Download Electromagnetism - Lecture 10 Magnetic Materials

Electromagnetism - Lecture 10
Magnetic Materials
• Magnetization Vector M
• Magnetic Field Vectors B and H
• Magnetic Susceptibility & Relative Permeability
• Diamagnetism
• Paramagnetism
• Effects of Magnetic Materials
1
Introduction to Magnetic Materials
There are three main types of magnetic materials with different
magnetic susceptibilities, χM :
• Diamagnetic - magnetization is opposite to external B
χM is small and negative.
• Paramagnetic - magnetization is parallel to external B
χM is small and positive.
• Ferromagnetic - magnetization is very large and non-linear.
χM is large and variable.
Can form permanent magnets in absence of external B
⇒ In this lecture Diamagnetism & Paramagnetism
Ferromagnetism will be discussed in Lecture 12
2
Magnetization Vector
The magnetic dipole moment of an atom can be expressed as an
integral over the electron orbits in the Bohr model:
Z
IAẑ
m=
atom
The current and magnetic moment of the i-th electron are:
e
evi
mi = IAẑ =
Li
I=
2πri
2me
The magnetic dipole density is the magnetization vector M:
M=
dm
e
= NA
< Li >atom
dτ
2me
This orbital angular momentum average is also valid in quantum
mechanics
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Notes:
Diagrams:
4
Magnetization Currents
The magnetization vector M has units of A/m
The magnetization can be thought of as being produced by a
magnetization current density JM :
Z
I
M.dl =
JM .dS
JM = ∇ × M
L
A
For a rod uniformly magnetized along its length the magnetization
can be represented by a surface magnetization current flowing
round the rod:
JS = M × n̂
The distributions JM and JS represent the effect of the atomic
magnetization with equivalent macroscopic current distributions
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Magnetic Field Vectors
Ampère’s Law is modified to include magnetization effects:
Z
I
B.dl = µ0 (JC + JM ).dS
∇ × B = µ0 (JC + JM )
L
A
where JC are conduction currents (if any)
Using ∇ × M = JM this can be rewritten as:
∇×(B − µ0 M) = µ0 JC
∇ × H = JC
B
H=
−M
µ0
B is known as the magnetic flux density in Tesla
H is known as the magnetic field strength in A/m
Ampère’s Law in terms of H is:
Z
I
H.dl =
JC .dS
L
A
6
∇ × H = JC
Notes:
Diagrams:
7
Relative Permeability
The magnetization vector is proportional to the external magnetic
field strength H:
M = χM H
where χM is the magnetic susceptibility of the material
Note - some books use χB = µ0 M/B instead of χM = M/H
The linear relationship between B, H and M:
B = µ0 (H + M)
can be expressed in terms of a relative permeability µr
B = µ r µ0 H
µr = 1 + χM
General advice - wherever µ0 appears in electromagnetism,
it should be replaced by µr µ0 for magnetic materials
8
Diamagnetism
For atoms or molecules with even numbers of electrons the orbital
angular momentum states +Lz and −Lz are paired and there is
no net magnetic moment in the absence of an external field
An external magnetic field Bz changes the angular velocities:
ω 0 = ω ∓ ∆ω
∆ω =
eBz
2me
where ∆ω is known as the Larmor precession frequency
Can think of as effect of magnetic force, or as example of induction
For an electron pair in an external Bz , the electron with +Lz has
ω 0 = ω − ∆ω, and the electron with −Lz has ω 0 = ω + ∆ω
For both electrons magnetic dipole moment changes in −z direction!
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Diamagnetic Magnetization
Change in orbital angular momentum of electron pair due to
Larmor precession frequency:
∆Lz = −2me r 2 ∆ω = −eBz r 2
and the induced magnetic moment of the pair:
e2
e
∆Lz ẑ = −
Bz r 2 ẑ
m=−
2me
2me
Averaging over all electron orbits introduces a geometric factor 1/3:
NA e 2 Z < r 2 > B
M = N A αM B = −
6me
where the atomic magnetic susceptibility is small and negative:
αM
e2 Z < r 2 >
=−
≈ −5 × 10−29 Z
6me
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Notes:
Diagrams:
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Notes:
Diagrams:
12
Paramagnetism
Paramagnetic materials have atoms or molecules with a net
magnetic moment which tends to align with an external field
• Atoms with odd numbers of electrons have the magnetic
moment of the unpaired electron:
e
L
m=
2me
• Ions and some ionic molecules have magnetic moments
associated with the valence electrons
• Metals have a magnetization associated with the spins of the
conduction electrons near the Fermi surface:
3Ne µ2B
B
M=
2kTF
F = kTF ≈ 10eV
where µB = eh̄/2me is the Bohr magneton
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Susceptibility of Paramagnetic Materials
The alignment of the magnetic dipoles with the external field is
disrupted by thermal motion:
N (θ)dθ ∝ e−U/kT sin θdθ
U = −m.B = −mB cos θ
Expanding the exponent under the assumption that U kT :
NA |m|2
B
M=
3kT
Paramagnetic susceptibility χM is small and positive.
It decreases with increasing temperature:
2
|m|
− αM
χM = N A
3kT
where the second term is the atomic susceptibility from the
diamagnetism of the paired electrons.
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Energy Storage in Magnetic Materials
The inductance of a solenoid increases if the solenoid is filled with a
paramagnetic material:
L = µr µ0 n2 πa2 l = µr L0
Hence the energy stored in the solenoid increases:
U=
1 2
LI = µr U0
2
The energy density of the magnetic field becomes:
1 B2
1
dUM
=
= B.H
dτ
2 µr µ0
2
These are HUGE effects for ferromagnetic materials
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Notes:
Diagrams:
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