University of Groningen
The Effect of a Magnetic Field on Linearly
Polarized Light in Lead Borate
Physics lab 3
Author(s):
Harmen de Beurs (s4134524)
Eelco Obdeijn (s4128184)
May 17, 2022
Supervisors:
Sytze Tirion
Contents
1 Abstract
2
2 Introduction
3
3 Theory
3.1 The Faraday effect . . . . .
3.2 Mathematical interpretation
3.3 Effect on the light beam . .
3.4 Small angle approximation .
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4
4
4
5
5
4 Materials and Methods
4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Measurement Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
6
6
5 Results
7
6 Discusion
6.1 External light sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Small angle approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Processed results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
9
9
9
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7 Conclusion
10
8 Bibliography
11
9 Appendix
9.1 Ampere to Tesla
9.2 Data . . . . . . .
9.3 Data Processing .
9.4 Error Analysis .
12
12
13
18
19
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1
Abstract
In this report the influence of a magnetic field on linearly polarized light of different wavelengths
is investigated. The dependence of the angle of polarization on the magnetic field is measured
for three different wavelengths. This was done by applying the magnetic field on the light beam
while it propagates through a Faraday glass. With these measurements the Verdet constants
were determined for the different wavelengths. Sequentially the eigenfrequency of the electrons
in the Faraday glass were calculated. Finally the difference in refraction between left and right
handed polarized light was calculated as well for each wavelength.
2
2
Introduction
This experiment examines the interaction between polarized light propagating trough Faraday
glass parallel to a magnetic field. Polarized light means that the oscillation of the wave happens
perpendicular to the motion, and in the same plane. The polarization plane can rotate right
(clockwise) or left (anti-clockwise). The Faraday effect shows that the interactions for right and
left polarized light with the glass atoms and the magnetic field can be different. The experimental
goal is: How does the polarization angle change with respect to the magnetic field strength? With
these polarization angles, the Verdet constant of our Faraday glass can be calculated. Is there
a relation between the wavelength and Verdet constant. Lastly we are interested in the relation
between the refractive indices for left and right polarized light. What is the difference between
these refractive indices.
3
3
Theory
3.1
The Faraday effect
Light propagates trough space as a wave and oscillates in a plane (planar). The Faraday effect
occurs when polarized light propagates trough a glass medium with a magnetic field parallel to
the light motion. Polarized light is when the oscillation is perpendicular to the direction of the
beam, and all photons oscillate in a single plane. The interactions of the incident beam with the
oscillations of free electrons in the medium will cause the plane of polarization to rotate around
the axis of motion. [1]
3.2
Mathematical interpretation
We will describe the motion of an electron around a "glass" molecule as in equation 1. [2]
d2 r
dr
= −mω02 r + qE0 e±iωt + iqB
(1)
dt2
dt
The movement of the electrons in the xy-plane is expressed as r=x+iy to prevent complex vector
calculations. The left side is the total force on the electron Ftot , where m is the mass of the
2
electron and ddtr2 the acceleration. The binding force between the electron and nucleon is expressed with the spring constant mω02 , where ω0 is the eigenfrequency of oscillation. The driving
force qE0 e±iωt is due to the electric field of the incident beam creates trough its polarization
rotation (at frequency ω). Here q is the electron charge and E0 is the electric field at t=0. As
the rotation of polarization can happen clockwise (-) and anti-clockwise (+), a ± is added. The
last part of the equation iqB dr
dt describes the Lorentz force due to the magnetic field B. With
this we can calculate the effect of clockwise and anti-clockwise polarized light on the refractive
index, which is expressed as in equation 2. To make the equation more manageable the binomial
approximation was used on the assumption that m(ω02 − ω 2 ) >> qωB [3] Here N is the amount
of contributing resonances (which is about the amount of valence electrons) per cm3 , m is the
mass and q the charge of an electron and ϵ0 is the permitivity of free space.
