Emmanuel O.
DP2 Physics
15 September 2012
DP Physics – Diffraction of Light
Data Collection and Processing
Raw data:
Qualitative data:
Red was the colour of light that was diffracted most by the diffraction grater, whereas blue
(adjacent to violent and indigo) was among the least diffracted.
Quantitative data:
Table 1: The distance between the straight-through to the diffracted blue, green and red
fringes ( ) at different distances between diffraction-grater and the screen (L).
Distance between diffractiongrater & screen (L)
/ cm
∆L= ±0.001m
0.300
0.400
0.500
0.600
0.700
Distance from the straight-through to
the blue, green and red fringe
/ cm
∆ = ±0.001m
Blue
Green
Red
0.045
0.052
0.064
0.044
0.050
0.066
0.045
0.051
0.066
0.044
0.052
0.064
0.046
0.050
0.066
0.057
0.067
0.087
0.060
0.070
0.089
0.058
0.068
0.088
0.058
0.067
0.089
0.058
0.068
0.087
0.073
0.088
0.113
0.074
0.088
0.113
0.072
0.087
0.112
0.073
0.089
0.114
0.072
0.087
0.112
0.084
0.096
0.128
0.086
0.098
0.130
0.085
0.097
0.129
0.084
0.096
0.129
0.085
0.098
0.129
0.097
0.120
0.159
0.100
0.121
0.160
0.099
0.120
0.158
0.097
0.120
0.158
0.098
0.121
0.160
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Emmanuel O.
DP2 Physics
15 September 2012
Arriving at the uncertainties:
For L and , the distances were measured the exact same metre ruler available at the
school Physics lab. The smallest division of this ruler is 0.001m, and the error is therefore
0.0005m. However, since this error is present when looking at placing the ruler on the
position of diffraction grater and also when reading the measurement off the ruler at the
screen, 0.0005m is multiplied by 2. Therefore, the error of the distances is 0.001m.
Data processing:
Exemplar Calculations
To process the raw data that will eventually enable me to find the wavelength of the various
colour fringes, I found the average. The average was found by simply summing the three
measurements and dividing the sum by five.
For Example for ‘Average distance from the straight-through to the blue fringe’ where
L=0.300m:
To obtain the Error in Average distance from the straight-through to the measured fringe, I
simply found the difference between the longest distance and the shortest for the L, and
divided the difference by two.
For example for the Error in Average distance from the straight-through to the blue fringe,
where L=0.300m:
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Emmanuel O.
DP2 Physics
15 September 2012
Table 2.1: The average distance between the straight-through and the blue fringe (
at different diffraction-grater to screen distances (L).
Distance between
diffraction-grater &
screen
/m
∆L = ±0.001m
Average distance from the
straight-through to the
blue fringe
0.300
0.400
0.500
0.600
0.700
0.045
0.058
0.073
0.085
0.098
/m
)
Error in Average distance from
the straight-through to the
blue fringe
∆
= ±m
0.001
0.002
0.001
0.001
0.002
Table 2.2: The average distance between the straight-through and the green fringe (
at different diffraction-grater to screen distances (L).
Distance between
diffraction-grater &
screen
/m
∆L = ±0.001m
Average distance from the
straight-through to the
green fringe
0.300
0.400
0.500
0.600
0.700
0.051
0.067
0.088
0.101
0.120
/m
Error in Average distance from
the straight-through to the
green fringe
∆
=±m
0.001
0.001
0.001
0.001
0.001
Table 2.3: The average distance between the straight-through and the red fringe (
different diffraction-grater to screen distances (L).
Distance between
diffraction-grater &
screen
/m
∆L = ±0.001m
Average distance from the
straight-through to the red
fringe
0.300
0.400
0.500
0.600
0.700
0.065
0.088
0.113
0.131
0.159
/m
)
) at
Error in Average distance from
the straight-through to the red
fringe
∆
=±m
0.001
0.001
0.001
0.001
0.001
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Emmanuel O.
DP2 Physics
15 September 2012
To be able to calculate the wavelength of the fringes, via plotting the average distance
between the straight-through and the coloured fringe (
screen distances (L), the diffraction grating equation:
) at different diffraction-grater to
Since n=1
The equation when plotted, is linear, hence the formula:
.
In the form
; y (dependent variable) is the average distance between the
straight-through and x (independent variable) is the distance of the diffraction-grater to
screen, whereas m (gradient) =
and the intercept, c, is 0.
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Emmanuel O.
DP2 Physics
15 September 2012
Fig. 1.1: Graph of the average distance between the straight-through and the blue fringe
(
) vs. different diffraction-grater to screen distances (L).
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Emmanuel O.
DP2 Physics
15 September 2012
Fig. 1.2: Graph of the average distance between the straight-through and the green fringe
(
) vs. different diffraction-grater to screen distances (L).
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Emmanuel O.
DP2 Physics
15 September 2012
Fig. 1.3: Graph of the average distance between the straight-through and the red fringe
(
) vs. different diffraction-grater to screen distances (L).
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Emmanuel O.
