Question

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 \theta 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?

Short answer

226 completely dark fringes occur. The most distant one is at approximately 82.8°.

Step-by-step solution

Understand the Problem

We are given the wavelength of light \( \lambda = 585 \ nm \) and the width of the slit \( a = 0.0666 \ mm \). Our task is to find the number of dark fringes on both sides of the central bright spot using the diffraction condition for single slits.

Recall the Condition for Dark Fringes

For a single-slit diffraction pattern, dark fringes (minimum intensity) occur at angles where \( a \sin \theta = m \lambda \), where \( m \) is an integer. Specifically, \( m = \pm 1, \pm 2, \ldots \). The maximum value for \( \sin \theta \) is 1.

Calculate Maximum Possible m

Set \( \sin \theta = 1 \) to find the maximum \( m \) using the formula for dark fringes:\[ a \cdot 1 = m \lambda \]\[ m = \frac{a}{\lambda} \]Convert \( a = 0.0666 \ mm \) to \( nm \) for consistency with \( \lambda \):\[ a = 0.0666 \ mm = 66600 \ nm \]Now compute \( m \):\[ m = \frac{66600 \ nm}{585 \ nm} \approx 113.85 \]Since \( m \) must be an integer, the maximum \( m \) is 113.

Determine Total Number of Dark Fringes

The total number of dark fringes includes both positive and negative values of \( m \), from \(-113\) to \(+113\). Don't forget to include \( m = 0 \) once (central bright spot), then:\[ 113 + 113 = 226 \text{ (sides)} \] adding the central point gives a total of 227 positions, so subtract the central spot:\[ 113 + 113 = 226 \text{ dark fringes} \]

Calculate Angle for Most Distant Dark Fringe

Using \( m = 113 \), calculate \( \theta \) for the most distant dark fringe:\[ a \sin \theta = 113 \lambda \]\[ \sin \theta = \frac{113 \times 585 \ nm}{66600 \ nm} \approx 0.992 \]Hence, \( \theta = \arcsin(0.992) \approx 82.8^\circ \).

Key concepts

Diffraction Pattern

In the realm of light and optics, a diffraction pattern is an intriguing phenomenon. It occurs when light bends around the edges of an obstacle, such as a slit, resulting in a unique arrangement of bright and dark bands on a screen. Think of it like the shadow effect of objects not blocking light completely, but rather bending it all around, creating a pattern. The central bright spot in this pattern is the largest and most intense, while the surrounding dark and light regions, known as fringes, vary in intensity. The arrangement is defined by the constructive and destructive interference of light waves. Constructive interference happens when light waves amplify each other, forming bright fringes. In contrast, destructive interference leads to dark fringes where light waves cancel each other out. These patterns, known for their symmetry, help us understand the wave nature of light.

Dark Fringes

Dark fringes are the lines on the screen where light and dark areas alternate in the diffraction pattern. In a single-slit diffraction experiment, dark fringes signify points of minimum intensity due to destructive interference. These fringes form where the path difference between light waves results in a complete wave cancellation. Mathematically, the condition for dark fringes is expressed by the formula:\[ a \sin \theta = m \lambda \]Where:

• \( a \) is the slit width.
• \( \lambda \) is the wavelength of light.
• \( m \) is the order number, an integer representing the fringe sequence.

The value of \( m \) determines the exact position of each dark fringe on the screen. Intriguingly, the number of visible dark fringes depends on the maximum possible value of \( m \), calculated from the physical constraints of the setup like slit width and wavelength.

Wavelength of Light

The wavelength of light is a crucial element in studying diffraction. It defines the color and type of light wave in the electromagnetic spectrum, typically measured in nanometers (nm).In this task, the given wavelength is 585 nm. The wavelength is vital because it directly influences the diffraction pattern produced. This is because it determines how far light bends around slits. A longer wavelength would result in more widely spaced fringes. The relationship between wavelength and diffraction can be visualized with the formula \( m \lambda = a \sin \theta \), linking wavelength, order number \( m \), and angle \( \theta \) for each fringe. Understanding wavelength and its role in diffraction helps grasp why different wavelengths scatter light in unique ways.

Angle of Diffraction

The angle of diffraction is the angle at which light deviates from its initial path due to passing through a narrow opening, like a slit. It is crucial in determining the positions of the dark and bright fringes in a diffraction pattern.For dark fringes, the angle is calculated using the formula:\[ a \sin \theta = m \lambda \]Here:

• \( \theta \) is the diffraction angle.
• \( a \) is the slit width.
• \( \lambda \) is the wavelength.
• \( m \) is the order of the fringe.

The maximum value of \( \sin \theta \) is 1, determining the highest possible angle. In this case, the most distant dark fringe occurs at an angle calculated using the highest integer \( m \) possible, which is where \( \sin \theta \approx 0.992 \), leading to an angle of about \( 82.8^\circ \). Recognizing these angles allows us to predict where light will vanish (dark fringes) on a screen.