14.7 Diffraction

Diffraction is the spreading of a wave around edges or through an opening, and it shows up most strongly when the opening is about the same size as the wavelength. For single-slit diffraction with monochromatic light, the path length difference between wavefronts sets up a pattern of bright and dark bands, with dark fringes where $a\sin\theta = m\lambda$.

Why This Matters for the AP Physics 2 Exam

Diffraction is part of physical optics, where you treat light as a wave and explain the bright and dark bands it produces. On the AP Physics 2 exam, you will analyze diagrams and intensity graphs of single-slit patterns, use functional dependence to predict how the pattern changes when you adjust wavelength, slit width, or screen distance, and connect those changes to the math. This kind of reasoning fits the multiple-choice section and the Translation Between Representations question, where you move between sketches, graphs, equations, and verbal descriptions of the same setup.

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Key Takeaways

• Diffraction is the spreading of a wave around an obstacle or through an opening, and it is strongest when the opening width is close to the wavelength.
• A single slit produces a wide central bright fringe with dimmer fringes and dark bands on either side.
• Dark fringes (minima) occur when the path length difference equals a whole number of wavelengths: $a\sin\theta = m\lambda$ for $m = 1, 2, 3, \ldots$
• For small angles ($\theta < 10^{\circ}$), use $a\left(\frac{y_{\min}}{L}\right) \approx m\lambda$ to relate slit width, wavelength, screen distance, and fringe position.
• The width of the central maximum is inversely proportional to slit width, so a narrower slit spreads the pattern wider.
• The shape of the opening changes the pattern: a rectangular slit gives bands, while a circular opening gives a central spot surrounded by rings.

Diffraction Definition

Diffraction is a wave behavior where waves spread out and bend around obstacles or through openings. This is what lets waves travel into regions that are not in the straight-line path of the source.

• Diffraction is the spreading of a wave around the edges of an obstacle or through an opening. It happens for all waves, but this topic focuses on monochromatic light passing through a single narrow slit and forming a pattern on a screen.
• It lets you hear sound around a corner even when you cannot see the source.
• It explains how light entering through a small opening can reach areas not in direct line of sight.
• It demonstrates the wave nature of light.

Diffraction vs Opening Size

How pronounced diffraction is depends on the size of the opening compared to the wavelength.

• Diffraction is most pronounced when the opening size is comparable to the wavelength of the wave.
• When opening width is much smaller than the wavelength: waves spread out in many directions.
• When opening width is much larger than the wavelength: little diffraction occurs, and waves travel mostly straight through.
• Visible light ($\lambda \approx 400$ to $700$ nm) shows almost no noticeable diffraction through a doorway but significant diffraction through microscopic openings.

Interference Patterns

When waves diffract through an opening, wavefronts from different parts of the opening interfere, creating a pattern of bright and dark bands.

• Wavefronts from different parts of the opening travel different distances to reach a given point on the screen.
• For single-slit diffraction, light from different parts of the slit interferes to form a wide central bright fringe and darker regions on either side.
• Dark fringes (minima) occur when $a\sin\theta = m\lambda$, where $m = 1, 2, 3, \ldots$
• The pattern of bright and dark bands (fringes) is characteristic of wave behavior.
• These patterns are strong evidence for the wave nature of light.
• The specific pattern depends on the shape of the opening and the wavelength of the wave.

Single-Slit Diffraction Setup

A single-slit experiment shows how waves behave when passing through a narrow opening.

• Monochromatic light with wavelength $\lambda$ passes through a narrow slit of width $a$.
• Light travels a distance $L$ to a viewing screen.
• Each point along the slit acts as a source of secondary wavelets, and these wavelets interfere to create bright and dark fringes on the screen.
• The central bright fringe is the widest and most intense.
• The interference at a point on the screen depends on the path length difference $\Delta D$ between wavefronts from different parts of the slit. For a slit of width $a$ observed at angle $\theta$ from the normal, the path length difference is $\Delta D = a\sin\theta$. Dark fringes occur when this path length difference equals $m\lambda$, so $a\sin\theta = m\lambda$ for $m = 1, 2, 3, \ldots$
• For small angles (where $\theta < 10^{\circ}$), this becomes: $a\left(\frac{y_{\min}}{L}\right) \approx m\lambda$
• Here $y_{\min}$ is the distance from the middle of the central bright fringe to the $m^{th}$ order of minimum brightness on the screen.

Diffraction Pattern Variations

The diffraction pattern depends on the shape of the opening, so analyzing the pattern helps you figure out the properties of the opening and the wave.

