---
title: "Magnetic field"
sort_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:

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