"
#
N q2
N q2
qωB
2
≈1+
1∓
nr,1 = 1 + (2)
ϵ0 m ω02 − ω 2 ± qωB
ϵ0 m ω02 − ω 2
m ω02 − ω 2
m
Based on equation 2 a relation between nl and nr was found, as shown in equation 3.
nr − n1 ≈
n2 − 1
qωB
n m ω02 − ω 2
4
(3)
3.3
Effect on the light beam
In equation 4 the electric field of the polarized light is written as the sum of the electric fields of
left (El ) and right (Er ) polarized light.
Esum = Er + El = E0 exp [−i (ωt − θr )] + E0 exp [i (ωt − θ1 )]
1
1
= 2E0 exp i (θr − θl ) cos ωt − (θr + θ1 ) .
2
2
(4)
The angle 21 (θr − θl ) describes the change in angle of the electric field of the polarized light
in a medium (with the subscript for left and right polarization). The polarization rotates with
the electric field, so using equation 3, equation 5 for the polarization angle θ was derived.
1
πd
(θr − θl ) =
(nr − n1 ) = BV d,
(5)
2
λ
The angles θr and θl in equation 5 have been written in terms of (nr , n1 , d and λ. Here d is the
distance of propagation trough the medium and λ the wavelength of the incident wave, and V is
the Verdet constant of the medium (dependant on the wavelength) given in equation 6 [2]
θ=
V =
3.4
n2 − 1
qω
qω 2
π n2 − 1
.
=
λ n m ω02 − ω 2
2n mc ω02 − ω 2
(6)
Small angle approximation
The angle of polarization θ of a light beam propagating trough glass rotates with a period of
1π 5, when increasing or decreasing the magnetic field strength B. If we take our y-axis to
be perpendicular to the motion of the beam, then the y-component of the polarization can be
expressed as cos(θ)2 . Now for a small interval around θ = 21 π, we can approximate cos(θ)2 as a
parabole (Figure 1). This is the small angle approximation, and can be written as equation 7
cos(θ)2 ≈ aθ2 + bθ + c
(7)
Figure 1: The small angle approximation. The horizontal axis is θ (rad), y is the axis perpendicular to the motion of light.
5
4
4.1
Materials and Methods
Experimental Setup
For this experiment the following setup was used Figure 2. Here all equipment was aligned such
that a light beam emitted by the light source propagates trough all components into the camera.
To prevent external light from disturbing the measurements, the experiment was conducted in
the dark. For the incident beam, three different wavelengths where used. A beam emitted by
a sodium lamp (1) (main wavelength 589nm), a mercury lamp (1) (main wavelength 550nm)
or the mercury lamp with a color filter (2) to create a beam of 448nm. First, the light beam
propagates trough the first polarizer (3), resulting in the beam being vertically polarized. After
this the polarized beam propagates trough the Faraday glass element (4) with index of refraction
n=1.60 and a length of d=76 mm, which is fitted into an electromagnet (4) with a variable and
reversible field. The source of this electromagnet can have input currents between 0 and 9 ampere
(creating a magnetic field parallel to the light beam). To create a negative magnetic field (anti
parallel), the input and output cable of the ampere source can be switched. The gauge graph
of this magnet can be found in Appendix 9.1. Then the light propagates through the second
polarizer (which is called the analyzer) and is rotatable with a scale to measure the polarization
rotation angle. Lastly the light falls onto a camera connected to a computer with software to
determine the intensity. The whole setup will be aligned on a rail.
Figure 2: experimental setup with the following elements mounted on a rail: light source (sodium
or mercury lamp) (1), a color filter (2), first polarizer (3), electromagnet (4), analyzer (5), camera
(6)
4.2
Measurement Plan
Measurement sets will be taken for different magnetic field strengths with input currents between
-9A to 9A with steps of 0.5A measurements. For each measurement set the analyzer will be turned
such to approximately minimize the intensity. Then the intensity will be measured using the
camera, thereafter will the analyzer be turned to +5 and -5 degrees with steps of 1 degree (a
total of 11 measurements per set). Over every set a quadratic function is plotted, instead of a
trigonometric function as in accordance with the small angle approximation 7. This will be done
for the sodium lamp and the mercury lamp with the green filter, for the blue filter the stepsize
for the current will be 2A (and a measurement when the magnet is turned off) and for the angle
will be taken -50 till 50 degrees around the darkest point with steps of 10 degrees.
