Wikipedia provides a mathematical explanation. Can I get the intuitive one? I'd like to, for example, understand a ferrite datasheet. These usually have graphs of H vs B, and the definition of permeability depends on understanding the relationship of H and B.

Also, I wonder: I was able to learn a great deal about electric fields before I knew what "fields" were. I learned about voltage and Ohm's law and so on, which a physicist might explain with a field, but which the electrical engineer explains with simpler concepts, like the difference between two points in a circuit. Is there a similar, simpler explanation of H vs B fields that is of more relevance to the electrical engineer, and less to the physicist?

  • \$\begingroup\$ I never knew about this, thanks for the question. My take on the wiki article is that H fields are from magnets, B fields are from current flowing in a wire. \$\endgroup\$ Commented Dec 27, 2013 at 12:52
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    \$\begingroup\$ geometrikal, you are totally wrong in your interpretation. H and B are present simultaneously in the same magnetic field. \$\endgroup\$
    – FiddyOhm
    Commented Jun 5, 2014 at 23:17
  • \$\begingroup\$ H is a bit like the number of magnetic field lines and B kinda is how tightly packed they are. More amps/more turns/shorter core means more field lines (bigger H - Aturns/m), higher permeability (measure of how easily those field lines can "flow") means they can be packed tighter together in the core (larger B - more intense magnetic field). I think H = Bcore area/length around core... \$\endgroup\$
    – Sam
    Commented May 28, 2016 at 6:04
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    \$\begingroup\$ Magnetic flux density B (tesla) is a response of the medium to an applied magnetic field strengh H (A/m). Permeability μ denotes how much the medium accepts to develop B when H is applied. B = µ x H, B is dependent on the medium. There is no magnetic field alone, only an electromagnetic field: Frequent confusion in litterature. \$\endgroup\$
    – mins
    Commented Sep 10, 2020 at 11:27
  • \$\begingroup\$ See this image showing the B-field, H-field and even the M-field of a bar permanent magnet. \$\endgroup\$
    – alejnavab
    Commented Feb 12, 2021 at 19:27

6 Answers 6


H is the driving force in coils and is ampere turns per metre where the metre part is the length of the magnetic circuit. In a transformer it's easy to determine this length because 99% of the flux is contained in the core. A coil with an air core is difficult as you might imagine.

I think of B as a by-product of H and B is made bigger by the permeability of the core.

In electrostatics, E (electric field strength) is the equivalent of H (magnetic field strength) and it's somewhat easier to visualize. Its units are volts per metre and also gives rise to another quantity, electric flux density (D) when multiplied by the permittivity of the material in which it exists: -

\$\dfrac{B}{H} = \mu_0\mu_R\$ and

\$\dfrac{D}{E} = \epsilon_0\epsilon_R\$

Regarding ferrite data sheets, the BH curve is the important one - it tells you the permeability of the material and this directly relates to how much inductance you can get for one turn of wire.

It will also indicate how much energy could be lost when reversing the magnetic field - this of course will always happen when ac driven - not all the domains in the ferrite return to produce an average of zero magnetism when the current is removed and when reversing the current the remaining domains need to be neutralized before the core magnetism goes negative - this requires a small amount of energy on most ferrites and gives rise to the term hysteresis loss.

Other important graphs in a ferrite data sheet are the permeability versus frequency graph and permeability versus temperature.

From personal experience of having designed a few transformers, I find them tortuous in that I never seem to naturally remember anything other than the basics each time I begin a new design and this is annoying - in this answer I had to double check everything except the units of H!

