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Categories: Electromagnetism, Physics.

Magnetic field

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 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 $\vb{E}$, $\vb{D}$ 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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