Faraday Generator. Construct a homopolar electric generator. Investigate the
electrical properties of the device and find its efficiency. Reporter: Markus
Kunesch.
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 • • • • • • • • •
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
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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
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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
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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
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Electromotive Force
∂B ⋅ dS + ∫ (v × B )⋅ dl E = −∫ ∂t
v... velocity B... magnetic field E ... electromotive force Team of Austria – Problem no. 14 – Faraday Generator
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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
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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] Team of Austria – Problem no. 14 – Faraday Generator
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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 Team of Austria – Problem no. 14 – Faraday Generator
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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