6. Magnetic Fields in Matter Matter becomes magnetized in a B field. Induced dipoles: Diamagnets Permanent dipoles : Paramagnets Ferromagnets

Magnetic dipoles are different from electric dipoles.

The dipoles are atomic current loops. The angular momenta l and s are quantized, i. e. they take fixed values, so does m.

Torque on a magnetic dipole (current loop): Force on a magnetic dipole:

Derivation for the square loop gives the general result.

Liquid oxygen is paramagnetic. Its dipoles are pulled into The 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 Magnetization Averaging over a small volume, which contains many atomic dipoles.

Diamagnetism The dipole moments of all atomic orbitals change, because the orbital motion is changed. The changehas 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 over a small volume containing many dipoles.

Bound surface currentBound volume current

Interpretation of the surface current Bound surface current

Interpretation of the bound volume current Bound volume current

Example 6.1 Field of the uniformly magnetized sphere.

The Auxiliary Field H The free current is at our disposal, the bound current is generated by the material. Auxiliary field 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 Magnetic susceptibility Permeability Permeability of free space

Example 6.2

Example 6.3 Solenoid filled with linear Material.

Boundary conditions

At surfaces between materials of different susceptibility:

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

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

Domains in an Fe-3% Si crystal observed in a scanning electron microscope. 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.