AcademySources of Magnetic Fields
Academy
Magnetic Materials
Level 1 - Physics topic page in Sources of Magnetic Fields.
Principle
Magnetic materials change the magnetic field because their atoms contain tiny magnetic dipoles. The response can be weak and linear, or strong and history-dependent.
Notation
\(\vec B\)
magnetic field
\(\mathrm{T}\)
\(\vec H\)
magnetizing field
\(\mathrm{A\,m^{-1}}\)
\(\vec M\)
magnetization, magnetic dipole moment per volume
\(\mathrm{A\,m^{-1}}\)
\(\mu\)
permeability of a material
\(\mathrm{N\,A^{-2}}\)
\(\mu_r\)
relative permeability
\(\chi_m\)
magnetic susceptibility
Method
Derivation 1: Separate free-current field from material response
The field inside matter is described using \(\vec H\) for the field produced by free currents and \(\vec M\) for the material's magnetization.
Material relation
\[\vec B=\mu_0(\vec H+\vec M)\]
Linear material
\[\vec M=\chi_m\vec H\]
Permeability form
\[\vec B=\mu\vec H=\mu_0\mu_r\vec H\]
Derivation 2: Classify the response
Diamagnetic materials weakly oppose an applied field. Paramagnetic materials weakly align with it. Ferromagnetic materials can align strongly through domains and can remain magnetized after the external field changes.
Derivation 3: Recognize hysteresis
In a ferromagnet, \(\vec B\) does not depend only on the current value of \(\vec H\). It also depends on magnetic history, producing a hysteresis loop.
Rules
General material field
\[\vec B=\mu_0(\vec H+\vec M)\]
Linear magnetic material
\[\vec B=\mu_0(1+\chi_m)\vec H=\mu_0\mu_r\vec H\]
Relative permeability
\[\mu_r=1+\chi_m\]
Examples
Question
A material has
\[\mu_r=500\]
If the same free current would produce \[B_0=2.0\times10^{-4}\,\mathrm T\]
in vacuum, estimate \(B\) in a linear-core model.Answer
For a linear material with the same geometry,
\[B=\mu_rB_0\]
Thus \[B=500(2.0\times10^{-4})=0.10\,\mathrm T\]
Checks
- Vacuum has \(\mu_r=1\).
- Diamagnetic response is weakly opposing; paramagnetic response is weakly aligning.
- Ferromagnetic response is strong and often nonlinear.
- Hysteresis means the present field depends on magnetic history, not only the present current.