# Electromagnetism - Lecture 10. Magnetic Materials

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1 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

2 Introduction to Magnetic Materials There are three main types of magnetic materials with different magnetic susceptibilities, χ M : Diamagnetic - magnetization is opposite to external B χ M is small and negative. Paramagnetic - magnetization is parallel to external B χ M is small and positive. Ferromagnetic - magnetization is very large and non-linear. χ M 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

3 Magnetization Vector The magnetic dipole moment of an atom can be expressed as an integral over the electron orbits in the Bohr model: m = IAẑ atom The current and magnetic moment of the i-th electron are: I = ev i 2πr i m i = IAẑ = e 2m e L i The magnetic dipole density is the magnetization vector M: M = dm dτ = N e A < L i > atom 2m e This orbital angular momentum average is also valid in quantum mechanics 3

4 Notes: Diagrams: 4

5 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 J M : M.dl = J M.dS J M = 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: J S = M ˆn The distributions J M and J S represent the effect of the atomic magnetization with equivalent macroscopic current distributions 5

6 Magnetic Field Vectors Ampère s Law is modified to include magnetization effects: B.dl = µ 0 (J C + J M ).ds B = µ 0 (J C + J M ) L A where J C are conduction currents (if any) Using M = J M this can be rewritten as: (B µ 0 M) = µ 0 J C H = J C H = B µ 0 M 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 = J C.dS L A H = J C 6

7 Notes: Diagrams: 7

8 Relative Permeability The magnetization vector is proportional to the external magnetic field strength H: M = χ M H where χ M is the magnetic susceptibility of the material Note - some books use χ B = µ 0 M/B instead of χ M = M/H The linear relationship between B, H and M: B = µ 0 (H + M) can be expressed in terms of a relative permeability µ r B = µ r µ 0 H µ r = 1 + χ M General advice - wherever µ 0 appears in electromagnetism, it should be replaced by µ r µ 0 for magnetic materials 8

9 Diamagnetism For atoms or molecules with even numbers of electrons the orbital angular momentum states +L z and L z are paired and there is no net magnetic moment in the absence of an external field An external magnetic field B z changes the angular velocities: ω = ω ω ω = eb z 2m e 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 B z, the electron with +L z has ω = ω ω, and the electron with L z has ω = ω + ω For both electrons magnetic dipole moment changes in z direction! 9

10 Diamagnetic Magnetization Change in orbital angular momentum of electron pair due to Larmor precession frequency: L z = 2m e r 2 ω = eb z r 2 and the induced magnetic moment of the pair: m = e 2m e L z ẑ = e2 2m e B z r 2 ẑ Averaging over all electron orbits introduces a geometric factor 1/3: M = N A α M B = N Ae 2 Z < r 2 > B 6m e where the atomic magnetic susceptibility is small and negative: α M = e2 Z < r 2 > 6m e Z 10

11 Notes: Diagrams: 11

12 Notes: Diagrams: 12

13 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: m = e L 2m e 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: M = 3N eµ 2 B 2kT F B ɛ F = kt F 10eV where µ B = e h/2m e is the Bohr magneton 13

14 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 : M = N A m 2 3kT Paramagnetic susceptibility χ M is small and positive. It decreases with increasing temperature: ( ) m 2 χ M = N A 3kT α M where the second term is the atomic susceptibility from the diamagnetism of the paired electrons. B 14

15 Energy Storage in Magnetic Materials The inductance of a solenoid increases if the solenoid is filled with a paramagnetic material: L = µ r µ 0 n 2 πa 2 l = µ r L 0 Hence the energy stored in the solenoid increases: U = 1 2 LI2 = µ r U 0 The energy density of the magnetic field becomes: du M dτ = 1 2 B 2 µ r µ 0 = 1 2 B.H These are HUGE effects for ferromagnetic materials 15

16 Notes: Diagrams: 16

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