Faraday Generator - IYPT

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

2

Experimental Setup


3

Experimental Setup


4

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


5

Experimental Setup


6

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]


7

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


8

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

9

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


10

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


11

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

12

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

13

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


14

Calculations r2

 r2  E =  ωB   2  r1 1 2 2 E = ωB r − (r − l ) 2 1 E = ωBl (2r − l ) 2

(

)


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

16

Further proof

∂B ⋅ dS + ∫ (v × B )⋅ dl E = −∫ ∂t v... velocity B... magnetic field E ... electromotive force

V


17

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


18

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


19

Eddy currents


20

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

21

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


22

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

23

Ad1


24

Ad2 Superconductor

∂B ⋅ dS + ∫ (v × B )⋅ dl E = −∫ ∂t


25

Ad3 Experimental Setup

21±0.05 cm


26

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


38

40

Angular v [rad/s]

R...10 to 15 Ω

27