Problem 27 - College Physics

Question

In a single-slit diffraction pattern using light of wavelength \(550 \mathrm{nm}\), the second-order minimum is measured to be at \(0.32^{\circ} .\) What is the slit width?

Step-by-Step Solution

Answer

The slit width is approximately 197 micrometers.

Step 1: Understand the Problem

We need to find the slit width given the wavelength of light and the angle for the second-order minimum in a single-slit diffraction pattern. The formula for the position of minima in single-slit diffraction is \( a \sin \theta = m \lambda \), where \( a \) is the slit width, \( \theta \) is the angle, \( m \) is the order of the minimum, and \( \lambda \) is the wavelength.

Step 2: Identify Given Values

From the problem, we are given that the wavelength \( \lambda = 550\, \text{nm} = 550 \times 10^{-9}\, \text{m} \), \( \theta = 0.32^{\circ} \), and \( m = 2 \).

Step 3: Convert Angle to Radians

Since diffraction formulas require angles in radians, convert \(0.32^{\circ}\) to radians: \( \theta (\text{radians}) = 0.32 \times \frac{\pi}{180} \approx 0.005585 \text{ radians}.\)

Step 4: Apply the Single-Slit Diffraction Formula

Use the formula for the position of minima: \( a \sin \theta = m \lambda \). Substitute the given values: \( a \sin(0.005585) = 2 \times 550 \times 10^{-9} \).

Step 5: Calculate the Slit Width

Rearrange the equation to solve for \( a \): \( a = \frac{2 \times 550 \times 10^{-9}}{\sin(0.005585)} \). Calculate \( a \approx 0.000197 \text{ meters} = 197 \text{ micrometers}.\)

Key Concepts

Diffraction Pattern

A diffraction pattern results from the interaction of waves when they encounter an obstacle or a slit. In single-slit diffraction, light waves pass through a narrow slit and spread out on the other side, creating a series of bright and dark bands on a screen. These bands are known as maxima (bright spots) and minima (dark spots). The dark areas are where destructive interference occurs, while the bright areas are where constructive interference happens. For a slit, these patterns depend heavily on how the wavefront bends around the edges of the slit. This bending of waves, known as diffraction, causes them to interfere with each other. In examining a diffraction pattern:

• The central bright band is usually the brightest and widest.
• Secondary maxima progressively diminish in intensity moving outwards.
• The condition for minima positions is determined by the geometry and properties of the incident light.

Wavelength

Wavelength is a fundamental concept in wave physics. It refers to the distance between consecutive crests (or troughs) of a wave. In the context of light, wavelength is a crucial factor determining the color that we perceive and the behavior of light waves.In our exercise, we address light with a wavelength of 550 nm. To use this value in calculations, we must convert it to meters, because the standard unit of length in physics is meters. Consequently:

• Wavelength, \( \lambda = 550 \text{ nm} = 550 \times 10^{-9} \text{ m} \).
• Wavelengths determine the pattern, as different wavelengths will create different spacing of bands.

The wavelength of light directly impacts the diffraction pattern produced.

Angle Conversion

In physics, particularly when dealing with wave optics, it's crucial to use radians for angle measurements in equations. This is because most trigonometric functions and diffraction equations are inherently connected to radians and not degrees.For our single-slit diffraction problem:

• The given angle for the second-order minimum was \( 0.32^{\circ} \).
• Convert degrees to radians using the conversion factor: \( 1^{\circ} = \frac{\pi}{180} \) radians.
• Hence, \( 0.32^{\circ} \approx 0.32 \times \frac{\pi}{180} = 0.005585 \) radians.

Converting angles properly ensures that calculations using formulas like trigonometric or diffraction equations yield accurate results.

Diffraction Formula

The single-slit diffraction formula is an essential equation to understand the positions of minima in a diffraction pattern. The formula is expressed as:\[ a \sin \theta = m \lambda \]Where:

• \( a \) is the slit width.
• \( \theta \) is the angle at which the minimum occurs.
• \( m \) is the order of the minimum (e.g., first-order, second-order, etc.).
• \( \lambda \) is the wavelength of the light.The formula provides a way to calculate the slit width if other values are known. In our problem, knowing the wavelength and angle for the second-order minimum, we rearrange the equation to find the slit width \( a \):\[ a = \frac{m \cdot \lambda}{\sin \theta} \]This rearrangement lets us directly obtain the slit width. It's crucial to use the angle in radians for accurate calculations.