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In two-slit interference, dark fringes are areas on the screen where the light waves cancel each other out, resulting in no light or a "dark" spot. This occurs when the path difference between the light coming from the two slits equals a half-integer multiple of the wavelength, specifically \(d \sin \theta = (m + \frac{1}{2})\lambda\). In this formula, \(d\) represents the slit separation, \(\theta\) is the angle to the fringe, and \(\lambda\) denotes the wavelength of the light. When the laser light in our scenario passed through the slits, the first dark fringes were observed at \(\pm 19.0^{\circ}\). To comprehend this, keep in mind that dark fringes are analogous to quiet places in an echoing room, where sound waves block each other out instead of constructing sound.

The intensity variation in a two-slit interference pattern is determined by how the light's brightness changes from one point to another on the screen. The intensity is at its maximum where the light waves from the two slits constructively interfere and is significantly reduced, or even becomes zero, where they destructively interfere. The formula for light intensity in this context is \(I = I_{max} \cos^2\left(\frac{\pi d \sin \theta}{\lambda}\right)\). This equation indicates how the intensity \(I\) varies across angles \(\theta\) and shows that the maximum intensity \(I_{max}\) is modified by the cosine squared term. If we want to find out at which angle the intensity drops to \(\frac{1}{10}\) of its maximum, we solve \(\cos^2 x = \frac{1}{10}\), which helps us find where this lessened intensity occurs.

Wavelength calculation is crucial in experiments like the two-slit interference. It helps us understand how the light's color and type affect the interference pattern observed. For our experiment, knowing the ratio \(d/\lambda\) aids in deducing the wavelength of the light used, given the slit separation and fringe positions. By using equations such as \(d \sin \theta = (m + \frac{1}{2})\lambda\) for dark fringes, and working backwards, we can calculate or estimate the wavelength if \(d\) and the angle \(\theta\) are known. Accurately calculating the wavelength can reveal properties about the light source.

The slit separation ratio, denoted as \(d/\lambda\), is a key parameter that affects the interference pattern in a two-slit experiment. It tells us how the physical spacing between the slits compares to the wavelength of the light passing through them. A higher ratio results in fringes that are closer together, while a lower ratio spreads them farther apart. In our given problem, we calculated \(d/\lambda\) to be approximately 1.535, meaning the slits are about 1.535 times the wavelength apart. This ratio provides insight into how finely or broadly the interference pattern is spread on the screen, which is essential for designing experiments and understanding wave propagation.