White light falls on two narrow slits separated by \(0.40 \mathrm{mm} .\) The interference pattern is observed on a screen 3.0 \(\mathrm{m}\) away. (a) What is the separation between the first maxima for red light \((\lambda=700 \mathrm{nm})\) and violet light \((\lambda=400 \mathrm{nm}) ? \quad\) (b) \(\mathrm{At}\) what point nearest the central maximum will a maximum for yellow light \((\lambda=600 \mathrm{nm})\) coincide with a maximum for violet light? Identify the order for each maximum.

Whenever waves overlap, they interact in different ways – this phenomenon is known as interference. Constructive interference occurs when the crest of one wave aligns with the crest of another, resulting in a wave of larger amplitude. Likewise, when troughs align, they reinforce each other too. This effect is commonly observed in Young's double-slit experiment, where light waves pass through two closely spaced slits and interfere to create a pattern of bright and dark fringes on a screen.

In technical terms, the mathematical condition for constructive interference is given by path difference of the waves being an integral multiple of the wavelength, symbolically \( m\lambda \) where 'm' is any integer (representing the 'order' of the fringe) and \( \lambda \) is the wavelength. This simple principle helps us understand complex natural phenomena, including the iridescence of a peacock's feather or the vibrant shimmer in a soap film.

• Waves reinforce each other at points of constructive interference.
• Bright fringes in interference patterns result from constructive interference.
• The path difference in constructive interference equals a whole number multiple of the wavelength.

The wavelength of light, symbolized as \( \lambda \), is a fundamental concept describing the distance over which the wave's shape repeats. It is a key factor in determining various optical phenomena, such as the color of the light which is visible to humans. For instance, violet light has a shorter wavelength than red light; they are approximately 400 nanometers and 700 nanometers respectively.

Wavelength plays a pivotal role in interference patterns, especially in Young's double-slit experiment. When light of a specific wavelength passes through the slits, the resulting pattern on a screen is directly related to \( \lambda \). Shorter wavelengths create a closer spacing of fringes, while longer wavelengths result in wider spaced fringes. This behavior can be described with the equation \( x = \dfrac{m\lambda D}{d} \) where 'x' is the fringe spacing, 'D' is the distance of the screen from the slits, and 'd' is the separation between the slits.

• Wavelength is inversely proportional to the energy and frequency of the light.
• The color of light is associated with its wavelength within the visible spectrum.
• Manipulating wavelength adjusts the interference pattern's fringe spacing.

The iconic Young's double-slit experiment demonstrates the wave nature of light, unveiling how light waves spread out and create an interference pattern when they pass through two closely-spaced slits. This experiment is monumental in physics because it provides a clear representation of the dual nature of light – exhibiting both particle and wave characteristics.

In the experiment, when monochromatic (single wavelength) light passes through the slits, the waves emanating from the slits meet on a screen to form an interference pattern consisting of alternating bright and dark fringes. The bright areas manifest due to constructive interference where the waves are 'in phase', reinforcing each other. Conversely, the dark areas denote zones of destructive interference, where waves are 'out of phase', canceling each other out.

This experiment not only clarifies concepts like constructive interference and wavelength but also sets the stage for understanding advanced topics in quantum mechanics, such as the principle of superposition. The correlations observed in the experiment apply to various fields, from engineering to medicine, influencing the technology behind lasers, optical instruments, and even modern quantum computers.

• Young's experiment is foundational in evidencing the wave behavior of light.
• The double-slit arrangement creates repeatable patterns of light and dark fringes.
• An interference pattern is direct proof of wave interaction.