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Coherent light is a fundamental concept in understanding many optical phenomena, such as the double-slit interference experiment. Coherent light refers to light waves that have a constant phase difference and the same frequency. This is essential for creating predictable interference patterns, like the alternating bright and dark fringes seen when coherent light passes through two slits.
The best example of coherent light is that produced by a laser. A laser emits a beam of light where the waves are in phase and have the same frequency and wavelength. This coherence is crucial for precise scientific experiments and for the creation of stable interference patterns. Without coherence, the interference pattern would be unpredictable or non-existent.
In this exercise, coherent light is used to ensure that the light waves from the two slits interfere constructively or destructively at consistent points on the screen, forming clear, predictable patterns of bright and dark fringes.

The wavelength of light is another critical concept for understanding interference patterns. In the given problem, the wavelength \[\lambda\]is calculated using the formula:\[\lambda = \frac{c}{f}\]where \(c\) is the speed of light \[3 \times 10^8\, \mathrm{m/s}\]and \(f\) is the frequency \[6.32 \times 10^{14}\, \mathrm{Hz}\].
Plugging these values into the formula gives a wavelength of \[4.75 \times 10^{-7}\, \mathrm{m}\]. This wavelength allows us to use the interference formulas to find the positions of the bright and dark fringes.
It's important to understand that the wavelength directly affects the spacing of these fringes on the screen. Longer wavelengths result in more spread-out fringes, while shorter wavelengths bring fringes closer together.

In a double-slit experiment, bright fringes are points of constructive interference where the light waves from each slit arrive in phase and reinforce each other. The position of these bright fringes can be calculated using the formula for bright fringes:\[d \cdot \sin(\theta) = m \cdot \lambda\]where \(d\) is the slit separation, \(\theta\) is the angle of the fringe, and \(m\) is the order of the fringe (in this case, 3 for the third bright fringe).
By using the small angle approximation, where \[\sin(\theta) \approx \tan(\theta) \approx \frac{x}{L}\], we can find the separation of the slits \(d\) as \[0.388\, \mathrm{mm}\]. This shows how the position and order of a bright fringe on the screen are directly tied to the wavelength and slit separation.
Bright fringes indicate that the light waves have traveled path lengths differing by an integer number of wavelengths, resulting in constructive interference.

Dark fringes are the regions of destructive interference in a double-slit interference pattern. These appear between the bright fringes and are points where the light waves from the slits arrive out of phase, effectively canceling each other out. In this way, they create a band of darkness on the screen.
The position of dark fringes can be determined using the formula:\[d \cdot \sin(\theta) = (m + 0.5) \cdot \lambda\]where \((m + 0.5)\) accounts for the half-wavelength path difference that results in cancellation of the light waves.
For the third dark fringe, by using values of \(m = 2.5\), \[d\], \[\lambda\], and the distance \[L\], we find that the third dark fringe occurs at roughly \[2.59\, \mathrm{cm}\] from the central bright fringe.
These positions of dark fringes provide insight into the path difference and aid in understanding wave interference in optics.