# Magnetic Flux - Misleading Definition or what?

• I
• ktmsud
In summary: I think the book does this well.In summary, the book does not provide a rigorous definition of the number of flux lines. The book does provide a semiquantitative way to depict how fields look, and assigns some physical reality to flux lines.
ktmsud
TL;DR Summary
Magnetic flux and magnetic flux density are defined in terms of number of magnetic field lines. is it appropriate?
Magnetic flux density(B) is defined as, magnetic force per unit pole strength and flux is defined as magnetic field passing normally through given surface. I think, I am clear about these definitions, and quantitative meaning they carry.
I usually find, in some texts, the definitions as in image below

As far as I know one can draw infinite number of magnetic field lines ( as you can draw one magnetic field line between two and repeat this process infinitely). Is there something that I am missing? If one can draw infinite number of lines then how can this definition give correct values for flux and flux density?

vanhees71 and Dale
I agree with you. I would use the flux density to define magnetic flux lines, but not vice versa. I would be skeptical about this book.

Do they provide a rigorous definition of the number of flux lines?

Peter034 and vanhees71
I agree with both you and Dale. Field lines are just imaginary tools to help visualize the magnetic field. I suppose if you set a given number of lines per unit area to represent the strength of the field you can then define total flux as the number of lines through a surface. (Though I wouldn't advise that.)

vanhees71
Me three. If one draws a finite number of lines of flux one can always construct an area pierced by none of them, but the flux there is not zero. As Doc Al says, making assumptions about a reasonable distribution of the lines of flux and a reasonably large and non-pathologically shaped area then it's a decent heuristic way to estimate flux density from a diagram of field lines, but no more than that.

I suspect that there's some Chinese Whispers going in here and if you trace explanations back in time you'll find one that sets out that this is a rough estimate, and/or workable for explaining to high schoolers why magnetism is weaker far from a magnet. But all the caveats got dropped aling the way.

vanhees71
Doc Al said:
I agree with both you and Dale.
Ibix said:
Me three.
Technically that would be "Me four." Just sayin'

Ibix
Dale said:
Do they provide a rigorous definition of the number of flux lines?
No they don't or maybe, I didn't understand well. "Magnetic field lines are defined as hypothetical closed curves through which a free unit magnetic northpole would move in the magnetic field."
I don't think this definition gives any idea about how many lines can be drawn in a particular region. Does it?

No, it doesn’t. I would be cautious in using that book.

vanhees71
I disagree very strongly . (Me one).
The lines of magnetic force were instituted as a semiquantitative way to sketch the field. Given the strength and shape of all pole faces, the knowledge that the lines "dislike each other" and emerge normal to the faces very good good sketch of the field can be made. Lines of magnetic force was a standard unit$$10^8~ lines~ of~ force=1Wb$$ so $$100 lines/{mm}^2=1T$$ if I did the arithmetic correctly.
This was when real men used sabres and physicists used pencils and french curves.

davenn
hutchphd said:
The lines of magnetic force were instituted as a semiquantitative way to sketch the field.
So that's the missing bit - a standard way to draw the lines and areas. I think there are still problems with large ratios of field density, because you can end up with small areas pierced by no lines, but I guess that's something you have to accept.

I still think the explanation is backwards - this may be a valid calculation technique, but explaining field density in terms of integral curves feels a bit like putting the cart before the horse.
hutchphd said:
This was when real men used sabres and physicists used pencils and french curves.
I've actually used a sabre, but never a French curve.

Dale, vanhees71 and hutchphd
Field lines are just a tool to qualitatively depict how "fields look". One should not try to interpret more into them than that. Quantitatively fields have to be defined properly in terms of mathematics (tensor calculus) in theory and physically operationally in terms of how they are measurable.

Gavran and berkeman
hutchphd said:
I disagree very strongly
What do you disagree with? Do you specifically think that the definitions in this book are good? Or are you just generally supportive of flux lines?

Personally, I have no issue with flux lines in general, but I think this specific book presents them very badly.

vanhees71
My point was that historically they have been assigned a (semi)quantitative reality for consideration and calculation. I think Michael Faraday ascribed to them some physical reality. In fact at some point a standard "line" was quantified relative to source strength.
I guess I don't see what the book did so badly in this context.

I must admit I never understood field lines quantitatively. How do you define "number of fieldlines going through a surface", while fields are defined on a continuum? The point of introducing fields after all is a description of physics in terms of a continuum theory.

Mathematically it's well defined as the surface integral of a vector field over a given surface that defines the flux of this field through this surfarce.

weirdoguy
You clearly have to define a certain amount of "charge" (magnetic or electric) to give rise to one field line. As I mentioned above the standard choice for a "line" of magnetic force was stipulated at one point. Of course Pemanent Magnets were without Niobium and Sumarium in those days so that definition probably would produce rather difficult drawings today

This does not help to understand, what you mean by giving "rise to one field line". The charge and current distributions are the sources of the electromagnetic field (with electric and magnetic components) but not of "field lines".

