Download - 16. Magnetic brakes
16. MAGNETIC BRAKES
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”When a strong magnet falls down a non ferromagnetic metal tube, it will experience a retarding force.
Investigate the phenomenon.”
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OutlineQualitative explanation
Quantitaive explanation• Force modeling• Dipole case• Equation of motion
Apparatus and experimental methods• Measurement of magnetic flux change• Measurement of magnet velocity
Comparison of experimental data and theory• Dependence of terminal velocity on magnet size• Terminal velocity for dipole• Dependence of terminal velocity on material conductivity• Equation of motion
What causes the force? • Magnet falls down under the influence of gravity
• Magnetic flux is changing in space
• Current is induced in the tube
• Currents act on magnet by force thatreduces the speed of magnetic fluxchange (reducing the magnet speed)
– magnetic momentvelocity of a magnet
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z
�⃗�
In th
e m
iddl
e th
ere
is n
o in
duce
d cu
rren
t
Dire
ction
of i
nduc
ed c
urre
nts
𝑣
𝑭𝒎
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BBzB
Force• According to Newton III. law:
• Because we know what is the force on a current loop we can write
∆ρ
ρm
𝐿=𝑁∆𝑧
z
F
G
d 𝑧
Bz
Bρ
Fz (Bρ)
Fρ (Bz) Bρ
Fz (Bρ)
Fρ (Bz)
force on magnetforce on currenttickness of the tubedz – lenght of a loopmean radius of the pipe
Assumption: no skin effect
Cu;
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Force
• Magnetic field component is causing the force in direction so the force can be written as
Using Maxwell equation we derive expresion for
𝑩𝝆=−𝟏
𝟐𝝅𝛒𝒎
𝒅𝝓𝒅𝒛
𝜀=𝑑𝜙𝑑𝑧
𝑣
𝒅𝑰=𝜟𝝆𝝈 𝒗𝟐𝝅 𝝆𝒎
𝒅𝝓𝒅𝒛
𝒅𝒛
𝑑𝐼=𝜎 Δ𝜌 𝜀2𝜋 𝜌𝑚
𝑑𝑧
𝒅𝑭𝒎=−𝒅𝑰 ∙𝟐𝝅 𝝆𝒎 ∙𝑩𝝆
Bz Bz
BρBρ
Fz (Bρ) Fz (Bρ)
Fρ (Bz) Fρ (Bz)
z
change of magnetic flux in z directionε – induced voltageσ – material conductivityR – restistance of the ring
Force
• After rearranging the term we obtain total force on a magnet caused by induced current
z1, z2 – coordinates of the tube edges
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𝑭𝒎=𝝈𝚫𝛒 𝒗𝟐𝝅 𝛒𝒎
∫𝒛𝟏
𝒛𝟐
( 𝒅𝝓𝒅𝒛 )𝟐
𝒅𝒛
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Equation of motion • II. Newton law for magnet states
• After solving for we obtain ()
Substitution:• Function is in form of
exponential decay• Terminal velocity can
be expressed as
Determing magnetic flux change• Because can be difficult to calculate we decided to
experimentaly determine it
𝐹=¿ ∙2𝜋𝜌 ∙𝐵𝜌
𝒅𝝓𝒅𝒛
=𝑭𝑵𝑰
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Wooden cube (in place to eliminate magnets impact on the scale)St
and
Screw for vertical dislacement (0.01 mm)
Coil same diameter as tube
Scale
Mag
net
Measured by scale
Property of coil N = 40Adjusted on current source I = 1A (DC)
ScaleCurrent source
Brass screw
Wooden block
Neodimium magnets
Coil
Adjustable
stand
Plastic tube
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10
Distance from magnet [mm]
0 5 10 15 20 25 30 35
Cha
nge
in m
agne
tic f
lux
in z
dire
ctio
n [T
m]
0,000
0,001
0,002
0,003
0,004
0,005
0,006
h = 0.6 cmh = 2.1 cm
Example of obtained data - magnetic flux change
2r = 4 mm
h
Velocity measurement
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Det
ecto
r co
ils
Coil used for releasing magnet
PC
ADC
Detector coils
Coil used for releasing magnet
Computer controlled power
supply
time [s]
-4 -2 0 2 4
volta
ge V
]
-0,15
-0,10
-0,05
0,00
0,05
0,10
0,15
h = 0.6 cmh = 2.1 cm
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Copper3 solenoids
