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

Team of Austria – Problem no. 14 – Faraday Generator 3

Experimental Setup

Team of Austria – Problem no. 14 – Faraday Generator 4

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

Team of Austria – Problem no. 14 – Faraday Generator 6

Results Voltage

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0

2

4

6

8

10

12

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

Team of Austria – Problem no. 14 – Faraday Generator 8

0

10

20

30

40

50

60

0 2 5 7 101215172022252730323537404245475052555760626567690

2

4

6

8

10

12

14

16

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

Team of Austria – Problem no. 14 – Faraday Generator 9

qWemf == E

∫ ⋅= dlW F

dlq ∫ ⋅= F1

E

WorkW...force iveelectromot ...field magnetic B...

velocityv...field electric E...

charge q...Force F...

E

Electromotive Force

Team of Austria – Problem no. 14 – Faraday Generator 10

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

Team of Austria – Problem no. 14 – Faraday Generator 11

( ) 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

Team of Austria – Problem no. 14 – Faraday Generator 12

( ) dldSt

⋅×+⋅∂∂

−= ∫∫ BvBE

force iveelectromot ...field magnetic B...

velocityv...

E

Comparison

Team of Austria – Problem no. 14 – Faraday Generator 13

( ) dldSt

⋅×+⋅∂∂

−= ∫∫ BvBE

( ) dldSt

⋅×+⋅∂∂

−= ∫∫ BvBE ( ) dl⋅×+= ∫ Bv0E

( ) dl⋅×= ∫ BvE V

force iveelectromot ...field magnetic B...

velocityv...

E

Calculations

Team of Austria – Problem no. 14 – Faraday Generator 14

( ) 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

Team of Austria – Problem no. 14 – Faraday Generator 17

( ) dldSt

⋅×+⋅∂∂

−= ∫∫ BvBE

V

force iveelectromot ...field magnetic B...

velocityv...

E

Determining the efficiency

Team of Austria – Problem no. 14 – Faraday Generator 18

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

Team of Austria – Problem no. 14 – Faraday Generator 21

( ) 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%

0

1

2

3

4

5

6

26 28 30 32 34 36 38 40

Angular v [rad/s]

Voltage [mV]