6
5
Results
All measured data points are listed in tables in Appendix 9.2. For each magnetic field strength
a measurement set was taken over which a quadratic curve was fitted. From these curves the
minima were determined. This gives the angle of rotation depending on the magnetic field for
each of the three measured wavelengths (for full analysis see 9.3).
Figure 3: Fitted lines for data: the angle of rotation (θ) for different strengths of the magnetic
field (B) for the three different wavelengths. Keep an eye on the differences in scales
The polarization angle θ is plotted against the strength of the magnetic field in figure 3, this
is done for each wavelength and a linear fit is added. By taking the slope of the fitted lines
(which can be found in Figure 3) and dividing it by the length of the piece of lead borate in
accordance with equation 5 we van calculate the different Verdet constant for the corresponding
incident beam wavelengths:
• for λ=589nm : Slope = 2.470 ± 0.005, V= 32.50 ± 0.07 rad T −1 m−1
• for λ=550nm : Slope = 2.986 ± 0.007, V = 39.3 ± 0.1 rad T −1 m−1
• for λ=448nm : Slope = 4.8 ± 0.3, V= 63.4 ± 4 rad T −1 m−1
7
With the Verdet constant for sodium lamp (589 nm) the eigenfrequency of the electrons was
be calculated using equation 6. This gives ω0 = 1.4217 ∗ 1015 ± 2 ∗ 1011 Hz.
The Verdet constant was used to determine the difference in refraction between the right
hand and the left hand with ∆n = BVπ λ for the largest rotation. This gives ∆n = 7.55 ∗ 10−7 ±
0.24 ∗ 10−7 . Using the eigenfrequency calculated for the sodium lamp, the Verdet constant will
be calculated for 448nm and 550nm using equation 6. This gives:
• for λ=550nm : V= 50.14 ± 0.04 rad T −1 m−1
• for λ=448nm : V= 81.40± 0.06 rad T −1 m−1
Finally with these Verdet constants the difference in diffraction will also be calculated for
these two wavelengths with the same formula as before, which gives:
• for λ=550nm :∆n = 1.08 ∗ 10−6 ± 0.04 ∗ 10−6
• for λ=448nm :∆n = 1.44 ∗ 10−6 ± 0.05 ∗ 10−6
8
6
6.1
Discusion
External light sources
In the darkened room of the experiment the surrounding light sources with a relatively small
intensity (for example a desk lamp) had low to no measurable effect on the measurements.
However, when changing the analyzer, ones hand moves relatively close to the camera. During
measuring we found that the light of external sources can be reflect by your hand onto the camera.
This alters the intensity measured, but is solvable by wearing black/non reflecting gloves.
6.2
Small angle approximation
By taking large angles for λ=448nm, the small angle approximation is considerably less precise.
This together with the lesser amount of measurements for λ=448nm resulted in much larger
errors, as can be seen in graph = 448nm in Figure 3. In future experimentation all measurement
sets should be taken in a similar fashion to the 550nm and 589nm light beams.
6.3
Processed results
From the results in Figure 3 the relation between the angle of rotation and the magnetic field
strength becomes clear, it is a increasing linear dependency for whom the slope increases as
the wavelength becomes shorter. The Verdet constants determined by the slopes (Figure 3)
is about a factor 1.3 smaller than the Verdet constants calculated using eigenfrequency of the
electrons (based on the measurement for λ = 589nm). The sodium lamp has a much more
centered spectrum than the mercury lamp (even with the colour filter). As a result the mercury
light source, light pollution of different wavelengths might alter the polarization angle, which
would explain the difference in the Verdet constant determined using the slopes and from the
eigenfrequency. With this reasoning we decided to determine the eigenfrequency of the electrons
using λ = 589nm, as the sodium lamp is more trustworthy.