𝜀=𝑑𝜙𝑑𝑧
𝑣
• Velocity measurement – coils detect passing magnet due to induction:
2r = 4 mm
h
h [mm]
0 5 10 15 20 25 30
v [m
s-1]
0,05
0,10
0,15
0,20
0,25
0,30
Al exp = 5.5 mm, = 1mmCu exp = 5 mm, = 1mmAl theoretical curveCu theoretical curve
Dependence of terminal velocity on cylindrical magnets height – Cu and Al
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Theoretical values numericaly calculated from:Substitution:
4 mm
h
Dipole• Dipole magnetic field in z direction (measured from the
magnet):
• If the magnet is far enough from the edges of the tube we can write
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Terminal velocity - dipole
• Spherical magnet – radius 2.5 mm• Aluminium tube
Theoretical prediction
Experimental result
𝒗 𝒕=𝟐𝟓𝟔𝒅𝒈
𝟏𝟓𝝅𝝈 (𝝁𝟎𝑴 )𝟐∙( 𝝆𝒎
𝒓 )𝟑
( 𝝆𝒎
𝜟𝝆 )d – magnet’s densityr – radius of a magnet vt – theoretical velocity
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Velocity - conductivity• Copper tube coled down to 77K in insulating box with liquid N2• As tube warms up magnet is released and velocity measured• Magnet was constantly kept on liquid nitrogen
temperature ≈ 77K (constant magnetization )• Conductivity measured directly - resistance of wire attached to
tubeInsulating box Coil used for
releasing magnet
Detector coils
Copper wire
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Dependence of final speed on material conductivity
𝑀 𝑐𝑜𝑙𝑑=1 .13𝑀𝑤𝑎𝑟𝑚
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Dependence of velocity on distance traveled
z [m]
0,0 0,1 0,2 0,3 0,4
velo
city
[ms
-1]
0,0
0,5
1,0
1,5
2,0
• Theoretical values calculated numericaly from
• Initial speed determined experimentally
• System of cylindrical magnet ( ) and weight
• Velocity experimentaly determined by using relation
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Conclusion• Theoretical model developed
Quantitaive No free parameters
Very good correlation with experimentAssumptions
No skin effect Constant direction and magnitude of magnetization of magnet Rotational symmetry of the tube
• ExperimentParameters changed
• Size and shape of magnet• Conductivity of material
Reference
1. Valery S Cherkassky, Boris A Knyazev, Igor A Kotelnikov, Alexander A Tyutm; Eur. J. Phys.; Breaking of magnetic dipole moving through whole and cut conducting pipes
2. Edward M. Purcell; Electricity and Magnetism Berkley physics course – volume 2, Second Edition; McGraw-Hill book company
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THANK YOU
Magnetic brakes Croatian team Domagoj Pluscec
IYPT 2014 TEAM OF CROATIA
Reporter: Domagoj Pluščec
Current
𝐼=𝜀𝑅
𝜀=− 𝑑𝜙𝑑𝑡
=− 𝑑𝜙𝑑𝑡
𝑑 𝑧0𝑑 𝑧0
= 𝑑𝜙𝑑 𝑧
𝑣
𝜀=𝑑𝜙𝑑𝑧
𝑣𝑅= 1𝜎
𝑙𝐴
= 1𝜎2𝜋 𝜌𝑚
Δ𝜌 ∙𝑑𝑧
𝒅𝑰=𝜟𝝆𝝈 𝒗𝟐𝝅 𝝆𝒎
𝒅𝝓𝒅𝒛
𝒅𝒛
v – magnets speed time change of magnetic fluxε – induced voltageσ – material conductivityl – ring perimeterA – area of ring cross sectionR – restistance of the ring - magnets position
- can be expresed as space change because magnetic flux around the magnet doesn’t depend on time
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Determing Bρ
• Maxwell equation (magnetic field lines are closed curves)
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𝑩𝝆=−𝟏
𝟐𝝅𝛒𝒎
𝒅𝝓𝒅𝒛
– magnetic flux
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3
2
Schematic representation of the magnetic flux through a closed surface (disc shape)
Terminal velocity• Terminal velocity is reached when gravitational force is
equal to magnetic force
Supstitution
𝒗𝟎=𝟐𝝅𝝆𝒎𝒎𝒎𝒈
𝚫𝝆𝝈𝒌
