9.2 Single-Slit Diffraction

Some students in a lab are performing a single-slit diffraction investigation. They have a green laser but they do not know the exact wavelength of the light. 

Describe which measurements the students can take and how they can use them to calculate the wavelength.

The students recorded the following information:

Calculate the wavelength of the laser and give its fractional uncertainty.

Single-slit diffraction patterns provide evidence for light as a wave. 

The images below show the diffraction of light around a small circular object. 

Suggest how the images provide evidence for light as a wave.

The image shows a diffraction pattern from a single rectangular slit.

Sketch the diffraction pattern if the slit width was 20% smaller.

A beam of electrons is incident normally to the plane of a narrow slit of width b. The beam of electrons can be observed to diffract over an angular area of 2θ.

The uncertainty in the position, Δx, and momentum, Δp, of an electron in the beam can be described by Heisenburg's uncertainty principle

This expression can be derived by considering the possible paths an electron might take as it passes through the slit. 

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Outline the effects on the range of diffraction angles and the uncertainties of position and momentum expected when 

The formula for the position of the first minimum of the diffraction pattern produced by a single slit is given by:

Using the conditions for interference, derive this formula. 

The frequency of monochromatic light passing through a slit is 2.44 × 10¹⁴ Hz. The screen is located 3.75 m from the single slit. The distance on the screen from the central maximum to the first minimum is 0.25 cm.

Find the slit width. 

Three diffraction patterns, x, y and z, are produced from three wavelengths of light, λ, 5λ and 10λ, passing through a slit of fixed width b. 

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The graph below shows the variation of light intensity, I with angular displacement θ on a screen placed far away from a narrow slit.

Draw the variation of light intensity, I with angular displacement θ on a screen when the narrow slit is been made even narrower. 

Plane wavefronts of monochromatic light of wavelength λ are incident on a rectangular slit of width b. The light is brought to a focus after passing through the slit on a screen at a distance D from the slit as shown in the diagram below. 

The width of the slit is comparable to the wavelength of the light and b ≪ D. The point X on the screen is opposite the centre of the slit.  

The variation of the intensity incident on the screen with angle θ is shown on the graph below.

Explain the shape of the intensity distribution in terms of the conditions for interference. 

The angular half-width of the central maximum for this intensity distribution is given by the expression φ =  where θ = φ. 

Derive an expression for the half-width, d, of the central maximum using the terms, D, λ and b.

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n a single-slit diffraction experiment, the slit width is 0.030 mm and the wavelength of the light incident on the slit is 525 nm.

Calculate the angle of diffraction for the second maxima.

The single slit of width b is replaced with rectangular slits, of width 2. The intensity distribution of a single slit is shown by the dotted line.

Draw a sketch within the central maxima of the single slit of the new intensity distribution on the screen. 

An experiment is set up where white light is incident on a single slit. 

The white light illuminates the slit. The first diffraction minimum for a wavelength of 630 nm is observed at 12°.  

(i)

Identify the first secondary maximum and the second diffraction minimum on the single slit intensity graph below.

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(ii)

Determine the wavelength of light for which the first secondary maximum occurs at an angle of 12°.

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Explain why the central maximum is white but the first secondary maximum at 12° is coloured.

Explain the pattern of the colours seen on the screen between 12° and the next minima.

A rectangular slit has width b and is comparable to the wavelength λ of plane wavefronts of monochromatic light incident upon it. After passing through the slit, the light is brought to a focus on a screen. 

The diagram shows a line normal to the plane of the slit, drawn from the centre of the slit to the screen is labelled PQ. The points X and Y are the first points of minimum intensity as measured from point Q. 

The diagram also shows two rays of light incident on the screen at point X. Ray RX leaves one edge of the slit and ray PX leaves the centre of the slit. The angle φ is small. 

On the diagram, label two angles of diffraction.

Derive an expression, in terms of λ, for the path difference RT between the rays RX and RT.

Describe the changes in the experimental setup that would decrease the width of the central bright maximum.

In a certain demonstration of single-slit diffraction λ = 400 nm, b = 0.12 mm, and the screen is a long way from the slit. 

Calculate the angular width of the central maximum of the diffraction pattern on the screen.