تجارب الليزر

Solution to Experimental Question 2
Section 1
i. A typical geometric layout is as shown below.
(a) Maximum distance from ruler to screen is advised to increase the spread of the di®raction
pattern.
(b) Note that the grating (ruler) lines are horizontal, so that di®raction is in the vertical direction.
SCREEN
70 mm LASER
RULER b
FRINGES
1400 mm
ii. Vis a vis the di®raction phenomenon, ¯ = ? y
1400 mm¢
The angle ¯ is measured using either a protractor (not recommended) or by measuring the value
of the fringe separation on the screen, y, for a given order N.
If the separation between 20 orders is measured, then N = §10 (N = 0 is central zero order).
The values of y should be tabulated for N = 10. If students choose other orders, this is also
acceptable.
N §10 §10 §10 §10 §10 §10 §10 §10 §10 §10
2y mm 39.0 38.5 39.5 41.0 37.5 38.0 39.0 38.0 37.0 37.5
y mm 19.5 19.25 19.75 20.5 18.75 19.0 19.5 19.0 18.5 18.75
Mean Value = (19:25 § 1:25) mm
i.e. Mean \spot" distance = 19:25 mm for order N = 10.
From observation of the ruler itself, the grating period, h = (0:50 § 0:02) mm.
Thus in the relation
N¸ = §h sin ¯
N = 10
h = 0:5 mm
sin ¯ ¯ =
y
1400 mm
= 0:01375
10¸ = 0:006875 mm
¸ = 0:0006875 mm
Since ¯ is small,
±¸
¸

±h
h
+
±y
y
10%
i.e. measured ¸ = (690 § 70) nm
The accepted value is 680 nm so that the departure from accepted value equals 1:5%.
Section 2
This section tests the student s ability to make semi-quantitative measurements and the use of judgement
in making observations.
i. Using the T = 50% transmission disc, students should note that the transmission through the tank
is greater than this value. Using a linear approximation, 75% could well be estimated. Using the
hint about the eye s logarithmic response, the transmission through the tank could be estimated to
be as high as 85%.
Any ¯gure for transmission between 75% and 85% is acceptable.
ii. Calculation of the transmission through the tank, using
T = 1 ? R = 1 ? µn1 ? n2
n1 + n2¶2
for each of the four surfaces of the tank, and assuming n = 1:59 for the perspex, results in a total
transmission
Ttotal = 80:80%
Section 3
With water in the tank, surfaces 2 and 3 become perspex/water interfaces instead of perspex/air interfacs,
as in (ii).
The resultant value is
Ttotal = 88:5%
Section 4
TRANSMISSION
FILTER
IN/OUT
LASER
TANK SCREEN
y x
~30 mm
~550 mm
Possible con¯guration for section 4 (and sections 2 and 3)
With pure water in the tank only, we see from Section 3 that the transmission T is
TWater 88%
The aim here is to determine the beam divergence (scatter) and transmission as a function of milk
concentration. Looking down on the tank, one sees
BEAM DIAMETER
2x = 2.00 mm
LASER
~30 mm 25 mm
35 mm
2x’
2q ’
i. The entrance beam diameter is 2:00 mm. The following is an example of the calculations expected:
With 0:5 mL milk added to the 50 mL water, we ¯nd
Scatterer concentration =
0:5
50
= 1% = 0:01
Scattering angle
2x0 = 2:2 mm ; 2µ0 =
2x0
30
= 0:073
Transmission estimated with the assistance of the neutral density ¯lters
Ttotal = 0:7 :
Hence
Tmilk =
0:7
0:88
= 0:79
Note that
Tmilk =
Ttotal
Twater
and Twater = 0:88 (1)
If students miss the relationship (1), deduct one mark.
ii. & iii. One thus obtains the following table of results. 2µ0 can be determined as shown above, OR by
looking down onto the tank and using the protractor to measure the value of 2µ0. It is important
to note that even in the presence of scattering, there is still a direct beam being transmitted. It is
much stronger than the scattered radiation intensity, and some skill will be required in measuring
the scattering angle 2µ0 using either method. Making the correct observations requires observational
judgement on the part of the student.
Typical results are as follows:
Milk volume (mL) 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
% Concentration 0 1 2 3 4 5 6 7 8
2x0 2.00 2.2 6.2 9.4 12 Protractor
2µ0 (Degrees) » 0 4 12 18 23 28 36 41 48
Tmilk 1.0 0.79 0.45 0.22 0.15 0.12 0.08 0.06 0.05
iii. From the graphed results in Figure 1, one obtains an approximately linear relationship between
milk concentration, C, and scattering angle, 2µ0 (= ?) of the form
? = 6C :
iv. Assuming the given relation
I = I0e?¹z = TmilkI0
where z is the distance into the tank containing milk/water.
We have
Tmilk = e?¹z
Thus
ln Tmilk = ?¹z ; and ¹ = constant £ C
Hence ln Tmilk = ?®zC.
Since z is a constant in this experiment, a plot of ln Tmilk as a function of C should yield a straight
line. Typical data for such a plot are as follows:
% Concentration 0 1 2 3 4 5 6 7 8
Tmilk 1.0 0.79 0.45 0.22 0.15 0.12 0.08 0.06 0.05
ln Tmilk 0 -0.24 -0.8 -1.51 -1.90 -2.12 -2.53 -2.81 -3.00
An approximately linear relationship is obtained, as shown in Figure 2, between ln Tmilk and C, the
concentration viz.
ln Tmilk ?0:4C = ?¹z
Thus we can write
Tmilk = e?0:4C = e?¹z
For the tank used, z = 25 mm and thus
0:4C = 25¹ or ¹ = 0:016C whence ® = 0:016 mm?1%?1
By extrapolation of the graph of ln Tmilk versus concentration C, one ¯nds that for a scatterer
concentration of 10%
¹ = 0:160 mm?1 :
0 2 4 6 8 10
% Concentration
0
10
20
30
40
50
f
Figure 1: Sample plot
0 2 4 6 8 10
% Concentration
-4
-3
-2
-1
0
ln Tmilk
Figure 2: Sample plot
Detailed Mark Allocation
Section 1
A clear diagram illustrating geometry used with appropriate allocations [1]
Optimal geometry used - as per model solution (laser close to ruler) [1]
Multiple measurements made to ascertain errors involved [1]
Correctly tabulated results [1]
Sources of error including suggestion of ruler variation
(suggested by non-ideal di®raction pattern) [1]
Calculation of uncertainty [1]
Final result [2]
Allocated as per:
§10% (612, 748 nm) [2]
§20% (544, 816 nm) [1]
§ anything worse [0]
Section 2
For evidence of practical determination of transmission rather than
simply \back calculating". Practical range 70 ? 90% [1]
For correct calculation of transmission
(no more than 3 signi¯cant ¯gures stated) [1]
Section 3
Correct calculation with no more than 3 signi¯cant ¯gures stated
and an indication that the measurement was performed [1]
Section 4
Illustrative diagram including viewing geometry used, i.e. horizontal/vertical [1]
For recognizing the di