I was studying electromagnetism and was going the basic definition and trying to understand their importance.

I could not figure the difference between The $H$-field and $B$-field at first. So I referred this. There it was stated that:

The H field (the magnetic field) is the field in vacuum. This field can induce a magnetization of ponderable matter and the total field (vacuum plus induced field) is the B field (magnetic induction)

With reference to those understanding, I tried to understand an image given in Wikipedia.

But I could not understand why the H-Field ( The magnetic field in a vacuum ) is from North(Red poles) To the South Pole inside a magnet. Whereas the B- Field direction is the opposite.

What is the H Field Trying to Convey? What is the Source of H Field and What is the source of B Field. I understood The M Feild is due to Magnetic domain alignment(I hope this is correct?)

And Outside a Magnet when to Consider H and when to consider B. And For magnetic Field by electric coils why B play a major Role?

Comparison of B, H and M inside and outside a cylindrical bar magnet.


3 Answers 3


The fact that the material is magnetic means that it has a magnetization, which is a source of Magnetic Field. But another interesting thing happens. Because it is made out of matter, the magnet has a different permeability $\mu$ than of vacuum, which is an obstacle to the field permeating the material. This means the resulting magnetic field will be weaker than the field generated by magnetization. The $\vec{H}$ field represents the difference between the resulting field and the one generated by magnetization only, and since magnetization is stronger than the resulting field insde a magnet, this points in the oposite direction.

The usefulness of the $\vec{H}$ field is better seen when trying to study Maxwell's Equations in matter. There, the use of $\vec{H}$ and $\vec{D}$ allows for the Maxwell's Equations to retain the same shape as they would have in vacuum.

  • $\begingroup$ How is retaining the same shape useful in Maxwell equations? $\endgroup$
    – VKJ
    Jul 12, 2019 at 13:47
  • 1
    $\begingroup$ Should you not compare $\mathbf{D}$ with $\mathbf{B}$? Afterall, the zero-divergence equations are set for these two fields $\endgroup$
    – Cryo
    Jul 12, 2019 at 16:16
  • $\begingroup$ Lucas, I tried to give the E and B fields a structure in One-dimensional structures of space. $\endgroup$ Jul 14, 2019 at 7:09
  • $\begingroup$ Retaining the shapes is useful in general because if you have solutions for one set of equations, you can use those solutions for another set that has the same format. @Cryo, well, the divergence of $\hat{D}$ is the bound current,not necessarily zero, as is the one for $\hat{B}$. The divergence of $\hat{H}$ is the same as of $-\hat{M}$. It is indeed non-zero near the edges (this field has monopoles), which is different from the true magnetic field. Perhaps this is as far as I can go, but I'll try to understand better the situation and find something better to add to the discussion. $\endgroup$ Jul 14, 2019 at 19:30

There are two basic equations for treating static magnetic fields in matter (In the following $\mathbf{J}=0$ is assumed). The first is $\mathrm{div}\mathbf{B}=0$ which essentially means that the magnetic flux density $ \mathbf{B}$ has no sources and $\mathbf{H}=\mathbf{B}-4\pi \mathbf{M}$ (here cgs-units are used) which defines $\mathbf{H}$ called magnetic field and finally $\mathbf{M}$ the magnetisation. $\mathbf{H}$ has to be considered as an auxiliary field since it is not fundamental as $ \mathbf{B}$ is. It is actually only important as soon as magnetized material comes into play (it also often used in Ampere's law, but as it is exchangeable with $ \mathbf{B}$ there, it looses its importance there). $\mathbf{H}$ is "kind of" based on the magnetisation which is an emerging phenomenon, so both quantities are not fundamental whereas $ \mathbf{B}$ is. So $\mathbf{H}$ does not correspond to the image -- we have of it often in mind -- being sourceless. $\mathbf{H}$ has sources, whereas the magnetic flux density $ \mathbf{B}$ has no sources. This has been the introduction.