9
7
Conclusion
With the experiment the following Verdet constants where obtained:
• λ=589nm: V= 32.50 ± 0.07 rad T −1 m−1
• λ=550nm: V= 50.14 ± 0.04 rad T −1 m−1
• λ=448nm: V= 81.40 ± 0.06 rad T −1 m−1
Which seem correct and were obtained with large enough amount of measurements to make
the error marge small to make them accurate and show a decreasing dependence as the wavelength
increases. To determine the type of relation the experiment should be conducted for more
wavelengths and for a broader range.
The difference in refraction was determined for the different wavelengths:
• for λ=589nm :∆n = 7.55 ∗ 10−7 ± 0.24 ∗ 10−7
• for λ=448nm :∆n = 1.44 ∗ 10−6 ± 0.05 ∗ 10−6
• for λ=550nm :∆n = 1.08 ∗ 10−6 ± 0.04 ∗ 10−6
Thus the difference in refractive indices decrease when the wavelength increases.
10
8
Bibliography
[1] David Jeffrey Griffiths. 9. Electromagnetic Waves. Cambridge University Press, 2018.
[2] The Faraday effect. University of Groningen, 2020.
[3] Frank S. Jr. Crawford. Waves berkeley physics course volume 3. Education of Development
Center, 1968.
11
9
9.1
Appendix
Ampere to Tesla
Figure 4 shows the Gauge graph of the electromagnet used in the experiment. The slope equals
137 G/A or 0.0137 T/A. Here 1 Tesla = 104 Gauss. Equation 8 can be used to convert the input
current to the magnetic field in Tesla’s.
B(T ) => 0.0137 ∗ I(A)
Notice that the units of equation 8 do not add up, but the units do.
Figure 4: The Gauge graph of the electromagnet
12
(8)
9.2
Data
The primary data is listed below in the Tables 1 until 6. Here the minimum intensity angle is an
estimate, and the + and - degree columns represent the range in degrees around this estimate.
The current is the input of the electromagnet 9.1.
Table 1: Measurements for λ = 589nm, from -5◦ until the estimated minimum intensity angle,
with error for the current ±0.1A and error for the angle ±0.5°.
Current
9
8.5
8
7.5
7
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
-5.5
-6
-6.5
-7
-7.5
-8
-8.5
-9
(-5°)
107
106
105
104
103
102
101
100
99
98
97
96
95
94
93
92
91
90
90
88
87
86
85
84
83
82
81
80
79
78
77
76
75
74
73
72
71
I
22.299
19.007
23.097
20.357
18.400
23.864
21.550
22.829
23.324
21.865
17.927
23.168
22.366
23.176
25.736
23.895
22.438
21.494
13.643
32.480
32.984
30.531
26.765
31.435
30.204
29.525
28.562
29.608
28.765
30.223
33.931
28.901
30.624
36.401
35.758
34.913
35.629
(-4°)
108
107
106
105
104
103
102
101
100
99
98
97
96
95
94
93
92
91
91
89
88
87
86
85
84
83
82
81
80
79
78
77
76
75
74
73
72
I
14.358
12.817
17.268
15.548
12.722
15.287
14.731
13.499
16.674
15.913
14.422
15.462
13.860
14.621
17.504
14.717
16.559
12.685
7.914
19.982
21.807
24.210
20.831
22.927
19.420
19.146
21.704
18.444
20.859
22.664
20.466
20.282
22.980
25.010
25.999
23.763
23.098
(-3°)
109
108
107
106
105
104
103
102
101
100
99
98
97
96
95
94
93
92
92
90
89
88
87
86
85
84
83
82
81
80
79
78
77
76
75
74
73
I
8.399
9.505
8.185
8.429
8.010
8.566
8.113
8.004
9.120
9.071
8.237
8.235
8.882
6.077
8.695
9.839
8.104
8.392
3.583
14.926
11.454
14.509
10.198
14.119
11.247
11.250
12.870
12.525
12.829
12.137
15.712
14.470
14.870
15.241
16.709
15.537
16.292
13
(-2°)
110
109
108
107
106
105
104
103
102
101
100
99
98
97
96
95
94
93
93
91
90
89
88
87
86
85
84
83
82
81
80
79
78
77
76
75
74
I
5.368
5.629
4.007
4.653
3.352
4.313
4.933
4.215
4.812
3.563
5.135
2.695
3.638
4.165
3.272
5.466
4.460
3.943
0.864
6.865
7.177
6.293
7.648
7.826
