# What is the difference between B and H?

• B
• Heisenberg7
Heisenberg7
A week ago, I started studying electromagnetism. I was introduced to a few new concepts and one of them was H. Now, in my book, they defined H as just magnetic field strength and B as magnetic induction. The thing is, I don't understand what those terms really are (in a physical way), let alone the difference between them. I know how they are connected and I've read at least a dozen definitions and watched a few videos but they always somehow manage to go past the explanation.

Welcome to PF.

You will get other better answers, but for me the easiest way to think about it is in terms of the magnetic permeability ##\mu##

https://en.wikipedia.org/wiki/Permeability_(electromagnetism)

ohwilleke and docnet
The magnetic field strength ## \vec H ## (the amount of magnetizing force) is a fundamental magnetic field. Electric current ## I ## generates the magnetic field strength ## \vec H ## and the magnetic flux density ## \vec B ## (the amount of magnetic force) is a response of the medium magnetized by ## \vec H ##.
The magnetic flux density is induced due to the magnetic field strength.

Heisenberg7 said:
Now, in my book, they defined H as just magnetic field strength and B as magnetic induction.
What are the definitions they gave? Can you quote or tell us which book and page?

The same question may be asked about the ##D## and ##E## which are related by the permittivity, ##\epsilon##, by ##D=\epsilon E##. The E field is volts per meter, the local rate of change of potential while ##D## is the charge per square meter. The permittivity, like ##\mu##, depends on the material and are often considered constant for many situations. Maxwells equations take a simple form in terms of E, B, D and H and make dealing with materials simpler.

ohwilleke and DaveE
martinbn said:
What are the definitions they gave? Can you quote or tell us which book and page?
It's actually a local book. I'm from Bosnia and Herzegovina so it wouldn't be of much help if I told you which book it is. For that reason, I'm just going to quote the book: B : "The quantitative measure with which we describe a magnetic field is the magnetic induction B."; H : "Strength of a magnetic field, unlike the magnetic induction, does not depend on the characteristics of the medium in which we have the magnetic field. Magnetic induction B and strength of the magnetic field are connected in this equation: ##B = \mu H##". Now this last definition is quite weird to me because it later on goes to say that for some materials ##\mu = \mu_r\mu_o## which means that it does depend on the medium in which we have the magnetic field (we use ##\mu_r## when we are not talking about vacuum so it doesn't make sense for us to say that it does not depend on the medium when it clearly does by this definition).

ohwilleke
Heisenberg7 said:
for some materials μ=μrμo
I would prefer to say that ## \mu = \mu_r \mu_o ## is always true. ## \mu_o ## is the permeability of free (empty) space and is always present. Some materials add their own factor, relative permeability ## \mu_r ##, which is dimensionless, like gain. This is really just a convenience in design so we don't always have to write out ##4 \pi \cdot 10^{-7} \frac{H}{m}##, which is ever present. Of course the "material" of free space has ## \mu_r = 1 ##.

ohwilleke
DaveE said:
I would prefer to say that ## \mu = \mu_r \mu_o ## is always true. ## \mu_o ## is the permeability of free (empty) space and is always present. Some materials add their own factor, relative permeability ## \mu_r ##, which is dimensionless, like gain. This is really just a convenience in design so we don't always have to write out ##4 \pi \cdot 10^{-7} \frac{H}{m}##, which is ever present. Of course the "material" of free space has ## \mu_r = 1 ##.
I agree. I did kinda word it in a wrong way.

ohwilleke and DaveE
The flip answer is "the difference between B and H is 4πM".

The seed of truth in that is that the thing that appears in the Lorentz force, B, comes from the magnetic field H and in the presence of materials, a magnetization (4πM).

dextercioby
Heisenberg7 said:
[...] to quote the book: B : "The quantitative measure with which we describe a magnetic field is the magnetic induction B."; H : "Strength of a magnetic field, unlike the magnetic induction, does not depend on the characteristics of the medium in which we have the magnetic field. Magnetic induction B and strength of the magnetic field are connected in this equation: ##B = \mu H##".
Now this last definition is quite weird to me because it later on goes to say that for some materials ##\mu = \mu_r\mu_o## which means that it does depend on the medium in which we have the magnetic field (we use ##\mu_r## when we are not talking about vacuum so it doesn't make sense for us to say that it does not depend on the medium when it clearly does by this definition).
You are right, the definitions are contradictory. In general, B and H are not simply proportional to each other. Increasing the current in a solenoid will not proportionally increase the magnetic induction (B) in the iron core after it has reached saturation (when all elementary magnets are aligned).

In homogeneous media there can be an approximate proportionality, similar to the relation between stress and strain in a solid, or between pressure and density in a gas. But they are independent, "conjugate" quantities. The product has the dimension of energy density. In the case of a weakly magnetized medium you can express the energy density as ## \frac 12 B \cdot H ##. But for an iron core in a solenoid the relation between B and H is much more complicated. What is traditionally called magnetic field (H) is the field due to the external current only, not including the microscopic currents due to the magnetization M of the medium. As @Vanadium 50 has indicated, the magnetic induction B, i.e. what really exerts forces on the electrons, can be written as $${1 \over \mu_0} B = H + M \ ,$$ (using SI units), and for a static magnetic field you have therefore $$\nabla \times H = j \qquad \text{or} \qquad {1 \over \mu_0} \nabla \times B = j + \nabla \times M \ .$$ And remember that the magnetization M is not always strictly proportional to what is causing it. (B or H?)

weirdoguy

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