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The wavelength calculation in a double-slit interference experiment is a fascinating process that helps us understand the nature of light. First, it's essential to know that the arrangement involves two slits a certain distance apart and a screen placed some distance behind the slits to capture the pattern created by interfering light waves. To find the wavelength of light used in this setup, we use the formula:\[ \lambda = \frac{d \, \Delta Y}{D \, \Delta m} \]where:- \( \lambda \) is the wavelength of the light.- \( d \) is the distance between the slits.- \( \Delta Y \) is the distance between the first and the fifth minima on the screen.- \( D \) is the distance from the slits to the screen.- \( \Delta m \) is the difference in the minima order, which indicates the number of dark fringes between two points on the screen.By plugging in the provided values, we can clearly see the step-by-step calculation leading to the wavelength as \( 500 \, \mathrm{nm} \). This demonstrates how precise measurements and known distances help determine fundamental properties of light.

An interference pattern emerges in a double-slit experiment due to the superposition of light waves originating from two slits. When light waves pass through these slits, they spread out and overlap, creating regions of constructive and destructive interference. Constructive interference occurs where the crests (or troughs) of the waves align, leading to bright fringes on the screen. Destructive interference happens when a crest of one wave meets a trough of another, resulting in dark fringes, known as minima. Such a pattern is observed as alternating bright and dark bands on the screen and serves as evidence of light's wave-like behavior. The calculations involving this pattern allow us to explore many properties of light, such as its wavelength. Understanding these patterns is crucial in fields like optics and helps in developing technologies like lasers and other optical devices.

Minima positions in an interference pattern are those where destructive interference occurs, resulting in dark spots on the screen. These positions can be precisely calculated using the order of minima, which refers to the sequence of dark fringes from the center outward.In practice, the position of any given minimum is calculated using the path difference between light reaching the point from two different slits and is given by:\[ d \sin \theta = (m + \frac{1}{2}) \lambda \]where:- \( d \) is the distance between the slits.- \( \theta \) is the angle from the central maximum to the minima.- \( m \) is the order number of the minima.- \( \lambda \) is the wavelength.In this exercise, the calculation focuses on the difference between the positions of the first and fifth minima. This involves determining the spacing (\( \Delta Y \)) between these minima to ultimately help in wavelength determination.

The light wavelength is a vital parameter that characterizes wave properties of light in a double-slit experiment. It determines how large each wave crest is and, by interacting with the slit spacing and screen distance, decides the interference pattern's nature. In this exercise, we find the wavelength to be \( 500 \, \mathrm{nm} \), signifying the light is likely visible. The visible spectrum generally ranges from about \( 400 \mathrm{nm} \) (violet) to \( 700 \mathrm{nm} \) (red). Thus, the light in this scenario probably appears green.Understanding the exact wavelength has vast applications:

• Identifying substances in spectrometry since different materials absorb different wavelengths.
• Developing optical technology, such as telescopes, microscopes, and lasers for diverse uses.

Grasping light's wavelength offers insight into both fundamental physics and practical technologies.