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Assumptions used in derivation of model
• Tube is rotational symetrical in z direction
• Thin tube and the magnet is falling with low velocities(model does not take into consideration skin effect)
• We didn’t take into account case in which magnet rotates in such a way that direction of its magnetic moment changes
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– skin depth – characteristic frequency
Cu;
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Detailed dipole model derivation 1/2• In theoretical modeling part we obtained
in this equation we need to find for dipole• Magnetic flux trought cross section of tube can be
expressed as
• And by using expresion for a field of dipole in z direction we obtain magnetic flux in z direction
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Detailed dipole model derivation 2/2• Magnetic flux in z direction
• From that we derive magnetic flux change
• By knowing magnetic flux change we can obtain force
• If the magnet is far enough from the edges of the tube we can write
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Equation of motion with initial velocity
After solving for we obtain:
Supstitution:
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Terminal velocity derived from energy conservation
After substituing terms for current and resistance we obtain
Supstitution
𝑃𝑔=𝑚𝑔𝑣0
𝒗=𝟐𝝅 𝝆𝒎𝒎𝒎𝒈
𝚫𝝆𝝈𝒌
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Apparatus properties - tubes
Copper Aluminium
5.5 mm 5 mm
1 mm 1 mm
0.999994 1.000022
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Apparatus properties - magnets • Neodimium magnets
• Magnetic permeability • Density
Cylindrical magnets
Spherical magnets
Distance from magnet [mm]
0 5 10 15 20 25 30
Fo
rce
[m
N]
0
5
10
15
20
25 a = 0.6 cma = 2.1 cm
Example of obtained data - force
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z [m]
0,000 0,005 0,010 0,015 0,020 0,025 0,030 0,035
k [T
2m
3]
0,0
2,0e-8
4,0e-8
6,0e-8
8,0e-8
1,0e-7
1,2e-7
1,4e-7
1,6e-7
h = 0.6 cmh = 2.1 cm
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𝑘=∫−∞
∞
( 𝑑𝜙𝑑𝑧 )2
𝑑𝑧
Determing magnetic flux change 2
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h [mm]
0 5 10 15 20 25 30
k *1
0-7[T
2m
3]
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
Function k(a) aproximated byon interval [2.81, 27.8] mm
𝑘=∫−∞
∞
( 𝑑𝜙𝑑𝑧 )2
𝑑𝑧
Determing magnetic flux change 3
Velocity measurement • Coils detect passing magnet due to induction:
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Aluminium, 2 solenoids
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Theoretical change of velocity
time [s]
0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18
velo
city
[m
s-1]
0,00
0,02
0,04
0,06
0,08
0,10
0,12
0,14
0,16
a = 3 mma = 6 mm
Supstitution:
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Cooling - change of the magnetization of the magnet
𝑣 ∝1
𝑀 2For our case:
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t [s]
0 1000 2000 3000 4000 5000
R [
]
0
2
4
6
8
10
R0=8.492
Heating of cooled tube
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Cooling - change of tube dimesion • Change of radius of the pipe
𝛼𝑐𝑢=16.6 ∙10−6𝐾 −1
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Tube with slit vs tube without slit
Tube Tube with slit Without tube
Time
1 m long aluminium tube
Magnet properties• Cylindrical magnet
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Superconductor
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Skin effect