4.934
5.789
5.936
6.729
7.331
7.330
7.449
8.018
8.032
9.799
9.914
11.342
11.010
(-1°)
111
110
109
108
107
106
105
104
103
102
101
100
99
98
97
96
95
94
94
92
91
90
89
88
87
86
85
84
83
82
81
80
79
78
77
76
75
I
3.017
1.700
1.686
1.407
1.466
1.868
1.686
1.122
1.120
0.858
1.202
0.930
1.114
0.395
0.936
0.786
1.268
0.864
0.022
1.899
2.104
2.814
1.763
2.639
1.963
3.143
2.944
3.513
3.087
4.576
3.351
4.305
4.672
5.596
4.942
6.266
5.792
θmin
112
111
110
109
108
107
106
105
104
103
102
101
100
99
98
97
96
95
95
93
92
91
90
89
88
87
86
85
84
83
82
81
80
79
78
77
76
I
2.173
1.381
1.376
1.248
0.872
1.114
0.900
0.417
0.552
0.316
0.361
0.172
0.136
0.042
0.002
0.048
0.002
0.002
0.004
0.130
0.204
0.426
0.154
0.183
1.446
0.606
0.567
0.825
1.027
1.445
1.783
2.090
2.416
2.979
3.493
3.645
3.944
Table 2: Measurements for λ = 589nm, from +5◦ to the estimated minimum intensity angle,
with error for the current ±0.1A and error for the angle ±0.5°.
Current
9
8.5
8
7.5
7
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
-5.5
-6
-6.5
-7
-7.5
-8
-8.5
-9
θmin
112
111
110
109
108
107
106
105
104
103
102
101
100
99
98
97
96
95
95
93
92
91
90
89
88
87
86
85
84
83
82
81
80
79
78
77
76
I
2.173
1.381
1.376
1.248
0.872
1.114
0.900
0.417
0.552
0.316
0.361
0.172
0.136
0.042
0.002
0.048
0.002
0.002
0.004
0.130
0.204
0.426
0.154
0.183
1.446
0.606
0.567
0.825
1.027
1.445
1.783
2.090
2.416
2.979
3.493
3.645
3.944
(1°)
113
112
111
110
109
108
107
106
105
104
103
102
101
100
99
98
97
96
96
94
93
92
91
90
89
88
87
86
85
84
83
82
81
80
79
78
77
I
3.449
2.217
2.664
2.909
2.012
2.368
2.223
2.274
1.315
0.840
1.040
1.203
0.780
0.365
0.318
0.100
0.120
0.173
2.308
0.028
0.123
0.046
0.295
0.216
0.631
0.436
1.102
1.081
1.371
1.412
1.786
1.804
2.353
2.892
2.803
2.964
3.360
(2°)
114
113
112
111
110
109
108
107
106
105
104
103
102
101
100
99
98
97
97
95
94
93
92
91
90
89
88
87
86
85
84
83
82
81
80
79
78
I
7.187
5.340
6.879
5.024
5.421
4.952
4.261
3.880
4.588
4.093
4.320
4.336
2.822
3.444
2.329
2.708
1.824
1.991
5.450
1.370
1.641
1.795
1.766
2.128
4.111
0.501
4.201
2.263
3.363
3.367
3.458
3.934
4.863
3.650
4.138
4.362
5.611
14
(3°)
115
114
113
112
111
110
109
108
107
106
105
104
103
102
101
100
99
98
98
96
95
94
93
92
91
90
89
88
87
86
85
84
83
82
81
80
79
I
11.452
8.265
10.986
8.037
7.098
8.327
9.003
7.229
8.527
8.411
9.560
8.123
7.910
6.433
6.291
6.826
6.247
5.306
9.286
4.672
5.601
6.548
6.106
6.862
8.734
7.666
8.273
6.896
7.794
8.268
7.000
6.803
8.471
6.655
8.016
7.320
8.686
(4°)
116
115
114
113
112
111
110
109
108
107
106
105
104
103
102
101
100
99
99
97
96
95
94
93
92
91
90
89
88
87
86
85
84
83
82
81
80
I
16.666
16.726
15.971
13.105
12.578
15.190
14.788
13.644
14.473
14.892
13.627
14.839
12.750
11.953
11.341
12.952
12.953
11.019
13.142
11.224
11.498
10.970
13.000
11.514
13.178
7.765
15.325
12.940
13.294
13.491
12.971
13.084
13.291
12.332
13.587
14.980
12.534
(5°)
117
116
115
114
113
112
111
110
109
108
107
106
105
104
103
102
101
100
100
98
97
96
95
94
93
92
91
90
89
88
87
86
85
84
83
82
81
I
24.927
22.500
24.532
23.937
19.469
21.296
21.310
22.088
21.773
20.913
21.801
22.950
21.532
20.304
20.878
19.494
19.206
16.128
25.171
19.289
17.261
17.204
18.779
20.194
20.694
20.024
22.655
19.907
20.292
21.421
19.442
21.696
19.465
20.273
18.474
19.253
22.085
Table 3: Measurements for λ = 550nm, from -5◦ to the estimated minimum intensity angle, with
error for the current ±0.1A and error for the angle ±0.5°.
