The aim of my coursework is to calculate the wavelength of red laser light using the diffraction grating formula and the Youngs double slit formula.

Calculation using the diffraction grating

These below steps show the calculation when using the diffraction grating which then will be followed by the calculation when using the slits as they both have a slightly different method.

This sketch below shows the part where the light builds maxima of fringes on the wall:

First of all we need to calculate θ by using trigonometry:

Tan θ = Distance between central maximum and 1st order spectrum

Distance between grating and wall

     θ = tan ˉ¹  Distance between central maximum and 1st order spectrum

Distance between grating and wall

To use the diffraction grating formula we need to find the spacing between each gap in the grating.

d =

d x sin θ = x λ 

Then we use the diffraction grating formula.

d = distance between each slit

θ = the angle calculated above

= the order of the maxima

 λ = wavelength to be calculated

Then we rearrange the formula to make λ the subject and we get.

λ = d x sin θ

   

Example with my results

Diffraction grating:                                                 300 lines per mm

Order of the maxima:                                                 1

Distance between central maximum and 1st order spectrum:                41cm

Distance between grating and wall:                                         2.0m

1. Calculate θ:

tan θ =

θ = tan ˉ¹ (0.205) = 11.59°

2. Calculate d:

d = x 10 ˉ⁶

3. Use diffraction grating formula:

λ = 3.33 x 10 ˉ⁶ x sin 11.59

                        1

Table from diffraction grating

From the graph it can be interpreted that with the use of lens, the closer the value is to the actual wavelength value. The assumption I made about the use of lens increasing accuracy turns out to be correct because the dots are smaller on the wall and therefore we can reduce the uncertainty in the measurement of the distance between the fringes.

Calculation for 2nd method with using slits

Young’s double slit formula =

λ = w x s

     d

λ = wavelength to be calculated

w = width of fringe which is measured by a ruler once maxima is marked on the wall

s = slit separation which is given on the slits used

d = distance between slit and wall

Slit no. 1:

Width of fringe:                                                         0.5cm

Slit separation:                                                         0.25mm

Distance between slit and wall:                                                 2.0m

λ = (0.5 x 10ˉ²) x (0.25 x 10ˉ³)

2

Slit no. 2:

Width of fringe:                                                                         0.4cm

Slit separation:                                                                         0.31mm

Distance between slit and wall:                                                         2.0m

λ = (0.4 x 10ˉ²) x (0.31 x 10ˉ³)

2

Slit no. 3:

Width of fringe:                                                                         0.7cm

Slit separation:                                                                         0.18mm

Distance between slit and wall:                                                         2.0m

λ = (0.7 x 10ˉ²) x (0.18 x 10ˉ³)

2

Slit no. 4:

Width of fringe:                                                                         1.1cm

Slit separation:                                                                         0.12mm

Distance between slit and wall:                                                         2.0m

λ = (1.1 x 10ˉ²) x (0.12 x 10ˉ³)

2

Table from variety of slits

Uncertainty

Uncertainty in ruler: ±0.5 mm, i.e.

• Uncertainty in distance between slit/grating and wall:

0.5 X 10ˉ³

       2        

• Uncertainty in distance between maxima on the wall when using diffraction grating:

                                       0.5 X 10ˉ³

                                       41 X 10ˉ²        

• Uncertainty in distance between maxima on the wall when using variety of slits:

1. 0.5 X 10ˉ³

0.4 X 10ˉ²

2. 0.5 X 10ˉ³

0.5 X 10ˉ²

3. 0.5 X 10ˉ³

0.7 X 10ˉ²

4. 0.5 X 10ˉ³

1.1 X 10ˉ²

The actual wavelength of red laser light is 632.8 nm.

From the graph drawn from the result of the slit experiment it can be concluded that as the slit separation decreased the closer you get to the actual value. At 0.18mm separation I got the closest value to the actual value. At 0.18mm the wavelength I got is 630nm with the actual value being 632.8nm.

Errors

• The fringes produced on the wall by the diffraction grating were too big and so it was difficult to decide from where to measure. This problem was improved by the use of lenses as the dots of light produced on the wall were sharper and smaller.
• The fringes produced on the wall by the variety of slits were too small so it was hard to mark on exactly and reading off measurements with a ruler was difficult. This problem could have been improved by using a magnification glass for clearer and sharper view of the fringes.
• If the converging and diverging lens was not straight behind each other, the light deflected in other directions.

Analysis of results achieved

I proved that the method of using lenses when doing the diffraction grating experiment increases accuracy. The difference of result I got from when using the lens and without the lens was massive and using the lens gave me a result close to the actual wavelength of red light, increasing the accuracy. I also came to an assumption that as you increase the lines per mm on a diffraction grating; the more closer you get to the actual wavelength. However as a variety of diffraction grating was not available to me, i couldn’t prove my assumption. I made the assumption after looking at the results I achieved from my second experiment with using the slits because as the distance between each slit decreased, the closer i got to the wavelength of red light.

I am going to use the results of the experiment I did of the diffraction grating with the use of lens as its closer to the actual wavelength value.

As you can see on the visible spectrum of the electromagnetic spectrum, we can see that my result i got from the grating experiment, with the use of lens, 653nm is in the red section.

I got even closer to the actual wavelength by using my second experiment with the use of variety of slits, all with different slit separations. The closest wavelength I got to the actual wavelength was 630nm at a slit separation of 0.18mm.