Light of wavelength 585 nm falls on a slit 0.0666 mm wide. (a) On a very large and distant screen, how many totally dark fringes (indicating complete cancellation) will there be, including both sides of the central bright spot? Solve this problem without calculating all the angles! (Hint: What is the largest that sin u can be? What does this tell you is the largest that m can be?) (b) At what angle will the dark fringe that is most distant from the central bright fringe occur?

Verified step by step guidance

1. Identify the given values: wavelength (λ) = 585 nm and slit width (a) = 0.0666 mm. Convert all units to meters for consistency.
2. Understand the concept of diffraction through a single slit and the condition for dark fringes, which is given by the equation a \sin(\theta) = m\lambda, where m is the order of the dark fringe (m = ±1, ±2, ±3, ...).
3. Determine the maximum value of m. Since the sine function has a maximum value of 1, set up the inequality a \sin(\theta) = m\lambda \leq a. Solve for m using m \leq \frac{a}{\lambda}.
4. Calculate the total number of dark fringes. Since m can be positive or negative, and includes values from -m to +m, the total number of dark fringes is 2m + 1, accounting for both sides of the central maximum.
5. For part (b), find the angle for the most distant dark fringe using \theta = \arcsin(\frac{m\lambda}{a}) where m is the maximum integer value calculated previously.

Key Concepts

Diffraction and Interference

Diffraction refers to the bending of waves around obstacles and the spreading of waves when they pass through narrow openings. In the context of light, when it passes through a slit, it creates an interference pattern of bright and dark fringes on a screen due to the constructive and destructive interference of light waves. This phenomenon is crucial for understanding how light behaves when encountering slits and is foundational for solving problems related to fringe patterns.

Dark Fringes and Order of Interference

Dark fringes in a diffraction pattern occur at specific angles where the path difference between light waves from different parts of the slit leads to destructive interference. The order of interference, denoted by 'm', represents the number of wavelengths by which the path difference differs from an integer multiple of the wavelength. The maximum value of 'm' is determined by the geometry of the setup and the wavelength of light, which is essential for calculating the number of dark fringes.

Sine Function and Its Limits

The sine function, sin(θ), is a mathematical function that relates the angle θ to the ratio of the opposite side to the hypotenuse in a right triangle. For diffraction problems, the maximum value of sin(θ) is 1, which occurs at θ = 90 degrees. This limit is critical in determining the maximum order of dark fringes (m) that can exist, as it sets a boundary for the angles at which dark fringes can be observed in the interference pattern.