Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Define diffraction and its conditions.
• Sketch diffraction patterns from single slits.
• Explain the relationship between wavelength, opening size, and diffraction.
• Apply superposition to explain diffraction patterns.
• Distinguish between sound and light diffraction with examples.

Lesson summary

Diffraction is a phenomenon that occurs when waves encounter an obstacle or an opening. Instead of simply stopping or reflecting, waves bend around the obstacle or spread out after passing through the opening. This bending is more pronounced when the size of the obstacle or opening is comparable to the wavelength of the wave. Understanding diffraction is crucial because it explains many everyday phenomena, from how sound travels around corners to how light creates patterns in optical instruments and in nature.

Lesson notes

2.1 What is Diffraction? Diffraction is the bending of waves as they pass around an obstacle or through an opening. It's a wave property, not a particle property. Imagine dropping a pebble into a still pond. The ripples (waves) will spread out in all directions. Now, imagine placing a small barrier in the pond. The waves will bend around the barrier, spreading into the "shadow" region behind it. This bending is diffraction. The amount of diffraction depends on two main factors: Wavelength (λ): The distance between two successive crests or troughs of a wave.

Size of the Obstacle/Opening (d): The width of the barrier or opening the wave encounters.

Important Relationship: Diffraction is most noticeable when the wavelength (λ) is comparable to the size of the obstacle or opening (d). If λ >> d, the wave almost completely bends around the obstacle. If λ << d, the wave mostly travels straight through with minimal bending. 2.2 Wavefronts: 2D and 3D Wavefront: An imaginary line or surface that represents points in a wave that are in phase (i.e., vibrating together). For example, all the crests of a water wave form a wavefront. 2D Wavefronts: These occur in two dimensions, like water waves on a surface. Imagine a long, straight stick vibrating up and down in water. It creates a series of parallel wavefronts moving outwards. 3D Wavefronts: These occur in three dimensions, like sound waves or light waves emitted from a point source. A point source creates spherical wavefronts that expand outwards in all directions. 2.3 Diffraction through a Single Slit Consider a plane wave (straight wavefronts) approaching a single slit.

Huygens' Principle: This principle states that every point on a wavefront can be considered as a source of secondary wavelets that spread out in all directions. The envelope of these wavelets forms the new wavefront. This principle is crucial for understanding diffraction.

At the Slit: According to Huygens' Principle, each point along the slit acts as a source of new wavelets. These wavelets interfere with each other (superposition).

Diffraction Pattern: The interference of these wavelets creates a diffraction pattern on a screen placed behind the slit. The pattern consists of a central bright fringe (maximum) surrounded by alternating dark (minimum) and bright fringes of decreasing intensity.

Central Maximum: The central bright fringe is the widest and brightest because the wavelets from all parts of the slit arrive at the centre of the screen almost in phase.

Dark Fringes (Minima): Dark fringes occur where the wavelets interfere destructively. For a minimum to occur at an angle θ, the path difference between the wavelets from the top and bottom of the slit must be equal to an integer multiple of the wavelength (nλ).

The condition for the first minimum is: ``` d*sin(θ) = λ ``` Where: d is the width of the slit θ is the angle of the minimum relative to the center λ is the wavelength of the wave 2.4 Diffraction around an Obstacle Similar to diffraction through a slit, waves also bend around obstacles. The amount of bending depends on the wavelength and the size of the obstacle. A smaller obstacle (relative to the wavelength) causes more pronounced diffraction. The pattern formed is similar to the slit diffraction but less distinct. 2.5 Diffraction of Sound and Light Sound Waves: Sound waves have relatively long wavelengths (e.g., a 1 kHz sound has a wavelength of about 34 cm in air). Because of these longer wavelengths, sound waves diffract easily around objects and through openings, which is why we can hear sounds even when we're not in a direct line of sight of the source. Imagine hearing a conversation around a corner – that's diffraction in action.

Light Waves: Light waves have much shorter wavelengths (e.g., visible light is around 400-700 nm). Because of these shorter wavelengths, diffraction of light is less noticeable in everyday situations.

However, it's crucial in optical instruments and phenomena like the colours seen in oil films or on CDs (caused by diffraction gratings). 2.6 Worked Examples Example 1: Diffraction of Sound A sound wave with a frequency of 500 Hz travels through a doorway that is 0.8 m wide. The speed of sound in air is 340 m/s. Calculate the angle at which the first minimum of the diffraction pattern occurs.

Solution: Calculate the wavelength: λ = v/f = 340 m/s / 500 Hz = 0.68 m Apply the condition for the first minimum: d*sin(θ) = λ 0.8 m * sin(θ) = 0.68 m sin(θ) = 0.68 m / 0.8 m = 0.85 Solve for θ: θ = arcsin(0.85) = 58.2° Comment: This means the first minimum occurs at a relatively large angle, indicating significant diffraction. Sound bends significantly around the doorway.

Example 2: Diffraction of Light Light with a wavelength of 600 nm passes through a single slit that is 2.0 x 10⁻⁶ m wide. What is the angle to the first dark fringe in the diffraction pattern?