Now comes the formal work. We will make use of $\mathrm{div}\mathbf{B}=0$ in $\mathbf{H}=\mathbf{B}-4\pi \mathbf{M}$:

$$ \mathrm{div}\mathbf{H}=\mathrm{div}\mathbf{B}-4\pi \mathrm{div} \mathbf{M} = -4\pi \mathrm{div}\mathbf{M}$$.

The only location where the magnetisation considerably changes is at the edge of the magnet, whereas inside the magnet we assume that it is constant. Furthermore let's consider an analogy of the electric field which fulfills the following equation:

$$ \mathrm{div} \mathbf{E}=4\pi \rho$$

where $\rho$ is the electric charge density. Comparing the equation of the magnetic field to the one of the electric field we see that the change of the magnetisation $\mathrm{div} \mathbf{M}$ serves a source of the magnetic field $\mathbf{H}$. This result to keep in mind is that both ends of the magnet can be seen as sources of the magnetic field $\mathbf{H}$ as charges are sources of the electric field $\mathbf{E}$. Now the designation of the quantity $\mathbf{H}$ as "magnetic field" is more comprehensible: it behaves as the electric field (with sources). The property of closed field lines is, however, reserved to the magnetic flux density $ \mathbf{B}$ .

By the way, the magnetisation depends only an applied external field $\mathbf{H}$. Ferromagnetic material have a remanent magnetisation which is nonzero even without external field $\mathbf{H}$ which is the case to be considered here. So in this case $\mathbf{M}$ can be considered independent from $\mathbf{H}$.

Summary: The source of $\mathbf{H}$ are the change of magnetisation $-\mathrm{div} \mathbf{M}$ whereas the magnetic flux density $ \mathbf{B}$ has no sources. $4\pi \mathbf{M}$ and $\mathbf{H}$ add up to give $ \mathbf{B}$. In magnetostatics $\mathbf{H}$ behaves similar to the electrical field in electrostatics. Things complicate a bit when currents come in, but this is not question now.

  • 1
    $\begingroup$ \div is the command for a division symbol - if you want the standard notation for the divergence operator, it would be given by \nabla \cdot $\endgroup$
    – J. Murray
    Jul 12, 2019 at 15:15
  • $\begingroup$ B is magnetic flux density which is different from Magnetic flux (|) $\endgroup$
    – VKJ
    Jul 12, 2019 at 16:18
  • 1
    $\begingroup$ @VKJ I mean magnetic flux density $\endgroup$ Jul 12, 2019 at 17:36
  • $\begingroup$ Thanks for the insight $\endgroup$
    – VKJ
    Jul 12, 2019 at 18:32
  • $\begingroup$ 'H has sources, whereas the magnetic flux B has no sources. ' I think this statement isn't correct. B-Field is resultant of H and the material through which H pass-through. Isn't? So indirectly H is the source of B.Right? $\endgroup$
    – VKJ
    Jul 14, 2019 at 9:27

I can be useful here to remove some confusion, may be was the student confusion, for sure it was my confusion when I started study magnetism. Now, here my considerations: There are 2 law of nature working here: Ampere’s law indicating that the H field circulation differ from zero only when the integration path includes some current The magnetic flux continuity law, indicating that the integral of the B field in a closed surface is zero The relation among the H and B field is by the permeability. Now the confusion: Let’s suppose a situation in which a current is generating a magnetic field and no magnetic material is present: let’s choose a closed integration path for the H field a compute the circulation. Ampere’s law will give a different from zero result for the circulation of the H field if some current is included in the path. Now assume somewhere a magnetizable material is placed inside the region we’re considering and ask ourselves two questions:

  1. is the H field changed because of the material? The answer is yes: the H field will be modified all around the region because of the inserted polarized material
  2. has the circulation of the H field around the same path considered above changed? The answer is no (as a consequence of Ampere's law) in conclusion: H fields produced from current are not conservative, H fields generated from permanent magnets are conservative. Now, here is the original question: “What is the H Filed Trying to Convey? What is the Source of H Field and What is the source of B Field. I understood The M Feild is due to Magnetic domain alignment(I hope this is correct?)” My answers are: The sources of H field are both electric current and permanent magnets. Yes, the M field is due to the alignment of magnetic domains.

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