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    \$\begingroup\$ You say E is equivalent to H, and D to B. I would rather say that E is equivalent to B, because they are linked repectively to total charges and total currents. While D is equivalent to H as D and H are linked respectively to free charges and free currents. If you base your argument only on the look of the equations, it is very weak: the shape of the equations only depends on conventions (e.g. signs of P and M). \$\endgroup\$
    – Benjamin T
    Commented Jul 22, 2018 at 6:48
  • \$\begingroup\$ @BenjaminT rather than leave a comment you should consider leaving a fully fledged answer to justify your thinking. \$\endgroup\$
    – Andy aka
    Commented Jul 22, 2018 at 9:32
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    \$\begingroup\$ No, because I do not answer OP question. I just disagree with a single of your sentences. Moreover I think my comment fully justify my thinking on that particular point. \$\endgroup\$
    – Benjamin T
    Commented Jul 22, 2018 at 9:36
  • \$\begingroup\$ @BenjaminT I think Andy aka meant that E and H are equivalent in the sense that when you build magnetic circuits, H can be treated the same way E is treated in an electrical circuit (as one does when designing inductors or transformers) i.e. they are treated the same way in a circuit model. He gave a practical EE answer not a physicists answer. However, saying E is equivalent to H just seems wrong, that's like saying voltage and current are equivalent because they are linked by Ohm's law (unless we have a very different definition of what the word "equivalent" means). \$\endgroup\$
    – Lucas
    Commented Dec 6, 2021 at 9:37

You are not the first to be befuddled by conventional explanations of B & H as they apply to practical electromagnetic devices such as ferrite inductor cores. I struggled for years with the standard explanations of the nature of B & H and their application in such devices. My salvation came from a single chapter in a largely forgotten book I happened upon in a used book store some twenty-odd years ago. I believe the book is now available on-line in pdf format. Try Google Books. The name of the book is "The Magnetic Circuit" by V. Karapetoff and was published around 1911 - yes, 110+ years ago! Nonetheless, magnetic principles were well understood at the time and the terminology has been essentially unchanged in the intervening decades.

If you read Chapter 1 very carefully you will be blessed with a very practical understanding of the magnetic field and all of its beautiful characteristics and its arcane terminology which is still in common use today (e.g. magnetomotive force, permeance, reluctance, flux vs flux density, etc.) The remaining chapters are also interesting, but not as well presented as Chapter 1, which I revere as a sparkling gem of engineering exposition.

It will also help your understanding if you construct a few simple air-core coils to experiment with as an aid to digestion of the basic concepts. Use a function generator to drive the coils and a smaller coil to sense the magnetic field and display it on an oscilloscope. The driven coils should be about 6-12 inches in diameter and the sense coil about 1/2" in diameter. A frequency of 1000 Hz is adequate. If you are really ambitious you should build the toroidal coil which the author uses as his main vehicle of explanation.

I'll end by giving my standard explanation of B & H: The simplest electrical circuit is a battery with a parallel connected resistor. Ohms Law can be learned solely from this simple arrangement of three elements - voltage source, resistance and wire - along with a voltmeter and ammeter. B & H can be analogously learned from the simplest magnetic circuit. This is a wire with a current (AC or DC) flowing through it.

The magnetic field produced by the current encircles the wire with a cylindrical formation of flux lines. "M" is the magnetomotive force analogous to the voltage of the battery in the Ohms Law example. "B" is the strength of the resulting magnetic flux field formed around the wire by that magnetomotive force M, and is analogous to the electrical current "I" in the Ohms Law example. The "resistor" is the permeability of the air surrounding the wire. The surrounding air forms a "collective" or "distributed" magnetic resistor of sorts around the wire. This "magnetic resistor" dictates a ratio of produced flux "B" for a given driving force (i.e. magnetomotive force) "M", which is in turn proportional to the value of the current flowing thru the wire, quite similar to Ohms Law. Unfortunately, we cannot purchase "magnetic resistors" in any value which suits our fancy. Nor is there a "Magnetomotive Force Meter" equivalent to our handy voltmeter available from Digikey. If you are fortunate enough to have a "flux meter" you could measure the "B" value of the flux lines surrounding the wire. So, imagine how you would decipher Ohms Law from the simple battery-resistor circuit I described above, if all you had to work with was an ammeter and did not know the value of the resistor or the voltage of the battery. It would be quite a puzzling intellectual exercise! This is the greatest practical burden to overcome when learning magnetic circuits - we simply don't have the basic magnetic measurement tools like we have for electricity.

