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Diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams traveling in different directions. The grating equation is fundamental to understanding how this diffraction process occurs. It is expressed as:

• \( d \sin \theta = m \lambda \)

Here, \( d \) represents the distance between each slit on the grating, known as the grating spacing.
\( \theta \) is the angle of diffraction, which the light maintains as it bends around the slits. The order of the diffraction is denoted by \( m \). The variable \( \lambda \) symbolizes the wavelength of the light.This equation tells us how light will spread when it strikes a grating.
By altering different parameters, one can control how the light diffracts. For normal incidence, where light hits the grating head-on, the angle is zero at the initial point.

Calculating the wavelength of light that can be observed through a diffraction grating involves manipulating the grating equation. In the scenario of seeking the longest wavelength, certain conditions optimize the process.
When the diffraction angle \( \theta \) reaches \( 90^\circ \), the sine of the angle is maximal, with \( \sin 90^\circ = 1 \).
This simplifies the equation to:

• \( d = m \lambda \)

Starting with the grating spacing \( d \), you calculate it by inverting the number of slits per length of the grating. If there are 9200 slits per centimeter, converting units yields:

• \( d = \frac{1}{9200 \times 100} \)

Once the spacing is known, substituting \( d \) and the given order \( m \) into the simplified equation allows solving for the desired wavelength.

In the world of diffraction, the term "order" refers to the number of wavefronts that have crested between adjacent slits. Higher orders equate to more wave crests and higher degrees of dispersion.
The third-order diffraction implies that the third crest of each previous wavefront strikes the grating simultaneously.
Imagine the initial beam of light dividing into multiple pathways, each at varied angles relative to the initial direction.For the given exercise, third-order diffraction messes with the concept by setting \( m = 3 \):

• The condition \( \theta = 90^\circ \) makes calculation straightforward.
• Using the derived values in the grating equation \( d = 3 \lambda \), one solves for the maximum possible wavelength.

This interesting play of light not only serves theoretical exercises but also finds practical use in devices like spectrometers and monochromators.