For what ratio of slit width to wavelength will the first minima of a single- slit diffraction pattern occur at \(\pm 90^{\circ} ?\)

When light encounters an obstacle or a slit that is comparable in size to its wavelength, it bends around the edges, a phenomenon known as diffraction. In the case of a single-slit diffraction, when a wavefront of light passes through a narrow opening, it spreads out and creates a pattern of bright and dark regions on a screen placed behind the slit. This pattern, referred to as a diffraction pattern, is composed of a central bright fringe with successive dark and bright fringes on either side.

The behavior of light in single-slit diffraction can be described using the Huygens-Fresnel principle, which states that every point on a wavefront can be considered as a source of secondary wavelets that spread out in all directions. These wavelets interfere with one another, resulting in a pattern of constructive and destructive interference; bright spots are a result of constructive interference, whereas dark spots are due to destructive interference.

The wavelength of light, often denoted by the Greek letter lambda (λ), is a fundamental concept in the study of diffraction. Wavelength is defined as the distance over which a wave's shape repeats. It is one of the factors that determines where the minima and maxima (the dark and bright fringes) will occur in a diffraction pattern. The relationship between slit width (𝑑), wavelength (λ), and the angle (θ) at which minima or maxima occur is given by the equation 𝑑*sin(θ)=n*λ, where 𝑛 is an integer representing the order number of the minima or maxima.

The wavelength is a crucial variable that affects the entire diffraction pattern. A longer wavelength will result in a broader pattern, whereas a shorter wavelength yields a more compact pattern. The exercise illustrates this relationship by asking for the ratio of slit width to wavelength, indicating how the size of the aperture relative to the wavelength of light can affect where the first minima appear in the diffraction pattern.

In the context of a single-slit diffraction pattern, minima are the dark bands where destructive interference occurs. Destructive interference happens when waves overlap out of phase; that is when the peak of one wave aligns with the trough of another, canceling each other out. The condition for the minima in a single-slit diffraction pattern is given by the equation 𝑑*sin(θ)=n*λ, where n is a nonzero integer (n = ±1, ±2, ±3, ...) which indicates the order of the minima.

The location of the first minima is useful for determining the properties of the diffracting slit and the light itself. In the provided exercise, the student is asked to find the ratio of the slit width to the wavelength that would result in the first minima occurring at a specific angle, ±90°. This implies an extreme scenario where the minima are as far apart as possible, given the constraints of the slit and the light's wavelength. Understanding the concept of minima is essential for analyzing and predicting the behavior of light in various applications, including optical instruments and diffraction gratings.