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To determine the longest wavelength observed through a diffraction grating, we utilize the grating equation: \[ d \sin \theta = m \lambda \]where:- \(d\) represents the distance between slits, known as the grating constant,- \(\theta\) is the angle of diffraction,- \(m\) signifies the order of diffraction,- \(\lambda\) denotes the wavelength.
In our example, the wavelength is calculated by rearranging the equation to solve for \(\lambda\): \[ \lambda = \frac{d \sin \theta}{m} \]For the longest wavelength, we assume \(\sin \theta = 1\), resulting in:\[ \lambda = \frac{d}{m} \]The calculated wavelength should then be converted to appropriate units like nanometers, by multiplying by \(10^9\). This process ensures accuracy in finding the longest wavelength for the specified diffraction order.

The term 'order of diffraction' refers to the number of wavelengths by which paths differ as they spread out from the grating. Indicated as \(m\) in diffraction equations, each integer value represents a distinct diffraction order. It is essentially a count of the constructive interference patterns observable when light diffracts through a grating.
Higher orders (larger \(m\)) correspond to greater angles, which are formed at larger wavelengths or when using gratings with fewer slits per unit area. For analyzing the longest wavelength in such scenarios, especially in multiple orders of diffraction like the third order \(m = 3\), it’s crucial to correctly apply the grating equation.

The grating constant \(d\) is the separation between two adjacent slits in a diffraction grating. It is pivotal in the grating equation, determining how light of different wavelengths will diffract. The value of \(d\) is calculated from the number of slits per unit length.
In the given problem, there are 6500 slits per centimeter. This converts to a grating constant in meters as: \[ d = \frac{1}{6500} \times 10^{-2} \mid \text{m} \] The conversion is essential because it aligns with standard units used in scientific calculations. A smaller \(d\) means more slits per unit length, which typically allows for a more defined diffraction pattern.

The angle of diffraction \(\theta\) defines how far a beam of light deviates as it passes through a grating. It is influenced by the wavelength of the light \(\lambda\), the order of diffraction \(m\), and the grating constant \(d\).
In normal incidence situations, like our exercise, light hits the grating directly, optimizing the diffraction spread. The angle varies with orders of diffraction, such that for the longest wavelength experiment, it's assumed that \(\theta\) reaches a maximum when \(\sin \theta = 1\). This implies that good knowledge of \(\theta\)'s relationships helps in practical settings, like designing optical instruments or understanding the course of light in diffraction phenomena.