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