Current
9
8.5
8
7.5
7
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
-5.5
-6
-6.5
-7
-7.5
-8
-8.5
-9
(-5°)
110
109
107
106
105
104
103
101
100
99
98
97
96
95
94
93
92
89
90
87
86
85
84
85
85
82
80
79
78
76
75
74
73
72
71
70
69
I
4.553
6.228
6.508
7.017
7.270
6.059
5.644
7.125
5.602
7.827
6.560
6.721
6.202
5.256
4.853
4.987
4.149
9.127
4.528
10.221
7.551
6.478
7.726
3.698
0.972
3.940
6.288
5.252
4.628
7.428
6.632
6.629
5.862
5.252
4.036
4.931
6.772
(-4°)
111
110
108
107
106
105
104
102
101
100
99
98
97
96
95
94
93
90
91
88
87
86
85
86
86
83
81
80
79
77
76
75
74
73
72
71
70
I
1.784
3.710
3.275
4.639
5.187
4.643
4.256
4.163
3.172
3.786
2.910
3.865
3.513
2.959
2.104
2.441
2.536
6.100
2.045
7.203
7.491
3.756
6.050
1.204
0.029
0.552
3.411
1.992
2.634
4.357
3.013
3.875
3.526
2.956
2.283
1.930
4.829
(-3°)
112
111
109
108
107
106
105
103
102
101
100
99
98
97
96
95
94
91
92
89
88
87
86
87
87
84
82
81
80
78
77
76
75
74
73
72
71
I
0.970
1.690
2.329
3.204
2.977
2.530
2.214
3.180
2.113
1.959
2.185
1.209
1.266
1.357
1.154
0.602
0.436
3.646
0.154
4.535
2.585
1.929
2.996
0.090
0
0.128
1.598
0.829
1.217
2.531
2.207
2.774
1.886
1.465
1.469
1.286
2.427
15
(-2°)
113
112
110
109
108
107
106
104
103
102
101
100
99
98
97
96
95
92
93
90
89
88
87
88
88
85
83
82
81
79
78
77
76
75
74
73
72
I
0.235
0.935
0.983
1.102
0.948
1.172
0.762
1.861
1.120
0.949
0.499
0.538
0.070
0.241
0.102
0.008
0
1.785
0
5.515
1.483
0.430
0.730
0
0
0
0.375
0.228
0.101
1.331
0.862
1.081
0.461
0.800
0.453
0.507
1.696
(-1°)
114
113
111
110
109
108
107
105
104
103
102
101
100
99
98
97
96
93
94
91
90
89
88
89
89
86
84
83
82
80
79
78
77
76
75
74
73
I
0.020
0.135
0.426
0.299
0.122
0.107
0.012
0.433
0.257
0.005
0.020
0.002
0
0
0
0
0
0.665
0
0.489
0.232
0
0.032
0
0.177
0
0
0
0
0.219
0.102
0.208
0.051
0.087
0.033
0.193
0.579
θmin
115
114
112
111
110
109
108
106
105
104
103
102
101
100
99
98
97
94
95
92
91
90
89
90
90
87
85
84
83
81
80
79
78
77
76
75
74
I
0.041
0.030
0
0.114
0
0
0.008
0.008
0
0.001
0
0
0
0
0
0
0
0
0
0.038
0
0
0
0.102
1.370
0
0
0
0
0
0.001
0.000
0.018
0.090
0.036
0.090
0.421
Table 4: Measurements for λ = 550nm, from +5◦ to the estimated minimum intensity angle,
with error for the current ±0.1A and error for the angle ±0.5°.
