Red light of wavelength of 700 nm falls on a double slit separated by \(400 \mathrm{nm}\). (a) At what angle is the firstorder maximum in the diffraction pattern? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

When a wave encounters an obstacle or a slit that is comparable in size to its wavelength, it will bend around it or spread out. This phenomenon is known as diffraction. In the context of light passing through a double slit, each slit acts as a source of waves that interfere with one another, creating a pattern of bright and dark fringes on a screen. This is called a double-slit interference pattern, or more simply, a diffraction pattern.

The pattern results from constructive interference (where waves combine to increase amplitude) at certain angles where a bright fringe is seen, and destructive interference (where waves cancel each other out) at other angles leading to a dark fringe. The position and spacing of these fringes depend on factors such as the wavelength of the light and the distance between the slits.

Understanding the intricacies of a diffraction pattern not only requires knowledge of the wavelength and slit separation but also appreciating the underlying principle of wave interference. For students, the challenge lies in translating the conceptual understanding of wave behavior into mathematical expressions that predict the pattern's characteristics.

The wavelength of light is a fundamental characteristic of electromagnetic waves and is defined as the distance between successive crests of a wave. It determines many properties of light, including its color. Visible light has wavelengths in the range of approximately 400 nm (violet) to 700 nm (red).

For the purpose of analyzing diffraction patterns, knowing the wavelength is crucial. It helps in calculating the angles where the bright and dark fringes occur on the diffraction pattern. The diffraction pattern is highly sensitive to wavelength changes; even a slight variation can markedly alter the pattern due to the precise dependency of fringe spacing and positions on the wavelength.

The error in the original exercise illustrates an important consideration when working with the diffraction formula: the wavelength must be appropriate for the slit separation to avoid nonsensical results. As educators, it is imperative to highlight the significance of wavelength in understanding optical phenomena, making sure that students grasp the relationship between physical properties of light and their observable consequences.

The angle of diffraction maximum, often denoted by θ, refers to the angle at which a bright fringe in the diffraction pattern is observed. According to the double-slit interference formula, this angle is directly related to the order of the maximum (m), the wavelength of the light (λ), and the distance between the double slits (d). In the provided exercise, the goal was to calculate the angle for the first-order maximum (m = 1).

However, the calculated value for the sine of the angle exceeded the maximum possible value for the sine function, which is ±1. This indicates an error in the assumptions or input parameters. In practical terms, an angle of diffraction maximum that doesn't yield a sine value between -1 and 1 is not physically possible, emphasizing the importance of checking the reasonableness of calculated results against the constraints of physical laws.

When teaching this concept, it's essential to guide students through the process of validation of their results, encouraging them to question each outcome. They must understand that the angle of diffraction is not only dependent on the interference formula, but must also adhere to the natural limits of trigonometric functions and the physical behavior of light.