Find the intensity as a fraction of the central peak intensity for the second secondary maximum in single-slit diffraction, assuming the peak lies midway between the second and third minima.

When a wave encounters an obstacle or a slit that is comparable in size to its wavelength, it bends around it. This phenomenon is called diffraction. In the case of light passing through a single slit, diffraction results in a pattern characterized by a central bright fringe with successive dark and bright fringes on either side.

This pattern arises due to the interference of light waves emerging from different parts of the slit. The central maximum is the brightest and stands at the zero degree angle to the slit, where constructive interference is at its strongest. The intensity diminishes for the fringes farther out from the center, as the conditions for constructive interference are less optimal the further the waves have to travel to interfere with each other.

The positions of the minima in the diffraction pattern are particularly important. They are found using the relationship \[ \theta_m = m \cdot \lambda / b \], where \(\theta_m\) is the angular position of the \(m\)-th minimum, \(\lambda\) is the wavelength, \(m\) is the order of the minimum, and \(b\) is the slit width. By examining these patterns, one can gain insights not only into the behavior of light but also into the properties of the material making up the slit.

The intensity of light in a diffraction pattern is not uniform; it varies across the pattern and is determined by the path difference between light waves from different parts of the slit. When these waves constructively interfere, we observe a maximum in the pattern. In contrast, destructive interference leads to minima.

To calculate the intensity at a particular point in the pattern, we use the formula \[ I = I_0 \left( \frac{\sin(\beta)}{\beta} \right)^2 \], where \(I\) is the intensity at the point of interest, \(I_0\) is the central maximum intensity, and \(\beta = \pi b \sin(\theta) / \lambda\). The variable \(\theta\) represents the angle from the central axis, and \(\beta\) is the phase difference between light waves from the upper and lower edges of the slit.

This formula is a result of the Huygens-Fresnel principle, which considers every point on a wavefront as a source of secondary spherical wavelets. The intensity calculation is key in determining how bright or dark any point of the diffraction pattern is compared to the central peak, thereby giving us a quantitative analysis of the pattern.

The wavelength of light is a fundamental characteristic that determines many of its properties, including its color and how it interacts with objects and materials. It is defined as the distance between two consecutive peaks or troughs in the wave and is usually measured in nanometers (nm) when discussing visible light.

In the context of diffraction, the wavelength is crucial because it is directly related to the spacing of the fringes in the diffraction pattern. A longer wavelength will result in fringes that are spaced further apart compared to a shorter wavelength for the same slit width. Also, the equation \(\theta_m = m \cdot \lambda / b\) ties the wavelength to specific angular positions of minima in the single-slit diffraction pattern, making it possible to calculate the wavelength if the other variables are known.

In summary, the wavelength is essential both in the conceptual understanding of light behavior and in practical calculations when analyzing diffraction patterns. It's fascinating how such a simple measure can hold so much information about the nature of light.