Electromagnetism - Lecture 12

• Ferromagnetism • Hysteresis & Permanent Magnets • Ferromagnetic Surfaces • Toroid with Ferromagnetic Core • Superconductivity • The Meissner Eﬀect

1 Ferromagnetism

Ferromagnetism occurs when the spins of conduction electrons in metals spontaneously align

Caused by an exchange interaction U ∝ S1.S2

The spontaneous alignment breaks down above a critical temperature, the Curie point T = TC

For T > TC the metal is paramagnetic: C χM = TC = λC BL = λM T − TC where BL is the local ﬁeld due to the spin-spin interactions

For T < TC the metal is ferromagnetic with a large magnetization

2 Magnetic Saturation

When all the spins are aligned the magnetization is saturated

2 −2λNeµB /kT MS = 2NeµBe 0 < T < TC

Examples of ferromagnetic materials:

Iron (Fe) TC = 1043K BS = 1.7T

Cobalt (Co) TC = 1288K BS = 1.4T

Nickel (Ni) TC = 627K BS = 0.5T

The relative permeability µr = B/µ0H of ferromagnets is very 3 5 large and has a wide range of values µr = 10 − 10

3 Notes:

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4 Magnetic Domains and Hysteresis

The direction of the spontaneous M in a ferromagnet is random A macroscopic sample contains many magnetic domains, in each of which M points in a diﬀerent direction. They are separated by domain walls A macroscopic ferromagnet can be unmagnetised if H = 0

Applying an external ﬁeld H deﬁnes a preferred direction for M ⇒ Domain walls move to favour the direction of H ⇒ Electron spins rotate into alignment with H

A hysteresis curve shows B = µ0(H + M) as a function of H

When H is removed this can leave a permanent magnet The movement of domain walls is not completely reversible

5 Notes:

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6 Ferromagnetic Surfaces

Surfaces || to M Surfaces ⊥ to M

H|| is continuous B⊥ is continuous 4 4 B|| decreases by µr ≈ 10 H⊥ increases by µr ≈ 10

Magnetization current JM = M × nˆ No magnetization current No ﬂux through surface Large ﬂux through surface

Example of a bar magnet with a magnetic dipole ﬁeld outside it.

For discussion - why is the direction of H|| at the middle of the surface of a bar magnet opposite to M?

7 Notes:

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8 Toroid with Ferromagnetic Core

Use Amp`ere’s law round a circular path at the centre of the core:

H.dl = J.dS IL ZA

H(2πR) = n(2πR)I H = nI B = µrµ0H If a small gap of length d is made in the core:

Hcore(2πR − d) + Hgapd = n(2πR)I

From boundary condition on B⊥ at edges of gap:

Bgap Bgap = Bcore = µrµ0Hcore Hgap = = µrHcore µ0

As a result of the gap Hcore is reduced but Hgap is large!

(2πR)nI µr(2πR)nI Hcore = Hgap = 2πR + (µr − 1)d 2πR + (µr − 1)d

9 Notes:

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10 Energy Stored in Toroid

Magnetic energy density: dU 1 M = B.H dτ 2 Energy stored in ferromagnetic core (without gap): 2 (2πR)πa 2 2 2 2 U = BH = µ µ0π Ra n I M 2 r As a result of the gap the energy stored in the ferromagnetic core is reduced because Hcore is reduced:

(2πR − d) 2 2 U = µ µ0 πa H core r 2 core ... but a lot of energy is stored in the gap!

d 2 2 2 d 2 2 U = µ0 πa H = µ µ0 πa H gap 2 gap r 2 core

11 Notes:

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12 Superconductivity

Superconductivity occurs when conduction electrons in metals with wavenumber spin = k ↑ and -k ↓ form a Cooper pair

Superconductivity breaks down above a critical temperature TC and above a critical magnetic ﬁeld strength HC Type II superconductors have two transition temperatures

Examples of Superconductors:

Type I Metals (Al,Pb,Sn,Zn...) TC a few K BC up to 1T

Type II Metal Alloys (NbTi) TC ≈ 10K BC = 15T

Type II Ceramics (YBa2Cu3O7) TC ≈ 100K BC up to 300T

13 Properties of Superconductors

• Perfect Conductivity No resisitivity ρ → 0, σ → ∞ and no electric ﬁeld E = 0 • Persistent Currents Any current density J is allowed J will continue to ﬂow for ever! • Perfect Diamagnetism

χM = −1, µr = 0 and no magnetic ﬁeld B = 0 There is no magnetic ﬂux inside a superconductor • Surface Magnetic Fields Can only have H tangential to surface

A non-zero H|| is associated with surface currents

14 The Meissner Eﬀect

A bar magnet levitates above the surface of a superconductor

Understood using method of images: To satisfy the boundary condition H tangential to surface, the dipole ﬁeld of the bar magnet has to be combined with the dipole ﬁeld of an image bar magnet an equal distance behind the surface

The relative orientation of the image magnet is not obvious! The lowest energy has the dipole moments parallel (not antiparallel)

Force between bar magnet and image magnet is repulsive The image bar magnet is equivalent to the eﬀect of physical surface currents that create H|| at the superconducting surface

15 Notes:

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