1. Problem 12. Rolling Magnets

2. Problem Investigate the motion of a magnet as it rolls down an inclined plane.

3. Outline Only rolling motion investigated!Only rolling motion investigated! Two distinct cases:Two distinct cases: Nonconducting planeNonconducting plane Conducting planeConducting plane Quasiinfinite planeQuasiinfinite plane Finite planeFinite plane Common parameters:Common parameters: Magnet propertiesMagnet properties Plane inclinationPlane inclination

4. The magnets Permanent Nd 2 Fe 14 B magnetsPermanent Nd 2 Fe 14 B magnets Field of magnetization 1.4 TField of magnetization 1.4 T Density 7500 kg/m 3Density 7500 kg/m 3 3 different sizes:3 different sizes: Diameter [cm] 2.541.000.95 Thickness [cm] 2.540.500.63

5. Case 1 – Nonconducting plate Wooden/plastic plateWooden/plastic plate Magnet influenced only by the Earth fieldMagnet influenced only by the Earth field Curved trajectoryCurved trajectory Parameters:Parameters: Plane inclinationPlane inclination Magnet propertiesMagnet properties Much less appealing than second case – not studied in detailMuch less appealing than second case – not studied in detail

6. Case 2 – Conducting plate Metal slab Growing flux Falling flux v – magnet velocity F d – drag force In conducting plate – eddy currents induced due to time-changing field fluxIn conducting plate – eddy currents induced due to time-changing field flux Eddy current field gradient – velocity- dependent drag on magnetEddy current field gradient – velocity- dependent drag on magnet

7. Conducting plate cont. Two subcases: Magnet moving far from the plate edges - ˝infinite˝ plateMagnet moving far from the plate edges - ˝infinite˝ plate

8. Conducting plate cont. Magnet getting near the edges – boundary effectsMagnet getting near the edges – boundary effects

9. 1. Infinite conducting plate First case much simpler:First case much simpler: Linear motionLinear motion Constant velocity (drag balances gravity) – simple reference system switchingConstant velocity (drag balances gravity) – simple reference system switching Main parameters:Main parameters: Magnet dimensions and magnetizationMagnet dimensions and magnetization Plate inclinationPlate inclination Plate conductivityPlate conductivity

10. Experiment Measurements:Measurements: Dependence of terminal velocity on plate inclination for several magnetsDependence of terminal velocity on plate inclination for several magnets Dependence of terminal velocity on plate conductivityDependence of terminal velocity on plate conductivity Aluminium plateAluminium plate Velocity measurement – solenoid systemVelocity measurement – solenoid system Conductivity modification – temperature changeConductivity modification – temperature change

11. 1. Velocity – inclination cont. Amplifier & ADC PC Detector solenoids

12. 1. Velocity – inclination cont. Velocity measurement – solenoids detect passingmagnet due to induction:Velocity measurement – solenoids detect passing magnet due to induction:

13. 2. Velocity - conductivity Conductivity change:Conductivity change: Cooling plate in insulating box to 73 K with liquid N 2Cooling plate in insulating box to 73 K with liquid N 2 As plate warms up magnet is released and velocity measuredAs plate warms up magnet is released and velocity measured Conductivity measured directly – resistance of wire attached to plateConductivity measured directly – resistance of wire attached to plate Temperature range 73 – 200 K Conductivity range 37 – 200 MS

14. Velocity – conductivity cont. Apparatus shematic:Apparatus shematic: Magnet detecting solenoids Liquid nitrogen Aluminium plate Magnet Styrofoam box Magnet insertion slit Temperature wire

15. Velocity – conductivity cont. The box Box inside with plate and solenoids

16. Theory The geometry in magnet reference system:The geometry in magnet reference system: M – magnetization vector F d – drag force x,y,z – magnet reference system x’,y’,z’ – plate reference system t - time

17. Theory cont. Induced field – from Maxwell equations in magnet reference systemInduced field – from Maxwell equations in magnet reference system For small velocities - field equation:For small velocities - field equation: j – current density σ – plate conductivity B 0 – field of magnet μ 0 – permeability of vacuum v – magnet velocity Induced field Source term – magnet field  Solution – power series in μ 0 σ v For small velocities – linear first term dominates!For small velocities – linear first term dominates!

