16. MAGNETIC BRAKES

1

”When a strong magnet falls down a non ferromagnetic metal tube, it will experience a retarding force.

Investigate the phenomenon.”

2

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

3

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

4

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;

5

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

6

𝑭𝒎=𝝈𝚫𝛒 𝒗𝟐𝝅 𝛒𝒎

∫𝒛𝟏

𝒛𝟐

( 𝒅𝝓𝒅𝒛 )𝟐

𝒅𝒛

7

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

8

 

 

 

 

 

 

 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

9

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

11

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

12

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

13

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

14

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

15

16

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

17

Dependence of final speed on material conductivity

𝑀 𝑐𝑜𝑙𝑑=1 .13𝑀𝑤𝑎𝑟𝑚

18

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

19

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

20

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

22

Determing Bρ

• Maxwell equation (magnetic field lines are closed curves)

1

𝑩𝝆=−𝟏

𝟐𝝅𝛒𝒎

𝒅𝝓𝒅𝒛

– magnetic flux

23

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

24

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

25

– skin depth – characteristic frequency

Cu;

26

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

27

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

28

Equation of motion with initial velocity

After solving for we obtain:

Supstitution:

29

Terminal velocity derived from energy conservation

After substituing terms for current and resistance we obtain

Supstitution

𝑃𝑔=𝑚𝑔𝑣0

𝒗=𝟐𝝅 𝝆𝒎𝒎𝒎𝒈

𝚫𝝆𝝈𝒌

30

Apparatus properties - tubes

Copper Aluminium

5.5 mm 5 mm

1 mm 1 mm

0.999994 1.000022

31

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

32

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

33

𝑘=∫−∞

∞

( 𝑑𝜙𝑑𝑧 )2

𝑑𝑧

Determing magnetic flux change 2

34

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:

35

Aluminium, 2 solenoids

36

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:

37

Cooling - change of the magnetization of the magnet

𝑣 ∝1

𝑀 2For our case:

38

t [s]

0 1000 2000 3000 4000 5000

R [

]

0

2

4

6

8

10

R0=8.492

Heating of cooled tube

39

Cooling - change of tube dimesion • Change of radius of the pipe

𝛼𝑐𝑢=16.6 ∙10−6𝐾 −1

40

Tube with slit vs tube without slit

Tube Tube with slit Without tube

Time

1 m long aluminium tube

Magnet properties• Cylindrical magnet

41

Superconductor

42

Skin effect