Current
9
8.5
8
7.5
7
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
-5.5
-6
-6.5
-7
-7.5
-8
-8.5
-9
θmin
115
114
112
111
110
109
108
106
105
104
103
102
101
100
99
98
97
94
95
92
91
90
89
90
90
87
85
84
83
81
80
79
78
77
76
75
74
I
0.041
0.030
0
0.114
0
0
0.008
0.008
0
0.001
0
0
0
0
0
0
0
0
0
0.038
0
0
0
0.102
1.370
0
0
0
0
0
0.001
0.000
0.018
0.090
0.036
0.090
0.421
(1°)
116
115
113
112
111
110
109
107
106
105
104
103
102
101
100
99
98
95
96
93
92
91
90
91
91
88
86
85
84
82
81
80
79
78
77
76
75
I
0
0.144
0
0.045
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.769
3.170
0.163
0
0.008
0.125
0
0.577
0.001
0.206
0.310
0.124
0.952
1.342
(2°)
117
116
114
113
112
111
110
108
107
106
105
104
103
102
101
100
99
96
97
94
93
92
91
92
92
89
87
86
85
83
82
81
80
79
78
77
76
I
0.023
0.366
0.001
0.016
0
0.057
0.003
0
0
0
0.002
0
0
0.016
0
0.013
0.074
0
0
0
0
0
0
2.131
4.873
0.723
0.046
0.745
0.442
0.059
0.577
0.319
0.680
0.943
0.358
1.508
2.573
16
(3°)
118
117
115
114
113
112
111
109
108
107
106
105
104
103
102
101
100
97
98
95
94
93
92
93
93
90
88
87
86
84
83
82
81
80
79
78
77
I
0.365
0.774
0.223
0.214
0.214
0.454
0.426
0.043
0.110
0.212
0.471
0.111
0.336
0.343
0.541
0.676
1.092
0
1.362
1.609
0
0.244
0.584
4.656
6.597
2.389
0.963
1.629
1.788
0.730
1.288
1.628
2.298
3.001
1.241
3.162
3.779
(4°)
119
118
116
115
114
113
112
110
109
108
107
106
105
104
103
102
101
98
99
96
95
94
93
94
94
91
89
88
87
85
84
83
82
81
80
79
78
I
0.933
2.371
0.480
0.833
1.504
1.341
1.305
0.592
0.452
1.054
1.434
1.188
1.692
1.550
1.387
1.957
1.750
0.102
3.470
3.935
1.068
1.638
1.764
6.827
10.153
3.937
3.344
2.646
2.420
2.167
2.628
2.704
3.170
4.333
4.671
5.145
6.420
(5°)
120
119
117
116
115
114
113
111
110
109
108
107
106
105
104
103
102
99
100
97
96
95
94
95
95
92
90
89
88
86
85
84
83
82
81
80
79
I
1.866
3.760
2.562
1.707
2.820
2.668
3.312
1.253
1.595
2.538
2.237
2.944
2.983
2.828
3.787
3.717
4.646
0.812
3.942
6.002
2.524
3.145
3.334
10.726
17.063
6.444
5.699
4.932
7.779
3.837
3.882
5.287
6.040
5.386
6.244
8.566
9.233
Table 5: Measurements for λ = 448nm, from -5◦ to the estimated minimum intensity angle, with
error for the current ±0.1A and error for the angle ±0.5°.
