Categories: Electromagnetism, Physics.

Magnetic field

The magnetic field B\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 qq:

F=qv×B\begin{aligned} \vb{F} = q \vb{v} \cross \vb{B} \end{aligned}

If an object is placed in a magnetic field B\vb{B}, and wants to rotate to align itself with the field, then its magnetic dipole moment m\vb{m} is defined from the aligning torque τ\vb{\tau}:

τ=m×B\begin{aligned} \vb{\tau} = \vb{m} \times \vb{B} \end{aligned}

Where m\vb{m} has units of J/T\mathrm{J / T}. From this, the magnetization M\vb{M} is defined as follows, and roughly represents the moments per unit volume:

MdmdV    m=VMdV\begin{aligned} \vb{M} \equiv \dv{\vb{m}}{V} \:\:\iff\:\: \vb{m} = \int_V \vb{M} \dd{V} \end{aligned}

If M\vb{M} has the same magnitude and orientation throughout the body, then m=MV\vb{m} = \vb{M} V, where VV is the volume. Therefore, M\vb{M} has units of A/m\mathrm{A / m}.

A nonzero M\vb{M} complicates things, since it contributes to the field and hence modifies B\vb{B}. We thus define the “free” auxiliary field H\vb{H} from the “bound” field M\vb{M} and the “net” field B\vb{B}:

H1μ0BM    B=μ0(H+M)\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 μ0\mu_0 is a known constant. It is important to point out some inconsistencies here: B\vb{B} contains a factor of μ0\mu_0, and thus measures flux density, while H\vb{H} and M\vb{M} do not contain μ0\mu_0, and therefore measure field intensity. Note that this convention is the opposite of the analogous electric fields E\vb{E}, D\vb{D} and P\vb{P}. Also note that P\vb{P} has the opposite sign convention of M\vb{M}.

Some objects, called ferromagnets or permanent magnets, have an inherently nonzero M\vb{M}. Others objects, when placed in a B\vb{B}-field, may instead gain an induced M\vb{M}.

When M\vb{M} is induced, its magnitude is usually proportional to the applied field strength H\vb{H}:

B=μ0(H+M)=μ0(H+χmH)=μ0μrH=μH\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 χm\chi_m is the volume magnetic susceptibility, and μr1+χm\mu_r \equiv 1 + \chi_m and μμrμ0\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 χm\chi_m.

If χm>0\chi_m > 0, the medium is paramagnetic, meaning it strengthens the net field B\vb{B}. Otherwise, if χm<0\chi_m < 0, the medium is diamagnetic, meaning it counteracts the applied field H\vb{H}.

For χm1|\chi_m| \ll 1, as is often the case, the magnetization M\vb{M} can be approximated by:

M=χmHχmB/μ0\begin{aligned} \vb{M} = \chi_m \vb{H} \approx \chi_m \vb{B} / \mu_0 \end{aligned}