DP2 Physics
15 September 2012
Graph 1.1 (Blue fringe)
Equation of the best fit line:
Calculating the uncertainty in the gradient using maximum and minimum lines:
Min. line equation:
Max. line equation:
The uncertainty in the gradient of the best fit line is calculated using the minimum and
maximum lines, with the formula:
.
(2)
As seen prior to the graphing, the gradient of the graphs is important in determining the
wavelength of the measured fringes. Using the error of the gradient, the maximum,
minimum and best fit wavelengths can be calculated.
(Re-arranged)
(3s.f.)
The wavelength of blue is 446nm according to the best fit line. To calculate the error in this
wavelength, the wavelength of blue as per the maximum and minimum lines, would be
used. The uncertainty in wavelength would be half the difference of the maximum gradient
(maximum line) and the minimum gradient (minimum line).
Uncertainty in
:
The uncertainty in the wavelength is found using the formula:
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Emmanuel O.
DP2 Physics
15 September 2012
So to find the maximum and maximum wavelength, the gradient and its calculated error
(i.e. (2)) are used:
(3s.f.)
(3s.f.)
(1s.f.)
Error in intercept:
Error in intercept is found by finding half the difference of the intercept of the maximum line
(c min) and the minimum line (c max).
6 (1s.f.)
6
The intercept uncertainty encompasses the origin, and reassures the possibility the intercept
was 0.000, as possible error may have resulted in the deviation.
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Emmanuel O.
DP2 Physics
15 September 2012
Graph 1.2 (Green fringe)
Equation of the best fit line:
Calculating the uncertainty in the gradient using max. and min. lines:
Min. line equation:
Max. line equation:
The uncertainty in the gradient of the best fit line is calculated using the minimum and
maximum lines, with the formula:
.
(3)
Subsequently, the maximum wavelength (largest gradient – maximum line), minimum
wavelength (smallest gradient – minimum line) and best fit wavelengths can be calculated.
(Re-arranged)
(3s.f.)
The wavelength of green fringe is 576nm as per the gradient of the best fit line. To calculate
the uncertainty in the wavelength of green, the same calculations will be done using (3).
Uncertainty in
:
The uncertainty in the wavelength is found using the formula:
First, finding the minimum and maximum wavelength:
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Emmanuel O.
DP2 Physics
(3s.f.)
15 September 2012
(3s.f.)
Error in intercept:
Error in intercept is found by finding half the difference of the intercept of the maximum line
(c min) and the minimum line (c max). Intercept of best fit line is 0.001 in terms of 3s.f.
1 (1s.f.)
Identical to the intercept of the blue fringe, the uncertainty of this intercept renders the
probability that the intercept may actually be 0.
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Emmanuel O.
DP2 Physics
15 September 2012
Graph 1.3 (Red fringe)
Equation of the best fit line:
Calculating the uncertainty in the gradient using max. and min. lines:
Min. line equation:
Max. line equation:
The uncertainty in the gradient of the best fit line is calculated using the minimum and
maximum lines, with the formula:
.
(4)
Subsequently, the maximum wavelength (largest gradient – maximum line), minimum
wavelength (smallest gradient – minimum line) and wavelength of the line of best fit can be
calculated.
(Re-arranged)
(3s.f.)
The wavelength of red fringe is 769nm using the gradient of the best fit line. To calculate
the uncertainty in this wavelength, the same calculations are to be done using (4).
Uncertainty in
:
The uncertainty in the wavelength is found using the formula:
First, finding the minimum and maximum wavelength:
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Emmanuel O.
DP2 Physics
(3s.f.)
15 September 2012
(3s.f.)
(1s.f.)
Error in intercept:
Error in intercept is found by finding half the difference of the intercept of the maximum line
(c min) and the minimum line (c max). Intercept of best fit line is 0.001 in terms of 3s.f.
3 (1s.f.)
Identical to the intercept of the blue fringe, the uncertainty of this intercept renders the
probability that the intercept may actually be 0.
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Emmanuel O.
DP2 Physics
15 September 2012
Therefore, in summary:
Table 3: The calculated wavelengths and their uncertainties when white diffracted by a
diffraction-grater (d=3.33x10-6) on a screen.
Colour of Fringe
Calculated wavelength
Error in calculated
Wavelength
∆
Blue
Green
Red
446
576
769
20
3
20
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Emmanuel O.
DP2 Physics
15 September 2012
Conclusion and Evaluation
Conclusion
To validate the wavelength found, I chose to compare the values to the literature found in
Tsokos’ Physics for the IB Diploma (Fig. 2)
Fig. 2: Captured image of tabulated frequency and wavelengths of visible light colours.
The accuracy of my obtained data could be determined by looking at the percentage
deviation of the data, as per the literature value. Since the wavelengths of the colours is a
range, I decided to take the median of the group, i.e. for Blue, where λ= (455 to 492) nm, I
assumed the literature value as 473.5nm.
Percentage deviation is found by calculating the difference in the literature wavelength and
experimental wavelength. This attained difference is then divided by the literature
wavelength and then multiplied by 100. Hence the result is the deviation in percentage.
Therefore, the percentage deviations are as follows:
Percentage deviation in
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Emmanuel O.
