Engineering matlab/physics

Fundamentals of Magnetic Levitation

Magnetic levitation is cool!

• Shanghai, Magnetic Levitation Train • Maximum normal operation speed = 431 km/h (268 mph)

Two magnets in anti-Helmholtz configuration:

N

N

S

S

• anti-Helmholtz configuration == two (identical) magnets with like poles facing one another

𝐵𝐵𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 =

0 0

− 2𝐵𝐵0 𝑑𝑑 𝑧𝑧 + 𝐵𝐵0

B = 0

Balance of forces on an object (at equillibrium):

N

N

S

S

�⃗�𝐹𝑚𝑚𝑚𝑚𝑚𝑚 = 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 𝜇𝜇0

𝑉𝑉 𝐵𝐵 � 𝛻𝛻 𝐵𝐵

�⃗�𝐹𝑚𝑚 = 𝜌𝜌𝑠𝑠 − 𝜌𝜌𝑚𝑚 𝑉𝑉�⃗�𝑔

𝑚𝑚�⃗�𝑎 = �⃗�𝐹𝑚𝑚𝑚𝑚𝑚𝑚 + �⃗�𝐹𝑚𝑚 + �⃗�𝐹𝑑𝑑

Object is stationary, if �⃗�𝐹𝑚𝑚𝑚𝑚𝑚𝑚 + �⃗�𝐹𝑚𝑚 = 0

�⃗�𝐹𝑚𝑚𝑚𝑚𝑚𝑚 + �⃗�𝐹𝑚𝑚 = 𝜌𝜌𝑠𝑠 − 𝜌𝜌𝑚𝑚 𝑉𝑉�⃗�𝑔 + 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 𝜇𝜇0

𝑉𝑉 𝐵𝐵 � 𝛻𝛻 𝐵𝐵 = 0

Along z-axis: − 𝜌𝜌𝑠𝑠 − 𝜌𝜌𝑚𝑚 𝑔𝑔 + 𝜒𝜒𝑠𝑠−𝜒𝜒𝑚𝑚 𝜇𝜇0

𝐵𝐵𝑥𝑥 𝜕𝜕𝐵𝐵𝑧𝑧 𝜕𝜕𝑥𝑥

+ 𝐵𝐵𝑦𝑦 𝜕𝜕𝐵𝐵𝑧𝑧 𝜕𝜕𝑦𝑦

+ 𝐵𝐵𝑧𝑧 𝜕𝜕𝐵𝐵𝑧𝑧 𝜕𝜕𝑧𝑧

= 0

Levitation height at equillibrium:

N

N

S

S

− 𝜌𝜌𝑠𝑠 − 𝜌𝜌𝑚𝑚 𝑔𝑔 + 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 𝜇𝜇0

𝐵𝐵𝑥𝑥 𝜕𝜕𝐵𝐵𝑧𝑧 𝜕𝜕𝑥𝑥

+ 𝐵𝐵𝑦𝑦 𝜕𝜕𝐵𝐵𝑧𝑧 𝜕𝜕𝑦𝑦

+ 𝐵𝐵𝑧𝑧 𝜕𝜕𝐵𝐵𝑧𝑧 𝜕𝜕𝑧𝑧

= 0

𝐵𝐵𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 =

0 0

− 2𝐵𝐵0 𝑑𝑑

𝑧𝑧 + 𝐵𝐵0

𝐵𝐵𝑧𝑧 𝜕𝜕𝐵𝐵𝑧𝑧 𝜕𝜕𝑧𝑧 = −

2𝐵𝐵0 𝑑𝑑 −

2𝐵𝐵0 𝑑𝑑 𝑧𝑧 + 𝐵𝐵0 =

4𝐵𝐵02

𝑑𝑑2 𝑧𝑧 − 2𝐵𝐵02

𝑑𝑑

𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 𝜇𝜇0

4𝐵𝐵02

𝑑𝑑2 𝑧𝑧 − 𝑑𝑑 2 = 𝜌𝜌𝑠𝑠 − 𝜌𝜌𝑚𝑚 𝑔𝑔

𝐵𝐵𝑧𝑧 𝜕𝜕𝐵𝐵𝑧𝑧 𝜕𝜕𝑧𝑧 =

4𝐵𝐵02

𝑑𝑑2 𝑧𝑧 − 𝑑𝑑 2

𝑧𝑧 = 𝜌𝜌𝑠𝑠 − 𝜌𝜌𝑚𝑚 𝑔𝑔𝜇𝜇0𝑑𝑑2

𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 4𝐵𝐵02 + 𝑑𝑑 2

Levitation height at equillibrium:

• The height at which the beads levitate is linearly proportional to the density of the beads

• Concentration of Mn2+ determines magnetic susceptibility of the medium

Measuring density with a ruler...

