Electromagnetism - Lecture 10

Home , Magnetic susceptibility, Magnetization

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:

m = IAˆz Zatom

The current and magnetic moment of the i-th electron are:

evi e I = mi = IAˆz = Li 2πri 2me

The magnetic dipole density is the magnetization vector M: dm e M = = NA < Li >atom dτ 2me This orbital angular momentum average is also valid in quantum mechanics

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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:

M.dl = JM.dS JM = ∇ × M IL ZA

For a rod uniformly magnetized along its length the magnetization can be represented by a surface magnetization current ﬂowing round the rod: JS = M × nˆ

The distributions JM and JS represent the eﬀect of the atomic magnetization with equivalent macroscopic current distributions

5 Magnetic Field Vectors

Ampère’s Law is modified to include magnetization effects:

B.dl = µ0 (JC + JM).dS ∇ × B = µ0(JC + JM) IL ZA where JC are conduction currents (if any)

Using ∇ × M = JM this can be rewritten as: B ∇×(B − µ0M) = µ0JC ∇ × H = JC 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:

H.dl = JC.dS ∇ × H = JC IL ZA

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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 = µ0M/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 = µrµ0H µr = 1 + χM

General advice - wherever µ0 appears in electromagnetism, it should be replaced by µrµ0 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:

0 eB ω = ω ∓ ∆ω ∆ω = z 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 0 0 ω = ω − ∆ω, and the electron with −Lz has ω = ω + ∆ω For both electrons magnetic dipole moment changes in −z direction!

9 Diamagnetic Magnetization

Change in orbital angular momentum of electron pair due to Larmor precession frequency: 2 2 ∆Lz = −2mer ∆ω = −eBzr and the induced magnetic moment of the pair: 2 e e 2 m = − ∆Lzˆz = − Bzr ˆz 2me 2me Averaging over all electron orbits introduces a geometric factor 1/3: 2 2 NAe Z < r > B M = NAαM B = − 6me where the atomic magnetic susceptibility is small and negative: 2 2 e Z < r > −29 αM = − ≈ −5 × 10 Z 6me

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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 m = L 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: 2 3NeµB M = B F = kTF ≈ 10eV 2kTF

where µB = e¯h/2me is the Bohr magneton

13 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 : N |m|2 M = A B 3kT

Paramagnetic susceptibility χM is small and positive. It decreases with increasing temperature: |m|2 χ = N − α M A 3kT M where the second term is the atomic susceptibility from the diamagnetism of the paired electrons.

14 Energy Storage in Magnetic Materials

The inductance of a solenoid increases if the solenoid is ﬁlled with a paramagnetic material:

2 2 L = µrµ0n πa l = µrL0

Hence the energy stored in the solenoid increases:

1 2 U = LI = µ U0 2 r

The energy density of the magnetic ﬁeld becomes: dU 1 B2 1 M = = B.H dτ 2 µrµ0 2

These are HUGE eﬀects for ferromagnetic materials

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