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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