The target is to take measurements to calculate the wavelength of red laser light by using the diffraction grating formula. Therefore I will use a variety of diffraction gratings. To improve accuracy I will always do the experiment with and in absence of two lenses. With this step I hope to get closer to the real wavelength.
The Set-up
Equipment list:
1. red laser
2. two metre rulers
3. wall or projector screen
4. double slit
5. slit holder
6. variety of diffraction gratings
7. diverging lens
8. converging lens
9. two lens holders
10. graph paper
11. cello tape, blue tack
12. marker pens
The light from the laser passes through the diverging lens and splits up. Afterwards the converging lens concentrates the light. This process gives a more focused and smaller dot on the wall which leads to higher accuracy. The grating causes the concentrated light to break up again. Maxima occur on the screen where the light is in phase. The dot in the centre is called central maximum or 0th order spectrum.
The next dots left and right from the central maximum are called 1st order spectrum; the next ones are called 2nd order spectrum and so on.
The measurements
1. Set up the equipment
2. Cut the graph paper into 4 stripes and glue them together to get one long stripe
3. Stick the long stripe with blue tack on the wall
4. Stick the laser to a fixed position and turn it on
5. Adjust the diverging and converging lens to obtain a focused and small dot
6. Insert the diffraction grating into the slit holder
7. Measure the distance between the grating and the wall
8. Mark the maxima on the graph paper with a pen
9. Measure the length of the distance between central maximum and 1st order spectrum
10. Change the grating and/or take away the lenses according to the next measurement
11. Change the graph paper
12. Repeat the process until you have finished all measurements
The Calculation
This sketch shows the part where the light builds maxima on the wall.
First we need to calculate ? by using simple trigonometry:
To use the formula for the diffraction grating we need to find the spacing between each gap in the grating.
Then we use the diffraction grating formula.
Where d is the distance between each slit, ? the angle calculated above, n the order of the maxima and ? the wavelength to be calculated. We rearrange the formula:
Example
Without lenses
Diffraction grating: 80 lines per mm
Order of the maxima: 1
Distance between central maximum and 1st order spectrum: 10.4cm
Distance between grating and wall: 2.0m
1. Calculate ?
2. Calculate d
3. Use diffraction grating formula
Results
Lines per mm
Wavelength without lenses
(nm)
Wavelength with lenses
(nm)
80
649
639
300
639
634
600
636
633
Uncertainty
Uncertainty in ruler: �0.5 mm, i.e.
* uncertainty in distance between grating and wall:
* uncertainty in the distance between maxima on the screen:
The actual wavelength of red laser light is 632.8 nm.
From the graph you can see that the more lines per mm the closer you get to the actual answer. With the use of lenses we can speed up the process. The assumption I made in the beginning turns out to be right. Because the dots are smaller on the wall and we can therefore reduce the uncertainty in the measurement of the distance between the central maximum and the 1st order spectrum the results get more accurate.
Errors
1. The dots produced by the diffraction gratings are too big. It is difficult to decide from where to measure – by using lenses this error decreased dramatically.
2. If the lenses are not straight behind each other the light is deflected in other directions.
3. The grating was not always aligned at right angles to the laser beam.
Interpretation
I proved that the method using lenses increases accuracy. So for my final wavelength I will use the results with lenses. I also showed that increasing the lines per mm on a diffraction grating gets closer to the actual wavelength. So I can use a weighted average:
If we have a look on the visible spectrum of the electromagnetic spectrum we can see that 634nm is in the red section.
In total I can say that there is a very small uncertainty included in my result. As stated in the uncertainty section, the uncertainty is always under 1%. There are very few errors which are exercisable, too.
I am satisfied with this experiment, because increasing the lines per mm would not give me a significant change in the wavelength. All in all I am very close to the actual wavelength.
Remember! This is just a sample.
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