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}