6. Magnetic Fields in Matter

Matter becomes magnetized in a B field.

Induced dipoles: Diamagnets

Permanent dipoles : ParamagnetsFerromagnets

Magnetic dipoles are different from electric dipoles.

The dipoles are atomic current loops.

spin 2

orbital 2

sm

llm

B

Bem

e

The angular momenta l and s are quantized, i. e. they take fixedvalues, so does m.

Torque on a magnetic dipole (current loop): BmN

Force on a magnetic dipole: )( BmF

Derivation for the square loop gives the general result.

Liquid oxygen is paramagnetic. Its dipoles are pulled intoThe inhomogeneous field of the permanent magnet.

Paramagnetism

The B field aligns the magnetic moment of the atoms/molecules.The thermal motion makes the orientation random.

Competition results in partial alignment

MagnetizationV

im

M

Averaging over a small volume, which contains many atomic dipoles.

Diamagnetism

The dipole moments of all atomic orbitals change, becausethe orbital motion is changed.

The change m has the opposite direction of B.

Much weaker than paramagnetism. Only important, if paramagnetism is zero.

A superconductor is a perfect diamagnet. Here, the superconducting Pendelum bob is repelled by the permanent magnet.

Field of a Magnetized Object

We consider the macroscopic field, which is the average overa small volume containing many dipoles.

'ˆ)'(

4)( 2

do

r

rrMrA

'ˆ)'(

4'

ˆ)'(

4)(

dd bobo

r

r

r

r rKrJrA

Bound surface current nMK ˆb

Bound volume current MJ b

Interpretation of the surface current

Bound surface current nMK ˆb

Interpretation of the bound volume current

Bound volume current MJ b

Example 6.1

Field of the uniformlymagnetized sphere.

The Auxiliary Field H

fb JJJ The free current is at our disposal,the bound current is generated bythe material.

MB

Ho

Auxiliary field

fJH Ampere’s law

Many other authors call H “magnetic field” and B “induction” or “flux density”.

Linear Media

For paramagnets and diamagnets there are the linear relations

HM m mMagnetic susceptibility

HB Permeability

oPermeability of free space

)1( mo

Example 6.2

Example 6.3

Solenoid filled with linear Material.

Boundary conditions

fbelowabove

belowabovebelowabove

KHH

MMHH

||||

)(

At surfaces between materials of different susceptibility:

0 H

Ferromagnetism

Unlike in paramagnetic material, there is a strong interactionbetween the spins of the atoms/molecules, which aligns them.

The ferromagnet is composed of domains with different orientationof M. In the unmagnetized state they compensate each other.

Domains in an Fe-3% Si crystal observed in a scanning electronmicroscope. The four colors indicate the four possible domain directions.

In the presence of an external field the domains with an M that is similar to H grow. Saturation is reached when only the best domain survived.

Hysteresis loop

Magnetic field lines on a cobalt magnetic recording tape. The Solid arrows indicate the encoded magnetic bits.