Current
1
3
5
7
9
0
-1
-3
-5
-7
-9
(-50°)
50
50
60
60
80
50
50
40
20
10
10
I
8.906
11.850
14.767
9.728
6.247
6.550
11.069
7.506
11.316
12.550
11.014
(-40°)
60
60
70
70
90
60
60
50
30
20
20
I
5.060
7.223
9.329
6.661
2.435
3.298
5.819
3.428
8.150
11.438
7.597
(-30°)
70
70
80
80
100
70
70
60
40
30
30
I
1.731
3.560
5.948
4.062
0.634
0.977
1.395
1.040
4.724
6.355
3.223
(-20°)
80
80
90
90
110
80
80
70
50
40
40
I
0.212
1.532
1.754
2.038
0.025
0.054
0
0
1.512
2.399
0.552
(-10°)
90
90
100
100
120
90
90
80
60
50
50
I
0
0.014
0.012
0.217
0
0
0
0
0.026
0.235
0
θmin
100
100
110
110
130
100
100
90
70
60
60
I
0
0
0
0
0
0
0
0
0
0
0
Table 6: Measurements for λ = 448nm, from +5◦ to the estimated minimum intensity angle,
with error for the current ±0.1A and error for the angle ±0.5°.
Current
1
3
5
7
9
0
-1
-3
-5
-7
-9
θmin
100
100
110
110
130
100
100
90
70
60
60
I
0
0
0
0
0
0
0
0
0
0
0
(10°)
110
110
120
120
140
110
110
100
80
70
70
I
0
0
0
0
0
0
0.108
0.014
0
0
0.221
(20°)
120
120
130
130
150
120
120
110
90
80
80
I
0.034
0.004
0
0
0
0.062
1.304
1.432
0.052
0.078
1.588
17
(30°)
130
130
140
140
160
130
130
120
100
90
90
I
0.723
0.151
0.455
0.000
0.460
1.556
4.920
3.537
0.737
1.295
3.288
(40°)
140
140
150
150
170
140
140
130
110
100
100
I
3.082
2.068
3.077
0.572
2.018
4.670
4.197
5.955
2.644
2.433
5.204
(50°)
150
150
160
160
180
150
150
140
120
110
110
I
4.479
6.333
5.181
4.895
4.463
5.183
7.009
8.247
4.266
6.832
8.415
9.3
Data Processing
The data was processed using python. For each magnetic field strength a measurement set was
taken over which a quadratic curve was fitted (see section 3.4). In Figure 5 one of these data
sets is presented as an example. Here python command [1] was used to fit the parabolic curve
ax2 + bx + c:
Parameters, covariance = curve_fit(parabolic, x, y, bounds=(-np.inf, np.inf))
[1]
Here x represents the angels from -5 to +5 degrees with respect to the minimum intensity
estimated. The measured intensity is represented with y. The variable parameters gives a string
of the values for a, b and c for the plotted line. The covariance represents the relation between
the variability of the parameters. From the covariance the error can be extracted using python
command [2], represented by the vertical bars in Figure 5:
error = np.sqrt(np.diag(covariance))
[2]
The derivative of ax2 +bx+c equals 2ax+b, so the angle of minimum intensity was calculated
using equation 9.
b
(9)
2a
This was done for all data sets. Lastly the minimum intensity was fitted against the magnetic
field strength using python commands [1] and [2]. Instead of a quadratic equation, a linear fit
was used in the form ax + b.
x=−
Figure 5: Measurement set for λ=589nm and an input current of 5.5A.
18
9.4
Error Analysis
Below the different equations for the errors are expressed. These errors correlate with the equations in 3.2
Verdet constant trough the slope
1 θ
∆
d B
(10)
(n2 − 1)qω 2
∆V
4nmcω0 V 2
(11)
∆V =
Eigenfrequency ω0
∆ω0 =
Difference in refraction between the right hand and the left hand waves
r
∆∆n =
(
∆B 2
∆V 2 BV λ
) +(
)
B
V
π
(12)
Verdet constant from the eigenfrequency
∆V =
2ω0 (n − 1)2 qω 2
∆ω0
nmc(ω02 − ω 2 )2
19
(13)