In an experiment to measure the wavelength of monochromatic light, the angles $ \Theta $ between a central bright image ($ n = 0 $) and the first and second order images to the left and right were measured. A source of monochromatic light and a diffraction grating of 500 lines per mm were used. Describe, with the aid of a labelled diagram, how the data were obtained. The following data were recorded. $ n \quad 2 \text{ (left)} \quad 1 \text{ (left)} \quad 1 \text{ (right)} \quad 2 \text{ (right)} $ $ \Theta (degrees) \quad 36.2 \quad 17.2 \quad 17.1 \quad 36.1 $ Use the data to calculate (i) the wavelength of the light (ii) the maximum number of images that could be observed. Explain what would happen to the positions of the images if (i) the wavelength of the light was decreased (ii) the diffraction grating was replaced with a diffraction grating of 300 lines per mm.

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In an experiment to measure the wavelength of monochromatic light, the angles $ \Theta $ between a central bright image ($ n = 0 $) and the first and second order images to the left and right were measured - Leaving Cert Physics - Question 3 - 2016

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In an experiment to measure the wavelength of monochromatic light, the angles $ \Theta $ between a central bright image ($ n = 0 $) and the first and second orde... show full transcript

Worked Solution & Example Answer:In an experiment to measure the wavelength of monochromatic light, the angles $ \Theta $ between a central bright image ($ n = 0 $) and the first and second order images to the left and right were measured - Leaving Cert Physics - Question 3 - 2016

Step 1

Describe, with the aid of a labelled diagram, how the data were obtained.

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Answer

To conduct the experiment to measure the wavelength of monochromatic light, a coherent light source, such as a vapor lamp, was used to illuminate a diffraction grating. The setup included a grating and a spectrometer.

A diagram should illustrate the following:

The arrangement of the laser (or vapor lamp) positioned to shine light onto the diffraction grating.
The distance ( D ) from the grating to the screen where the diffraction pattern is observed.
The angles ( \Theta_L ) and ( \Theta_R ) marked for the first and second-order images on both left and right sides of the central maximum, indicating how angles are measured from the central bright image.

Step 2

Use the data to calculate (i) the wavelength of the light

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Answer

To calculate the wavelength ( \lambda ) of the light using the provided data, we can use the formula derived from the diffraction grating equation:

$d = 2 \times 10^{-6} \text{ m} \quad (d \text{ is the grating spacing})$
$n \sin \Theta = d \; \text{, for } n = 2\, \Theta = 36.2^{\circ} \text{ (left)} \; (d = 1 / 500 \times 10^{-3} m)$

First, compute ( d ) from the number of lines per mm: $d = \frac{1}{500 \times 10^{3}} = 2 \times 10^{-6} \text{ m}$
Then calculate ( \lambda ) using: $\lambda = \frac{d}{n \sin \Theta}$
Using ( n = 2 ) and ( \Theta = 36.2^{\circ} ): $\lambda = \frac{2 \times 10^{-6}}{2 \sin(36.2^{\circ})} \approx 5.9 \times 10^{-7} \text{ m}$

Step 3

Use the data to calculate (ii) the maximum number of images that could be observed.

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Answer

The maximum angle ( \Theta_{max} ) for observable images in a diffraction pattern is given by:

$\Theta_{max} = 90^{\circ}$

Given that: ( n_{max} = \frac{d}{\lambda} = 3 ) for this configuration, the maximum number of observable images can be calculated as follows: [ n_{max} = 3 ]
Combining both sides produces: [ 3 + 3 + 1 = 7 ]
Therefore, the maximum number of images that could be observed is 7.

Step 4

Explain what would happen to the positions of the images if (i) the wavelength of the light was decreased.

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Answer

If the wavelength of the light decreases, the angles ( \Theta ) for the maxima will also decrease, meaning the positions of the images will be closer together. This is due to the relationship observed in the diffraction equation where:

$n \lambda = d \sin(\Theta)$

A smaller ( \lambda ) results in a smaller value of ( \Theta ) for each order of maximum observed.

Step 5

Explain what would happen to the positions of the images if (ii) the diffraction grating was replaced with a diffraction grating of 300 lines per mm.

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Answer

Replacing the diffraction grating with one of 300 lines per mm would increase the grating spacing, ( d ). As a result, according to the diffraction equation:

$n \lambda = d \sin(\Theta)$
A larger ( d ) would allow for larger angles ( \Theta ) for each order of maximum, leading to the images being further apart. Thus, the overall spacing between the images would increase.

In an experiment to measure the wavelength of monochromatic light, the angles \( \Theta \) between a central bright image (\( n = 0 \)) and the first and second order images to the left and right were measured - Leaving Cert Physics - Question 3 - 2016

Worked Solution & Example Answer:In an experiment to measure the wavelength of monochromatic light, the angles \( \Theta \) between a central bright image (\( n = 0 \)) and the first and second order images to the left and right were measured - Leaving Cert Physics - Question 3 - 2016

Describe, with the aid of a labelled diagram, how the data were obtained.

Use the data to calculate (i) the wavelength of the light

Use the data to calculate (ii) the maximum number of images that could be observed.

Explain what would happen to the positions of the images if (i) the wavelength of the light was decreased.

Explain what would happen to the positions of the images if (ii) the diffraction grating was replaced with a diffraction grating of 300 lines per mm.

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