A double-slit experiment is to be set up so that the bright fringes appear \(1.27 \mathrm{cm}\) apart on a screen \(2.13 \mathrm{m}\) away from the two slits. The light source was wavelength 500 nm. What should be the separation between the two slits?

The double-slit experiment is a classic demonstration of the wave-like properties of light, which shows that light can interfere with itself to produce patterns of light and dark bands known as an interference pattern. This phenomenon occurs when coherent light waves pass through two closely spaced slits and overlap on a distant screen. The bright fringes, or bands, appear where the waves from each slit reinforce one another, also called constructive interference. In contrast, the dark fringes emerge where the waves cancel each other out, known as destructive interference. The pattern's regularity and spacing provide valuable insights into the nature of the light used, including its wavelength.

The positions of these fringes are not arbitrary but relate directly to the light's wavelength, the slit separation, and the distance from the slits to the screen. Understanding the interference pattern is crucial for interpreting the results of the double-slit experiment and for applications in fields such as optics and wave physics.

The wavelength of light is the distance between consecutive peaks or troughs of a wave. It is a fundamental property that determines various characteristics of light, including its color in the visible spectrum. Wavelength is commonly measured in units such as nanometers (nm) for visible light and in other units like meters for different contexts. In our textbook exercise, the wavelength is given as 500 nm, falling within the green portion of the visible spectrum.

When conducting a double-slit experiment, the wavelength governs the spacing of the interference pattern on the screen. The relationship is such that a longer wavelength would result in a wider spacing of the fringes, while a shorter wavelength leads to closer fringes. By analyzing the pattern created, one can deduce the wavelength of the source light if the slit separation and distance to the screen are known. This concept is central in understanding and applying the formula for the double-slit interference pattern.

The slit separation, denoted as 'd' in the double-slit formula, represents the distance between the two parallel slits in the experiment. This dimension is a critical factor influencing the resulting interference pattern. When the slit separation is comparable to the light's wavelength, the resulting interference effects are readily observable.

In the context of the problem at hand, the slit separation is what we are asked to find, given the wavelength of the light and the distance between the bright fringes on the screen. It is important to realize that the slit separation affects not only the width of the fringes but also their intensity, with narrower slits leading to more pronounced diffraction and thus a more significant impact on the interference pattern. The manipulation of slit separation allows us to control the pattern for various applications or experiments that involve wave properties of light.