Ahhhh, but nobody can lay it out exactly like good old Karapetoff - whoever he was and where ever he now rests!

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    \$\begingroup\$ you introduced M but didnt clarify H \$\endgroup\$ Commented May 27, 2017 at 16:37
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    \$\begingroup\$ I have never seen the magnetomotive force being written by a capital em letter (\$M\$), but instead by a script capital ef letter (\$\mathcal{F}\$). The magnetization field is usually denoted by \$\mathbf M\$. \$\endgroup\$
    – alejnavab
    Commented Feb 12, 2021 at 19:19
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    \$\begingroup\$ "whoever he was": en.wikipedia.org/wiki/Vladimir_Karapetoff \$\endgroup\$
    – gwideman
    Commented Oct 3, 2021 at 15:28

\$ B = \mu_c\times H\$

B is the magnetic flux density and is unique to the material. Higher \$\mu_c\$ means more magnetic flux density under the same magnetic field.

H is the magnetic field strength and is an absolute quantity.


Short version: Both B and H come from either magnets or current.

One (H) is straight "ampere turns", (no : Andy is correct : ampere-turns per metre) the other (B) is H times the permeability of the magnetic circuit. For air or vacuum, this is 1 so B=H. For iron, B=permeability(large number) * H.

(EDIT to clarify : as Phil says, B is actually H * the permeability of free space : which is 1 in CGS units, and a constant ( \$\mu_0\$ ) in SI units. In either system it is multiplied by the "relative permeability" of magnetic materials like iron)

For a more complex scenario like a motor, involving iron pole pieces, iron bars in a rotor, and air gaps, each section has its own permeability, length and area, so while you know ampere-turns, figuring out the magnetic flux in each area (the air gap between poles and rotor for example) and thus the torque you can expect from the motor becomes a complex accounting process.

You might think increasing permeability to increase magnetic flux for the same current is a good thing - and you'd be right up to a point : the B-H relationship is non-linear (above a certain B, permeability decreases (crudely, when all the magnetic domains are already aligned) - this is known as saturation of a magnetic core - or of one component in the magnetic circuit of a transformer or motor. For example, if one component saturates before the others, increase its cross sectional area or change its material. In some materials, the B-H curve also has hysteresis, i.e. the material becomes magnetised and stores previous state : this is why it can act as computer storage or audio tape.

Designing magnetic circuits is as much an art as designing electrical circuits, and too often neglected.

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    \$\begingroup\$ I think B=H is true in a vacuum only if using CGS units (gauss, oerstead), and even then, B and H have different units. Confusing, since you are otherwise using SI units. \$\endgroup\$
    – Phil Frost
    Commented Dec 28, 2013 at 3:03
  • \$\begingroup\$ Yeah magneto motive force (MMF) is just ampere turns and totally equaivalent to volts (EMF) in electrostatics. H is equivalent to E (volts per metre) and B (mag) is equivalent to D (lectrics). Whay or why are caps so much easier to get yer head around. Happy new year (shortly) Brian \$\endgroup\$
    – Andy aka
    Commented Dec 31, 2013 at 23:34

As I see it, H is the magnetic field caused by the current in the coil. It assumes no ferromagnetic core is inserted. If inserting ferromagnetic core, the magnetic field gets stronger in the core and thus there was a need to describe that net magnetic field, denoting it by B. Since there was a need to distinguish between them, H was called field intensity and B was called flux density.


I think,H is a absolute quantity which does not vary with the material and remain constant for same deriving force(eg.current carrying wire or magnet).But the value of B depends upon the material .Value of B depends upon how much magnetic field of lines ,any material allows to pass through it.Hence mu_0 is a conversion factor which relates the total applied magnetic field H(which is absolute) to field lines any material allows through them(which varies from material to material).


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