Physical meaning of the $\vec H$ field in electromagnetism

Question

For a constant magnetic field in (non-ferromagnetic) isotropic matter one distinguishes two fields $\vec B$ and $\vec H$. They are related $$\vec B=\mu \vec H,$$ where $\mu$ is a constant depending on the matter. As far as I understand, the physical meaning of the field $\vec B$ is that it is the macroscopic magnetic field in the matter which can actually be measures by its action on moving macroscopic charges.

However the physical meaning of $\vec H$ is less clear to me. How it can be measured at least in principle? Equivalently, how $\mu$ can be measured?

In literature $\vec H$ is defined by $$\vec H=\vec B-4\pi \vec I,$$ where $\vec I$ is the magnetization vector. Thus, equivalently, it would suffice to explain how to measure $\vec I$.

Answer 1

H is the driving field of the coil. Took me awhile to appreciate. B is the total field.

Answer 2

The figure and the equation in the following description are borrowed from Wikipedia's description of solenoid coil article. However, the red circle in the figure was added by me. And am using diffrent unit system.

Suppose we have a solenoid coil with an iron core as shown in the figure (although the iron core is not explicitly depicted in the figure).

The magnetic flux density $\vec{B}$ in the iron core is expressed by the following equation (see Wikipedia), \begin{equation} B_{\text{iron}}=\mu_0\mu_r\frac{NI}{l},\tag{1} \end{equation} where $B_{\text{iron}}$ is the magnetic flux density in the iron-core, $l$ is the length of the solenoid, $\mu _{0}$ is the magnetic permeability of vacuum, $\mu_r$ is the relative magnetic permeability of iron, $N$ the number of turns, and $I$ the electric current.

In Equation (1), some of the physical quantities are directly measurable. The current $I$ can be measured with an electric circuit ampere meter. Direct measurement of $B_{\text{iron}}$ may be very difficult, but from our knowledge of Maxwell's equations, we know that the normal component of B is continuous inside and outside the iron. Thus \begin{equation} B_{\text{iron}}=B_{\text{space point}}, \tag{2} \end{equation} where the space point is a point very close to the iron core and at the center axis, an example of this space point is the red circle point in the figure. $B_{\text{space point}}$ can be measured with a magnetometer.

From equations (1) and (2), $\mu_r$ can be calculated: \begin{equation} \mu_r=B_{\text{space point}}\frac{1}{\mu_0}\frac{l}{NI} \tag{3}. \end{equation} The result of this calculation can be regarded as a measurement. Thus, \begin{align*} H_{\text{iron}}&=\frac{B_{\text{iron}}}{\mu_0\mu_r}=\frac{NI}{l},\tag{4} \\ \mu_0M&=B_{\text{iron}}-\mu_0H_{\text{iron}}=\mu_0(\mu_r-1)\frac{NI}{l},\tag{5} \end{align*} where $H_{\text{iron}}$ is the magnetic field and $\mu_0M$ (or $M$) is the magnetization of iron core.

Of course, this measurement method has a large measurement error, so the actual measurement method needs to be devised.

Answer 3

The electromagnetic fields are such that they arise from sources and the interaction of those source fields with the various media present. The interaction is quantified by constitutive relations. If the interaction between the source field and the respective media is simple, i.e. linear, then the constitutive relation is a constant of proportionality. The constant is unity for the vacuum (no interaction with media, field has the character of source only) and some other number for other media; like iron for example. In general for non-linear media, the constitutive relation may not be characterized by a simple direct proportionality by a constant and may rather be an independent function.

Answer 4

I am stating all the notations and significances so that it is simpler to understand.

$B$ is called magnetic flux density, and is defined as: The total magnetic lines of force i.e. magnetic flux crossing a unit area in a plane at right angles to the direction of flux.

$M$ is the magnetic field intensity and it is defined as: the force experienced by a unit north pole of one Weber strength (in SI) or one Maxwell (in CGS), placed at that point in the field.

Now it is definitely true that $B=H$ but that's only for $\mu=1$; i.e. in vacuum, the force experienced by a unit north pole, at a certain point in the field, equals the number of flux (or field lines) that crosses a unit area in a plane at right angles to the direction of flux.

Well, there is no such notion of Macroscopic Magnetism or such, rather it is better to put together as, B-Field or Magnetic Flux Density, is the Amplitude of the Magnetic Field, and Magnetic Field Intensity or H-Field is the measure of the Magnetomotive Force which is required to create an unit Flux Density within a certain material per unit length of it.

Now, the experiment of measuring permeability or $\mu$ of any materiel is done by Darcy's Law, that states that, the rate of fluid flow through a porous medium is proportional to the potential energy gradient within that fluid, and molten core of known coefficient of viscosity is used as a fluid and corresponding potential gradient is measured for the rate of fluid flow, through some porous materiel. (I am not much of an Experimentalist type, you can check the complete process here: http://wwwciv.eng.cam.ac.uk/geotech_new/publications/TR/TR326.pdf)

And the magnetizing vector or $M$, empirically, is the measure of how any magnetic material gets magnetized. And this concept can be well understood by: $M=\chi_mH$ and this $\chi_m$ is the Susceptibility of the material.

And for understanding purposes, it can be stated as: Higher a value of Susceptibility of any material, higher it is 'Susceptible' (or has tendency) to get more magnetized by a definite field (H).