PX157 - C10c - paramagnetic materials

$μ_{r} > 1 :$ magnetic field is enhanced
permanent atomic $\vec{μ}$ , but randomly oriented (thermal)
the external ${\vec{B}}_{0}$ causes them to align
typically, $μ_{r} \approx 1.00001 - 1.003$
eg: $A l$ , $C u$ (metals where magnet doesn't stick)

magnetization vector

magnetization vector, $\vec{M}$ , for bound currents:

\vec{M} = \frac{1}{V o l .} \sum_{j} {\vec{μ}}_{j}

units: $C m^{- 1} s^{- 1}$
the magnetic field inside a paramagnetic material:

\vec{B} = {\vec{B}}_{0} + μ_{0} \vec{M}

H-field for contributions of free currents:

\vec{H} = \frac{{\vec{B}}_{0}}{μ_{0}}

∴ \vec{B} = μ_{0} (\vec{H} + \vec{M})

also, $\vec{B} = μ_{r} {\vec{B}}_{0} :$

\vec{B} = μ_{0} μ_{r} \vec{H}

rewriting for $\vec{M} :$

\vec{M} = (μ_{r} - 1) \vec{H} = χ_{m} \vec{H}

- where, $χ_{m} =$ magnetic susceptibility

\vec{B} = {\vec{B}}_{0} + χ_{m} {\vec{B}}_{0} = (1 + χ_{m}) {\vec{B}}_{0} ⟹ μ_{r} = 1 + χ_{m}