( \sin \theta \), which when substituted into the grating equation helps unveil the wavelength of light. The angle of diffraction is essential in numerous applications, such as spectrometry, to understand the composition of light sources.