Faraday Generator

Team of Austria
Markus Kunesch, Julian Ronacher, Angel
Usunov, Katharina Wittmann, Bernhard Zatloukal
Reporter: Markus Kunesch
14. Faraday Generator
Construct a homopolar electric generator. Investigate the electrical
properties of the device and find its efficiency.
Team Austria
powered by:
IYPT 2008 – Trogir, Croatia
Overview
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Introduction
Experimental Setup
Results – Voltage / angular velocity
Theory – The Lorentz Force
Theory – The electromotive force
Comparison
Determining the efficiency
Eddy currents
Conclusion
Team of Austria – Problem no. 14 – Faraday Generator
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Experimental Setup
Team of Austria – Problem no. 14 – Faraday Generator
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Experimental Setup
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Experimental Setup
Angular velocity
0-50 (±0.017) rad/s
Radius of disk
1.5 , 6, 21 (±0.05) cm
Material of disk
V
Strength of magnets
127, 371, 6, 200 (±0.5) mT
Velocity of magnets
0-50 (±0.017) rad/s
Shape of magnets
Position of contacts
Team of Austria – Problem no. 14 – Faraday Generator
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Experimental Setup
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Results Voltage
Voltage [mV]
16
Error: ±0.05 mV
14
12
10
8
6
4
2
0
0
4
Time
8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 [s]
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Results angular velocity
Angular v [rad/sec]
60
50
Error: ±0.017 rad
Voltage [mV]
Error: ±0.05 mV
16
14
12
40
10
30
8
20
6
4
10
2
0
Time
0
0 2 5 7 10121517202225273032353740424547505255576062656769 [s]
0 6 12 18 24 30 36 42 48 54 60 66
Team of Austria – Problem no. 14 – Faraday Generator
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F... Force
Theory – Lorentz Force
F = q(E + v × B )
q... charge
E... electric field
v... velocity
B... magnetic field
E ... electromotive force
W... Work
W
emf = E =
q
W = ∫ F ⋅ dl
1
E = ∫ F ⋅dl
q
Team of Austria – Problem no. 14 – Faraday Generator
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Electromotive Force
1
E = ∫ F ⋅ dl
q
F = q(E + v × B )
1
E = ∫ q(E + v × B )⋅ dl =
q
F... Force
q... charge
E... electric field
v... velocity
B... magnetic field
E ... electromotive force
= ∫ E ⋅ dl + ∫ (v × B )⋅ dl
Team of Austria – Problem no. 14 – Faraday Generator
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Electromotive Force – Stokes Theorem
E = ∫ E ⋅ dl + ∫ (v × B )⋅ dl
∂B
∇×E = −
∂t
F... Force
q... charge
E... electric field
v... velocity
B... magnetic field
E ... electromotive force
∇...Nabla operator
∫ E ⋅ dl = ∫ (∇ × E)⋅ dS
∂B
= −∫
⋅ dS
∂t
Team of Austria – Problem no. 14 – Faraday Generator
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Electromotive Force
∂B
⋅ dS + ∫ (v × B )⋅ dl
E = −∫
∂t
v... velocity
B... magnetic field
E ... electromotive force
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Comparison
∂B
⋅ dS + ∫ (v × B )⋅ dl
E = −∫
∂t
∂B
E = − ∫ 0 ⋅ dS + ∫ (v × B )⋅ dl
∂t
E=
(
)
v
B
×
⋅
dl
∫
V
v... velocity
B... magnetic field
E ... electromotive force
Team of Austria – Problem no. 14 – Faraday Generator
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Calculations
(
)
v
×
B
⋅
dl
∫
E = ∫ (rω × B )⋅ dl
E=
v... velocity
B... magnetic field
E ... electromotive force
ω...angular velocity
r...radius
r2
r

E = 2 ⋅  ωB 
 2
 r1
2
Team of Austria – Problem no. 14 – Faraday Generator
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Calculations
r2
 r2

E =  ωB 
 2
 r1
1
2
2
E = ωB r − (r − l )
2
1
E = ωBl (2r − l )
2
(
)
Team of Austria – Problem no. 14 – Faraday Generator
v... velocity
B... magnetic field
E ... electromotive force
ω...angular velocity
r...radius
l...length of magnet
l
15
Comparison
Voltage [mV]
6
Average error: 6.9%
5
4
3
2
1
0
26
28
30
32
34
36
38
40
Angular v [rad/s]
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Further proof
∂B
⋅ dS + ∫ (v × B )⋅ dl
E = −∫
∂t
v... velocity
B... magnetic field
E ... electromotive force
V
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Determining the efficiency
Eout
η=
Ein
Ein = Ekin
2
V
P(out ) =
R
η ...efficiency
E out/in ...Energy out(in)put
V...Voltage
R...Resistance
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Kinetic Energy
η ...efficiency
E
...Energy out(in)put
Mω 2 R 2
M...mass
Ekin =
ω...angular velocity
2
R...Resistance
Ekin lost
−1
P(in ) =
= 0.292 ± 0.076 Js
t
P(in )
= 0.000075 ± 0,0000002489%
P (out )
out/in
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Eddy currents
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Conclusion
• Full mathematical analysis of the problem
• The Voltage output is best calculated using:
∂B
E = −∫
⋅ dS + ∫ (v × B )⋅ dl
∂t
• Voltage is obtained when:
– Only the disk is rotating
– Magnet and disk are rotating
– Only the external circuit is rotating
– The external circuit and the
magnet are rotating
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Conclusion
• A description of the phenomenon is possible in every
inertial frame – even in the rotating system!
• The efficiency is extremely poor – especially when using
an inhomogene magnetic field.
• More Voltage or Current is obtained with:
– Stronger magnets
– Higher angular velocity
– Smaller internal resistance
– A bigger magnet
– A bigger disk
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References
• Am. J. Phys. Vol. 46 (7), July 1978, M.J. Crooks, D.B. Litvin,
P.W.Matthews, R. Macaulay, J. Shaw
• Am. J. Phys. Vol. 55 (7), July 1987, R. D. Eagleton
• Taschenbuch der Physik, Stöcker H., Wissenschaftlicher Verlag
Harri Deutsch, Frankfurt am Main, 2005
• Mathematik für Physiker, Dr. rer. Nat. Helmut Fischer, Dr. rer. Nat.
Helmut Kaul, B. G. Teubner, 2005
• Homopolar generator,
http://www.physics.brown.edu/physics/demopages/Demo/em/demo/
5k1080.htm
• The homopolar generator,
http://farside.ph.utexas.edu/teaching/plasma/lectures/node70.html
• http://sciencelinks.jp/jeast/article/200123/000020012301A0808251.php
• Homopolar Disk Generator, http://jnaudin.free.fr/html/farhom.htm
Team of Austria – Problem no. 14 – Faraday Generator
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Ad1
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Ad2 Superconductor
∂B
⋅ dS + ∫ (v × B )⋅ dl
E = −∫
∂t
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Ad3 Experimental Setup
21±0.05 cm
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Ad4 Voltage - EMF
Voltage [mV]
6
5
4
3
2
1
0
26
28
Average error: 6,9%
30
32
34
36
Vmeassured
V =
Rinternal
1−
Rinternal + R1
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40
Angular v
[rad/s]
R...10 to 15 Ω
27