𝑧𝑧 = 𝜌𝜌𝑠𝑠 − 𝜌𝜌𝑚𝑚 𝑔𝑔𝜇𝜇0𝑑𝑑2

𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 4𝐵𝐵02 + 𝑑𝑑 2

𝜌𝜌𝑠𝑠 = 𝑎𝑎𝑧𝑧 + 𝑏𝑏

𝑎𝑎 ≝ 4 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 𝐵𝐵02

𝑔𝑔𝜇𝜇0𝑑𝑑2

𝑏𝑏 ≝ 𝜌𝜌𝑚𝑚 − 2 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 𝐵𝐵02

𝑔𝑔𝜇𝜇0𝑑𝑑

• Density of sample depends linearly on levitation height at equilibrium (as well as other system parameters)

Kinetics of magnetic levitation:

N

N

S

S

�⃗�𝐹𝑚𝑚𝑚𝑚𝑚𝑚 = 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 𝜇𝜇0

𝑉𝑉 𝐵𝐵 � 𝛻𝛻 𝐵𝐵

�⃗�𝐹𝑚𝑚 = 𝜌𝜌𝑠𝑠 − 𝜌𝜌𝑚𝑚 𝑉𝑉�⃗�𝑔

𝑚𝑚�⃗�𝑎 = �⃗�𝐹𝑚𝑚𝑚𝑚𝑚𝑚 + �⃗�𝐹𝑚𝑚 + �⃗�𝐹𝑑𝑑

�⃗�𝐹𝑑𝑑 = −6𝜋𝜋𝜋𝜋𝜋𝜋�⃗�𝑣

− 𝜌𝜌𝑠𝑠 − 𝜌𝜌𝑚𝑚 𝑉𝑉𝑔𝑔 + 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 𝜇𝜇0

4𝐵𝐵02

𝑑𝑑2 𝑉𝑉 𝑧𝑧 − 𝑑𝑑 2 − 6𝜋𝜋𝜋𝜋𝜋𝜋

𝑑𝑑𝑧𝑧 𝑑𝑑𝑡𝑡 = 0

𝑑𝑑𝑧𝑧 𝑑𝑑𝑡𝑡 =

8 9 𝜋𝜋2𝐵𝐵02

𝜇𝜇0𝑑𝑑2𝜋𝜋 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 𝑧𝑧 + −

2𝜋𝜋2

9𝜋𝜋 𝜌𝜌𝑠𝑠 − 𝜌𝜌𝑚𝑚 𝑔𝑔 + 2𝐵𝐵02

𝜇𝜇0𝑑𝑑 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚

Kinetics of magnetic levitation:

𝜉𝜉 = 8 9 𝜋𝜋2𝐵𝐵02

𝜇𝜇0𝑑𝑑2𝜋𝜋 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚

𝜁𝜁 = − 2𝜋𝜋2

9𝜋𝜋 𝜌𝜌𝑠𝑠 − 𝜌𝜌𝑚𝑚 𝑔𝑔 + 2𝐵𝐵02

𝜇𝜇0𝑑𝑑 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚

𝑑𝑑𝑧𝑧 𝑑𝑑𝑡𝑡

= 𝜉𝜉𝑧𝑧 + 𝜁𝜁

� 𝑧𝑧𝑖𝑖

𝑧𝑧𝑓𝑓 1

𝜉𝜉𝑧𝑧 + 𝜁𝜁 𝑑𝑑𝑧𝑧 = � 0

𝑐𝑐0

𝑑𝑑𝑡𝑡 1 𝜉𝜉 � 𝑧𝑧𝑖𝑖

𝑧𝑧𝑓𝑓 1

𝜉𝜉𝑧𝑧 + 𝜁𝜁 𝑑𝑑 𝜉𝜉𝑧𝑧 + 𝜁𝜁 = � 0

𝑐𝑐0

𝑑𝑑𝑡𝑡

1 𝜉𝜉 log𝑐𝑐 𝜉𝜉𝑧𝑧𝑓𝑓 + 𝜁𝜁 − log𝑐𝑐 𝜉𝜉𝑧𝑧𝑐𝑐 + 𝜁𝜁 = �

𝑧𝑧𝑖𝑖

𝑧𝑧𝑓𝑓 1

𝜉𝜉𝑧𝑧 + 𝜁𝜁 𝑑𝑑 𝜉𝜉𝑧𝑧 + 𝜁𝜁 = � 0

𝑐𝑐0

𝑑𝑑𝑡𝑡 = 𝑡𝑡0

𝑡𝑡0 = 1 𝜉𝜉 log𝑐𝑐

𝜉𝜉𝑧𝑧𝑓𝑓 + 𝜁𝜁 𝜉𝜉𝑧𝑧𝑐𝑐 + 𝜁𝜁

Kinetics of magnetic levitation:

𝜉𝜉 = 8 9 𝜋𝜋2𝐵𝐵02

𝜇𝜇0𝑑𝑑2𝜋𝜋 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚

𝜁𝜁 = − 2𝜋𝜋2

9𝜋𝜋 𝜌𝜌𝑠𝑠 − 𝜌𝜌𝑚𝑚 𝑔𝑔 + 2𝐵𝐵02

𝜇𝜇0𝑑𝑑 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚

𝑡𝑡0 = 1 𝜉𝜉

log𝑐𝑐 𝜉𝜉𝑧𝑧𝑓𝑓 + 𝜁𝜁 𝜉𝜉𝑧𝑧𝑐𝑐 + 𝜁𝜁

• If 𝑧𝑧𝑓𝑓 = ℎ (equilibrium levitation height), then 𝑡𝑡0 = ∞ (the object could never reach exact equilibrium)

• Neglected thermal motion of particles as implicit assumption… how large an object has to be to levitate?

Homework assignment:

• Read and comprehend “Measuring Densities of Solids and Liquids Using Magnetic Levitation: Fundamentals” by Mirica et al., JACS 2009, 131, 10049–10058

• Reference material PDFs (located in “Week 6 magnetic particles” folder on Blackboard):

– MAGNETIC FIELD AND MAGNETIC FORCES

– SOURCES OF MAGNETIC FIELD