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: m = IA^z Zatom The current and magnetic moment of the i-th electron are: evi e I = mi = IA^z = 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 3 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: M:dl = JM:dS JM = r × 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 × n^ The distributions JM and JS represent the effect of the atomic magnetization with equivalent macroscopic current distributions 5 Magnetic Field Vectors Amp`ere's Law is modified to include magnetization effects: B:dl = µ0 (JC + JM):dS r × B = µ0(JC + JM) IL ZA where JC are conduction currents (if any) Using r × M = JM this can be rewritten as: B ∇×(B − µ0M) = µ0JC r × 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 Amp`ere's Law in terms of H is: H:dl = JC:dS r × H = JC IL ZA 6 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 = 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 ! = ! ∓ ∆! ∆! = z 2me where ∆! 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 ! = ! − ∆!, and the electron with −Lz has ! = ! + ∆! 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 ∆! = −eBzr and the induced magnetic moment of the pair: 2 e e 2 m = − ∆Lz^z = − Bzr ^z 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 10 Notes: Diagrams: 11 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 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 jmj2 M = A B 3kT Paramagnetic susceptibility χM is small and positive. It decreases with increasing temperature: jmj2 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 15 Notes: Diagrams: 16.