You have clear meanings with the math describing the corresponding Maxwell equations. To make it simple, consider statics only. Then you have
$$\vec{\nabla} \times \vec{E}=0, \quad \vec{\nabla} \cdot \vec{E}=\rho, \quad \vec{\nabla} \times \vec{B}=\frac{1}{c} \vec{j}, \quad \vec{\nabla} \cdot \vec{B}=0.$$
using Heaviside-Lorentz units, which are usually cleaner than SI units if it comes to fundamental issues.

These equations you can of course bring into "integral form" by applying Gauß's and Stokes's theorem:
$$\int_{C} \mathrm{d} \vec{r} \cdot \vec{E}=0, \quad \int_{\partial V} \mathrm{d}^2 \vec{f} \cdot \vec{E}=Q_V, \quad \int_{\partial A} \mathrm{d} \vec{r} \cdot \vec{B}=\frac{1}{c} I_{A},$$
where ##C## is an arbitrary closed curve, ##V## an arbitrary volume with its boundary surface ##\partial V## and ##A## an arbitrary surface with its boundary ##\partial A##.

How do you define from this continuum notions of fields (densities of) field lines in a quantitative way? For me getting rid of all these hand-waving arguments with "field lines" and using the clear definitions in terms of fields was a revelation!

ktmsud said:
If one can draw infinite number of lines then how can this definition give correct values for flux and flux density?
You're right, it can't. But what it can do is tell you the relative values for the flux and density.

ktmsud said:
TL;DR Summary: Magnetic flux and magnetic flux density are defined in terms of number of magnetic field lines. is it appropriate?

Magnetic flux density(B) is defined as, magnetic force per unit pole strength and flux is defined as magnetic field passing normally through given surface. I think, I am clear about these definitions, and quantitative meaning they carry.
I usually find, in some texts, the definitions as in image below
View attachment 327903

As far as I know one can draw infinite number of magnetic field lines ( as you can draw one magnetic field line between two and repeat this process infinitely). Is there something that I am missing? If one can draw infinite number of lines then how can this definition give correct values for flux and flux density?
The Serway definition seems a lot easier and over there you may know the magnetic flux density as the magnetic force applied to a charged particle moving normal to the flux. Then F=qv ⋅ B. So far that is Serway formula 26.1, Raymond Serway, Physics for Scientists and Engineers, 1983, page 538. This seems to translate through my series of equivalents until B = F/qv.

I am old enough to have been taught my physics using the cgs system of units, where 4 pi lines of force originate from a unit magnetic pole. Seemed quite a good system at the time.

Lines of force for me were always a very dubious idea. To be honest, I never understood it, while the ideas of flow/flux (locally the divergence of a vector field) and vortices (locally the curl of a vector field) have a pretty intuitive meaning and also an accurate mathematical meaning.

The electromagnetic field is operationally defined by the action on (point charges) in terms of the Lorentz force,
$$\vec{F}=q (\vec{E}+\vec{v} \times \vec{B}),$$
here written in SI units. Of course, the choice of the units has no physical significance at all in all this business.

vanhees71 said:
Lines of force for me were always a very dubious idea. To be honest, I never understood it, while the ideas of flow/flux (locally the divergence of a vector field) and vortices (locally the curl of a vector field) have a pretty intuitive meaning and also an accurate mathematical meaning.
In his "Lectures on Theoretical Physics" Sommerfeld introduces the dielectric displacement ## D ## in exactly this "dubious" way, as the density of electric lines of force. These lines do not begin or end at the boundaries of dielectric media, so there is some geometrical prescription of how to draw them. Surely Sommerfeld didn't think that these lines cannot be given an "accurate mathematical meaning", it's just a different description of the same underlying physics.

hutchphd said:
You clearly have to define a certain amount of "charge" (magnetic or electric) to give rise to one field line. As I mentioned above the standard choice for a "line" of magnetic force was stipulated at one point.
Exactly. It gave rise to the mixture of electrostatic and electromagnetic units of charge that characterizes the Gaussian system of units. Much of present day metrology depends on the Josephson effect and the magnetic flux unit ## h/2e = 2.067 833 848 × 10^{-15} ~\rm Vs ##. So there is a way to count field lines.

Of course Sommerfeld draws fields in terms of field lines. Otherwise he gives precise mathematical definitions, and the flux of field through a surface is the area integral with an arbitrarily chosen direction of the surface-normal vectors. If it's a closed boundary of a volume by convention the normal vectors point out of the volume under consideration. The flux of a vector field ##\vec{V}## is defined as a surface integral, ##\int_A \mathrm{d}^2 \vec{f} \cdot \vec{V}##.

Nobody is arguing about these definitions. But do you really see a mathematical conflict with the description in terms of field lines? In an incompressible liquid the velocity would also have to be proportional to the density of flow lines. Or is it that you feel that there can be only one correct way of describing things?

vanhees71
It's a qualitative picture, but not more!

And no less.

What more do you need to define a vector but magnitude and direction?

I referred to the claim that the density of field lines had something to do with field strength. Fields are a continuum concept and you cannot simply "count field lines". What you can calculate is the flux through a surface.

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