A five-slit system with \(7.5-\mu \mathrm{m}\) slit spacing is illuminated with 633 -nm light. Find the angular positions of (a) the first two maxima and (b) the third and sixth minima.

When light passes through a diffraction grating, it forms a pattern composed of bright spots (maxima) and dark spots (minima). This occurs due to constructive and destructive interference of light waves that have traveled different paths after passing through the slits. To determine where these maxima and minima will occur, we can apply the equations given in the problem.

For maxima, we use the equation:
d = d sin(θ_m) = mλ

where d is the distance between the slits (slit spacing), θ_m is the angle at which a maximum occurs, m represents the order of the maximum, and λ is the wavelength of the light. For minima, the formula changes slightly to:
d sin(θ_min) = (m + 1/2)λ
With these formulas, the angles can be calculated by inputting the different values of m. Thus, for a five-slit system with a known slit spacing and light wavelength, we can substitute the given data into these equations to determine the angular positions for the maxima and minima patterns observed.

The crucial relationship between slit spacing (d) and wavelength (λ) defines the diffraction pattern that is observed. By understanding this relationship, we can predict how changing one variable will affect the position of the bright and dark fringes.
When the slit spacing is comparable to the wavelength of the light, a notable diffraction effect is observed. If d is much larger than λ, then diffraction patterns become less visible. Conversely, as the wavelength approaches the size of the slit spacing, the diffraction angles increase, making the patterns more spread out. It's important to note that since the wavelength of visible light ranges from about 400 nm to 700 nm, slit spacings in this range are usually chosen for visible light diffraction experiments to yield measurable and observable patterns.

The equations for maxima and minima incorporates both d and λ, and only when these are accurately known, can the accurate angular positions of the fringes be calculated.

To determine the angular positions of the fringes produced by a diffraction grating, the pertinent equations must be solved with respect to the angle θ. We start by isolating θ in both the maxima and minima equation:
d sin(θ_m) = mλ      and      d sin(θ_min) = (m + 1/2)λ
Solving for θ in these equations typically involves the use of the inverse sine function, also known as arcsine. This process can be summarized as follows:

• Substitute the known values of d, λ, and m into the equation.
• Solve for sin(θ_m) or sin(θ_min).
• Calculate θ by taking the arcsine of the result obtained in step 2.

Remember to ensure that your scientific calculator or computation software is set to the correct unit (degrees or radians) when calculating the angles. This method allows for precise determination of the angle at which a maximum or minimum will appear on a screen or detector placed to capture the diffraction pattern.

With a deep understanding of these concepts, one can adeptly navigate through problems involving light diffraction through gratings, making calculations of diffraction angles clear and approachable.