DP2 Physics
15 September 2012
Percentage deviation in
Percentage deviation in
The calculated wavelengths for the three different colours of visible light are fairly accurate
as apart from the blue fringe, the calculated green and red fringes fall within the range
provided in Tsokos’ Physics for IB Diploma.
From the graphs drawn above, I can conclude that when diffracting white light, the distance
between the diffraction-grater and the screen (L) is linearly related to the distance from the
straight-through to the measured fringe (x). The gradient of this linear relationship is the
wavelength divided by the slit width of the diffraction-grater. Note: units of these quantities
has to be uniform, preferably metres.
Moreover, considering intercepts and their errors, the best fit lines are actually supposed to
cut through the origin. Therefore, upon the assumption that is true, then the linear
relationship of these two quantities could now be described as a direct relationship. This is
because the best fit line would be in the form, y=mx, as c=0.
This supports the theory
(5)
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Emmanuel O.
DP2 Physics
15 September 2012
Simply put, when (5) or (1) is re-arranged and represented, it will lead to the other, in other
words:
Since n=1, then
This evidently shows that this experiment supports the theory put forward.
This could also be thought of differently but still provide the same answer. The equation (5)
is the diffraction grating equation. Imagine the same set-up but the only moving apparatus
is the screen (back and forth), i.e. how far it is from the diffraction grater. Clearly,
everything will be constant except the distance from the straight-through to coloured fringe.
And since these quantities will increase simultaneously (i.e. linearly), whilst having an
intercept of 0 (y=mx), then they must be directly proportional.
Evaluation
Personally, I believe my results are fairly conclusive, accurate and consistent. This because
the calculated wavelength do not deviate, in percentage, as accuracy is evident as the line
of best fit cuts through all error bars in all the drawn graphs, and the values attained from
calculating and manipulating the results, are found within the literature values. Moreover, I
am confident of the presented data as obtaining five replicates for each value of L,
diffraction-grater to screen distance, resulted in errors of low magnitude.
Among the errors existent in the experiment’s data, one is the parallax error. Parallax error
refers to apparent inclination of a line sight that is biased against the true line of light, hence
resulting in poor or incorrect measurement of a value. On the graphs drawn, there are
points that are outliers or do not conform to the line of best fit; these are random errors.
Parallax errors were present as the experiment was set-up on the table, which forced me to
measure the data from the sides. This could have altered the measured distance from the
straight-through to the measured fringe, hence affecting the calculated wavelength of colour
measured. Although this error appears to be existent throughout the experiment, it may
prominent in other sets of data.
Moreover, parallax error is coupled with the lack of clarity of exactly where the coloured
fringes were. This worsened as the distance between the screen and diffraction grater was
decreased, as the spread on the screen in terms of width of the continuous spectrum
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Emmanuel O.
DP2 Physics
15 September 2012
decreased. This made it harder to recognize exactly where the red, green and blue fringes
were located. This is a systematic error that could have affected the values attained, and
this is evident for the blue fringe. In the continuous spectrum of visible light, blue is least
distinct in comparison to red and green, as distinguishing between blue from violet, indigo,
etc., at poor brightness is virtually impossible. This could be explained by the fact that
different colours have different amounts of constructive interferences at different distances
from the screen. Nevertheless, the clarity of the fringes does not affect the identified
relationship as linearly related but the wavelengths instead.
Also, a prevalent error present in the experiment is the small angle approximation. As it
name puts it, the approximation is intended to be made for small angles of about times ten
power negative four. Instead, the small angle approximation was applied to the experiment
where the angles clearly do not conform to the assumption that sinϴ ϴ. This
approximation increases the value of the angle which affects the distance from the straightthrough to the measured fringe. In this case then, it means that the calculated wavelength
would be greater than the true wavelength of the coloured fringe.
Improvements
In order to eliminate or diminish the parallax error in the experiment, I would simply use a
setsquare coupled with performing the experiment on the ground, where I would be
measuring the fringes from a view of the top. This would decrease the degree of the
parallax error as the setsquare ensures the measurement is aligned parallel to the colour
fringe, whilst changing my position orients my angle of measurement to being in the
appropriate line of sight.
To improve the clarity of the visible light spectrum displayed on the screen, the first
mitigation measure would be to perform the experiment in a dark room, as opposed to
performing it in broad daylight. This will reduce the interference of the white light, and
enable the pattern on the screen to be much clearer. In addition to the dark room, I could
use a different diffraction grater, one with a decreased slit width. The decreased slit width
would result in greater diffraction, hence the intensity of the principal maxima to increase
their overall brightness and visibility. This will help distinguish the colours on the spectrum.
However, some of the colours might be hard to locate, hence having an image of a
spectrum or display of colours to refer to, would aid identify exactly where they lie.
Also, in order to find the wavelength of the coloured fringes more accurately, I could diffract
monochromatic (red, green and blue), that will interfere constructively and destructively to
showing a pattern of bright fringes of that colour equally spaced out from the dark fringes.
This would mean I could conveniently and accurately identify the fringe separation, and thus
accurately determining the wavelength of the light.
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