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Team of AustriaMarkus Kunesch, Julian Ronacher, Angel
Usunov, Katharina Wittmann, Bernhard Zatloukal
IYPT 2008 – Trogir, Croatia
14. Faraday Generator
Construct a homopolar electric generator. Investigate the electrical properties of the device and find its efficiency.
Reporter: Markus Kunesch
Team Austriapowered by:
Overview
• Introduction• Experimental Setup• Results – Voltage / angular velocity• Theory – The Lorentz Force• Theory – The electromotive force• Comparison• Determining the efficiency• Eddy currents• Conclusion
2Team of Austria – Problem no. 14 – Faraday Generator
Experimental Setup
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Experimental Setup
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Experimental Setup
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V
Angular velocity
0-50 (±0.017) rad/s
Radius of disk
1.5 , 6, 21 (±0.05) cm
Material of disk
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
Experimental Setup
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Results Voltage
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0
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8
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14
16
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68Time [s]
Voltage [mV] Error: ±0.05 mV
Results angular velocity
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0
10
20
30
40
50
60
0 2 5 7 101215172022252730323537404245475052555760626567690
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0 6 12 18 24 30 36 42 48 54 60 66
Time [s]
Angular v [rad/sec] Voltage [mV] Error: ±0.05 mVError: ±0.017 rad
Theory – Lorentz Force
( )BvEF ×+= q
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qWemf == E
∫ ⋅= dlW F
dlq ∫ ⋅= F1
E
WorkW...force iveelectromot ...field magnetic B...
velocityv...field electric E...
charge q...Force F...
E
Electromotive Force
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dlq
⋅= ∫F1E ( )BvEF ×+= q
( )
( ) dldl
dlqq
⋅×+⋅=
=⋅×+=
∫ ∫
∫BvE
BvE1E
force iveelectromot ...field magnetic B...
velocityv...field electric E...
charge q...Force F...
E
Electromotive Force – Stokes Theorem
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( ) dldl ⋅×+⋅= ∫ ∫ BvEE
11
t∂∂
−=×∇BE
( )∫∫ ⋅×∇=⋅ dSdl EE
∫ ⋅∂∂
−= dStB
operator ...Nablaforce iveelectromot ...field magnetic B...
velocityv...field electric E...
charge q...Force F...
∇E
Electromotive Force
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( ) dldSt
⋅×+⋅∂∂
−= ∫∫ BvBE
force iveelectromot ...field magnetic B...
velocityv...
E
Comparison
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( ) dldSt
⋅×+⋅∂∂
−= ∫∫ BvBE
( ) dldSt
⋅×+⋅∂∂
−= ∫∫ BvBE ( ) dl⋅×+= ∫ Bv0E
( ) dl⋅×= ∫ BvE V
force iveelectromot ...field magnetic B...
velocityv...
E
Calculations
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( ) dl⋅×= ∫ BvE
( ) dlr ⋅×= ∫ BωE
2
12
2E2 r
r
r
⋅= Bω
r...radius velocity...angular
force iveelectromot ...field magnetic B...
velocityv...
ωE
Calculations
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2
12
E2 r
r
r
= Bω
( )( )22
21 lrrB −−= ωE
l
( )lrBl −= 221ωE
magnet of l...lengthr...radius velocity...angular
force iveelectromot ...field magnetic B...
velocityv...
ωE
Comparison
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Average error: 6.9%
0
1
2
3
4
5
6
26 28 30 32 34 36 38 40
Angular v [rad/s]
Voltage [mV]
Further proof
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( ) dldSt
⋅×+⋅∂∂
−= ∫∫ BvBE
V
force iveelectromot ...field magnetic B...
velocityv...
E
Determining the efficiency
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in
out
EE
=η kinin EE =
( )R
VoutP2
=
anceR...ResisteV...Voltag
out(in)putEnergy...Ency...efficie
out/in
η
Kinetic Energy
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2
22RMEkinω
=
( ) 1076.0 0.292 −±== JstlostEinP kin
( )( ) 89%0,00000024 0.000075 ±=outPinP
anceR...Resist velocity...angular
M...massout(in)putEnergy...E
ncy...efficie
out/in
ω
η
Eddy currents
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Conclusion
• Full mathematical analysis of the problem• The Voltage output is best calculated using:
• 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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( ) dldSt
⋅×+⋅∂∂
−= ∫∫ BvBE
Conclusion
• A description of the phenomenon is possible in everyinertial frame – even in the rotating system!
• The efficiency is extremely poor – especially when usingan 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/j-east/article/200123/000020012301A0808251.php
• Homopolar Disk Generator, http://jnaudin.free.fr/html/farhom.htm
23Team of Austria – Problem no. 14 – Faraday Generator
Ad1
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Ad2 Superconductor
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( ) dldSt
⋅×+⋅∂∂
−= ∫∫ BvBE
Ad3 Experimental Setup
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21±0.05 cm
Ad4 Voltage - EMF
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Ω
+−
= 1510...1
1internal
internal
meassured toR
RRR
VVAverage error: 6,9%
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26 28 30 32 34 36 38 40
Angular v [rad/s]
Voltage [mV]