18. Theory cont. Needed for force – y - componentNeeded for force – y - component Numerical integration yields:Numerical integration yields: Magnet radius [cm] 0.5 Magnet thickness [cm] 0.5 Counductivity [MS] 29.85 Upper plate boundary z = 0 Semiinfinite plate Magnet centre of mass z = 0.5 cm Section y = 0

19. Theory cont. The currents are obtained by differentiation:The currents are obtained by differentiation:

20. Theory cont. Drag force – for small velocitiesDrag force – for small velocities Terminal state – balance between gravity and drag force:Terminal state – balance between gravity and drag force: Λ – calculated constant σ – plate conductivity v – magnet velocity Diameter [cm] 2.541.000.95 Thickness [cm] 2.540.500.63 Λ ·10 9 [kg/sS] Λ ·10 9 [kg/sS]52.8(4)1.19(3)2.95(2) ζ – magnet mass g – acceleration of gravity φ – plate inclination

21. Results and comparation cont. For two magnets – dependence of terminal velocity on sin φ linear!For two magnets – dependence of terminal velocity on sin φ linear! Diameter [cm]1.0 Thickness [cm]0.5 Counductivity [MS]29.85 Plate thickness [cm]1.0

22. Results and comparation cont. Diameter [cm]2.54 Thickness [cm]2.54 Counductivity [MS]29.85 Plate thickness [cm]1.0

23. Results and comparation cont. For third magnet – dependence of terminal velocity on 1/conductivity linear:For third magnet – dependence of terminal velocity on 1/conductivity linear: Diameter [cm]0.95 Thickness [cm]0.63 Plate angle [°]28.5 Plate thickness [cm]1.0

24. Results and comparation cont. From three measurements the coefficient Λ is obtained:From three measurements the coefficient Λ is obtained: Agreement is very good – justification of linearization!Agreement is very good – justification of linearization! Λ·10 9 ExperimentTheory 1.21 ± 0.02 1.19(3) 53.2 ± 0.2 52.8(4) 2.97 ± 0.02 2.95(2)

25. 2. Boundary effects Close to edge – nonsymmetric induced currents:Close to edge – nonsymmetric induced currents:

26. 2. Boundary effects cont. Repulsive force occursRepulsive force occurs Magnet follows a quasiperiodical trajectoryMagnet follows a quasiperiodical trajectory Exact modeling very difficultExact modeling very difficult

27. Theory Acting on the magnet rolling motion:Acting on the magnet rolling motion: GravityGravity Earth field torqueEarth field torque FrictionFriction From the side From above x,y – unit vectors m – magnetic moment

28. Theory cont.  Trajectory equation: x 0 – initial x – position of magnet R – magnet radius ρ – magnet density D – magnet thickness Trajectory – portion of circleTrajectory – portion of circle For different initial angles numerical solution neccesaryFor different initial angles numerical solution neccesary

29. Theory cont. Linear acceleration while rolling:Linear acceleration while rolling: Special case: magnetic moment initially normal to Earth field – simple trajectorySpecial case: magnetic moment initially normal to Earth field – simple trajectory Magnetic field torque:Magnetic field torque: g – acceleration of gravity φ – plate inclination θ – angle between magnetic moment and Earth field vector m – magnetic moment vector B E – Earth field vector I – moment of inertia of magnet

30. Results and comparation

31. Theory cont. Magnet ↔ an array of infinitely thin dipolesMagnet ↔ an array of infinitely thin dipoles Force on one dipole:Force on one dipole:  Force on magnet in our geometry: R – magnet radius D – magnet thickness B iny – y - component of induced field ε - parameter B in – induced field x m, y m,z m – dipole coordinates