---
title: "Magnetic field"
date: 2021-07-12
categories:
- Physics
- Electromagnetism
layout: "concept"
---

The **magnetic field** $\vb{B}$ is a vector field
that describes magnetic effects,
and is defined as the field that correctly predicts
the [Lorentz force](/know/concept/lorentz-force/)
on a particle with electric charge $q$:

$$\begin{aligned}
    \vb{F}
    = q \vb{v} \cross \vb{B}
\end{aligned}$$

If an object is placed in a magnetic field $\vb{B}$,
and wants to rotate to align itself with the field,
then its **magnetic dipole moment** $\vb{m}$
is defined from the aligning torque $\vb{\tau}$:

$$\begin{aligned}
    \vb{\tau} = \vb{m} \times \vb{B}
\end{aligned}$$

Where $\vb{m}$ has units of $\mathrm{J / T}$.
From this, the **magnetization** $\vb{M}$ is defined as follows,
and roughly represents the moments per unit volume:

$$\begin{aligned}
    \vb{M} \equiv \dv{\vb{m}}{V}
    \:\:\iff\:\:
    \vb{m} = \int_V \vb{M} \dd{V}
\end{aligned}$$

If $\vb{M}$ has the same magnitude and orientation throughout the body,
then $\vb{m} = \vb{M} V$, where $V$ is the volume.
Therefore, $\vb{M}$ has units of $\mathrm{A / m}$.

A nonzero $\vb{M}$ complicates things,
since it contributes to the field
and hence modifies $\vb{B}$.
We thus define
the "free" **auxiliary field** $\vb{H}$
from the "bound" field $\vb{M}$
and the "net" field $\vb{B}$:

$$\begin{aligned}
    \vb{H} \equiv \frac{1}{\mu_0} \vb{B} - \vb{M}
    \:\:\iff\:\:
    \vb{B} = \mu_0 (\vb{H} + \vb{M})
\end{aligned}$$

Where the **magnetic permeability of free space** $\mu_0$ is a known constant.
It is important to point out some inconsistencies here:
$\vb{B}$ contains a factor of $\mu_0$, and thus measures **flux density**,
while $\vb{H}$ and $\vb{M}$ do not contain $\mu_0$,
and therefore measure **field intensity**.
Note that this convention is the opposite of the analogous
[electric fields](/know/concept/electric-field/)
$\vb{E}$, $\vb{D}$ and $\vb{P}$.
Also note that $\vb{P}$ has the opposite sign convention of $\vb{M}$.

Some objects, called **ferromagnets** or **permanent magnets**,
have an inherently nonzero $\vb{M}$.
Others objects, when placed in a $\vb{B}$-field,
may instead gain an induced $\vb{M}$.

When $\vb{M}$ is induced,
its magnitude is usually proportional
to the applied field strength $\vb{H}$:

$$\begin{aligned}
    \vb{B}
    = \mu_0(\vb{H} + \vb{M})
    = \mu_0 (\vb{H} + \chi_m \vb{H})
    = \mu_0 \mu_r \vb{H}
    = \mu \vb{H}
\end{aligned}$$

Where $\chi_m$ is the **volume magnetic susceptibility**,
and $\mu_r \equiv 1 + \chi_m$ and $\mu \equiv \mu_r \mu_0$ are
the **relative permeability** and **absolute permeability**
of the medium, respectively.
Materials with intrinsic magnetization, i.e. ferromagnets,
do not have a well-defined $\chi_m$.

If $\chi_m > 0$, the medium is **paramagnetic**,
meaning it strengthens the net field $\vb{B}$.
Otherwise, if $\chi_m < 0$, the medium is **diamagnetic**,
meaning it counteracts the applied field $\vb{H}$.

For $|\chi_m| \ll 1$, as is often the case,
the magnetization $\vb{M}$ can be approximated by:

$$\begin{aligned}
    \vb{M}
    = \chi_m \vb{H}
    \approx \chi_m \vb{B} / \mu_0
\end{aligned}$$