• A narrow rectangular slit produces a central bright fringe with dimmer side fringes.
• A circular opening produces a central bright spot surrounded by rings.
• By examining the pattern shape and spacing, you can infer the size of the opening and the wavelength.

Visual Representations of Patterns

Intensity graphs and diagrams help you analyze diffraction patterns and connect them to wave properties.

• Intensity graphs show how brightness varies across the pattern.
• The central maximum is much brighter than the secondary maxima.
• The width of the central bright fringe is inversely proportional to the slit width, so narrower slits produce wider patterns.
• The spacing between fringes is directly proportional to wavelength.
• Measuring fringe spacing lets you calculate either the wavelength or the slit width.
• Comparing observed patterns with predictions supports the wave model of light.

How to Use This on the AP Physics 2 Exam

Problem Solving

• Identify what you are given: wavelength $\lambda$, slit width $a$, screen distance $L$, fringe order $m$, and fringe position $y_{\min}$.
• For minima, start from $a\sin\theta = m\lambda$. If the angle is small ($\theta < 10^{\circ}$), switch to $a\left(\frac{y_{\min}}{L}\right) \approx m\lambda$.
• Keep units consistent. Convert nanometers and millimeters to meters before plugging in.
• Remember $m$ counts dark fringes here, so the first dark fringe is $m = 1$, not $m = 0$.

Translation Between Representations

• Be ready to move between a sketch of the slit and screen, an intensity graph, the equation, and a verbal description of the same pattern.
• Use functional dependence to predict changes. If you double the wavelength, $y_{\min}$ doubles. If you double the slit width $a$, $y_{\min}$ is cut in half. If you double $L$, $y_{\min}$ doubles.
• Connect the math to the picture: a wider central maximum on the graph means a narrower slit or a longer wavelength.

Common Trap

• Do not mix up the single-slit minima condition $a\sin\theta = m\lambda$ with the double-slit maxima condition $d\sin\theta = m\lambda$ from a later topic. In single-slit, the formula gives dark fringes.

Common Misconceptions

• Diffraction does not require a special setup. Any wave passing an edge or opening diffracts; you just notice it most when the opening is close to the wavelength in size.
• The equation $a\sin\theta = m\lambda$ gives dark fringes (minima) in single-slit diffraction, not bright fringes. This trips up students who memorize it as a maxima condition.
• A narrower slit does not make a smaller pattern. A narrower slit spreads the pattern wider because central-fringe width is inversely proportional to slit width.
• The central maximum is wider than the other fringes, not the same width. It is about twice as wide and much brighter.
• Diffraction does not change the wavelength or frequency of the light. It only redistributes where the energy lands on the screen through interference.

Practice Problem 1: Single-Slit Diffraction

Solution

For a single-slit diffraction pattern, the position of the first dark fringe (m=1) is given by: $a\sin\theta = m\lambda$

For small angles, you can use the approximation: $\sin\theta \approx \tan\theta \approx \frac{y}{L}$

Where y is the distance from the center to the first dark fringe, and L is the distance to the screen.

Substituting into the equation: $a \cdot \frac{y}{L} = m\lambda$

For the first dark fringe (m=1): $a \cdot \frac{3.16 \times 10^{-3} \text{ m}}{2.0 \text{ m}} = 1 \cdot 632.8 \times 10^{-9} \text{ m}$

$a \cdot 1.58 \times 10^{-3} = 632.8 \times 10^{-9}$

$a = \frac{632.8 \times 10^{-9}}{1.58 \times 10^{-3}} = 4.0 \times 10^{-4} \text{ m} = 0.40 \text{ mm}$

Therefore, the width of the slit is 0.40 mm.

Practice Problem 2: Diffraction and Opening Size

Solution

First, calculate the wavelength of the sound wave: $\lambda = \frac{v}{f} = \frac{340 \text{ m/s}}{680 \text{ Hz}} = 0.50 \text{ m}$

Diffraction is most pronounced when the opening size is comparable to the wavelength. Since the wavelength is 0.50 m, diffraction would be more noticeable for a 0.5 m opening than for a 5 m opening. The 0.5 m opening is comparable to the wavelength, so the wave will spread significantly after passing through. The 5 m opening is much larger than the wavelength, so the wave will pass through with relatively little spreading.

Related AP Physics 2 Guides

• 14.2 Periodic Waves
• 14.3 Boundary Behavior of Waves and Polarization
• 14.6 Wave Interference and Standing Waves
• 14.1 Properties of Wave Pulses and Waves
• 14.4 Electromagnetic Waves
• 14.5 The Doppler Effect