Chapter 6 - Magnetic Fields in Matter

Reference "Introduction to Electrodynamics" (5e) by David Griffiths.

If a material is placed in a $\vec{B}$-field, it will acquire a magnetic polarization (or magnetization). Different materials acquire different polarizations depending on their atomic structures.

• Paramagnets: magnetization parallel to applied $\vec{B}$, materials with odd # of electrons.
• Diamagnets: magnetization antiparallel to applied $\vec{B}$, materials with an even # of electrons.
• Ferromagnets: magnetization persists on the material even after the applied $\vec{B}$-field is removed, and is determined by the whole "magnetic history" of the object.

All electrons act as magnetic dipoles.

Imagine a magnetic dipole as pointing from south to north (the Gilbert model). It's an inaccurate model at small scales according to Griffiths, but he recommends it for intuition.

Torques and Forces on Magnetic Dipoles

Magnetic dipoles will experience some torque $N$ in an applied field, where $$\vec{N}=\vec{m}\times \vec{B}$$ and $m$ is the magnetic dipole moment. For paramagnetic materials (odd electrons), $m$ will be roughly in the same direction as the applied $\vec{B}$-field.

For a current loop, $m=Iab$ where $a,b$ are side lengths.

In a uniform field the net force on the dipole is zero, though this is not the case for nonuniform fields.

For an infinitesimal loop with dipole moment $\vec{m}$ in field $\vec{B}$, the force on the loop is $$\vec{F}=\nabla (\vec{m}\cdot \vec{B})$$

Diamagnets

Diamagnetism affects all materials, but is much weaker than paramagnetism, so is most easily observed in materials with an even number of electrons.

When an external $\vec{B}$-field is applied to a material, individual electrons will speed up according to $$\Delta v=\frac{eRB}{2m_e}$$ where $R$ is the "radius" of the electron from the nucleus of an atom. This increase in orbital speed will change the dipole moment $$\Delta \vec{m}=-\frac{1}{2}e(\Delta v)(R)\hat{z}=-\frac{e^2{R}^2}{4m_e}\vec{B}$$ this change in the dipole moment is antiparallel to $\vec{B}$ as shown above.

Magnetization

Magnetization $\vec{M}$ is $$\vec{M}=\text{magnetic dipole moment }\vec{m}\text{ per unit volume}$$ For a paramagnet, perhaps suspended above a solenoid, the magnetization would be positive/upward, and force downward. For a diamagnet, the magnetization would be instead downward, and force upward.

In general in a nonuniform field, paramagnets are attracted into the field, and diamagnets are repelled away.

Note: $\vec{M}$ is an average over a wildly complex set of infinitesimal dipoles and "smooths out" the dipole into a macroscopic view.

Note: both diamagnetism and paramagnetism are quite weak compared to, for instance, ferromagnetism, and so are often neglected in experimental calculations.

Field of a Magnetized Object

For some single dipole, the magnetic vector potential is $$\vec{A}(\vec{r})=\frac{{\mu }_0}{4\pi }\frac{\vec{m}\times \hat{R}}{R^2}$$ For a magnetized object with magnetization $\vec{M}$, $$\vec{A}(\vec{r})=\frac{{\mu }_0}{4\pi }\int {\frac{\vec{M}({\vec{r}}^{\prime })\times \hat{R}}{R}}^{2}d{\tau }^{\prime }$$ Alternatively, we can look at the object in terms of its volume current density ${\vec{J}}_b$ and surface current density ${\vec{K}}_b$, where $${\vec{J}}_b=\nabla \times \vec{M}{\vec{K}}_b=\vec{M}\times \hat{n}$$ $$\vec{A}(\vec{r})=\frac{{\mu }_0}{4\pi }{\int }_V\frac{{\vec{J}}_b({\vec{r}}^{\prime })}{R}d{\tau }^{\prime }+\frac{{\mu }_0}{4\pi }{\oint }_S\frac{{\vec{K}}_b({\vec{r}}^{\prime })}{R}d{A}^{\prime }$$

This means the potential (and therefore magnetic field) is the same as would be made by some volume current ${\vec{J}}_b$ throughout the material plus the surface current ${\vec{K}}_b$ on the boundary.

This means we needn't integrate all the infinitesimal dipoles, but rather just determine the bound currents ${\vec{J}}_b$ and ${\vec{K}}_b$ and find the field they produce.

Note: $\nabla \cdot {\vec{J}}_b=0$