# Magnetic field lines and magnetic flux density

I'm trying to understand the relationship between the "number" of field lines passing through a region and the magnetic force in this region.

I understand that the drawings are of course conceptual: we cannot draw "all" the field lines (although can be visualized with iron fillings).

Also the magnetic field B⃗

(measured in Teslas) and called "magnetic flux density". This is often considered as the "strength" of the field (the magnitude of the vector at a given point of the vector field).

I also read that one can consider that the strength of the magnetic field B

is proportional to the number of field lines in the region. With the vector field approach one would rather say that the vector at point (x,y,z)
in space have a given magnitude and that this magnitude is the "strenght" of the magnetic field.

How can we put the 2 together? Can we say that there really is physically a certain number of field lines passing by a region of space?

Dale
Mentor
2020 Award
How can we put the 2 together?
They are just two different ways of plotting the same data, so you could easily draw both together in the same plot. Basically you would have a standard vector field plot where some of the vectors are joined together to form streamlines.

Thanks, I understand that, this is not really my question though. My question is rather, can we consider there is physically "more lines" passing through a region? Can we say "informally" : if more field lines pass through an area of x square meters then the field is stronger?

Dale
Mentor
2020 Award
Certainly. It is simply another way to describe the same thing. Talking about field strength vectors is no more nor less physical than talking about flux lines.

On another forum someone says that "there is a field line passing through every point in space " ... what would that mean? It doesn't seem to make sense to me... we would need to have less field lines in some regions otherwise it would be all the same

Mister T
Gold Member
On another forum someone says that "there is a field line passing through every point in space " ... what would that mean? It doesn't seem to make sense to me... we would need to have less field lines in some regions otherwise it would be all the same
Disregarding the places where the field is zero, does it make sense to you that there is a field at every point? The direction of that field is tangent to the field line that passes through that point. As you said in an earlier post, we typically draw only a representative number of those lines.

No it does not make sense that there is a field line at every point, which is why I ask the question... However, I know space is continuous, formulas use integrals, etc. But the "conceptual" idea of field line at every point seems contradictory with the idea of saying that there are denser regions than others (i.e. more field lines in region A than region B) ... ?

hutchphd
Homework Helper
Does it help if we say that we can imagine field lines at every point in space and calculate using that assumption ?
Making life a little more complicated, there was a unit of magnetic flux in the obsolete CGS system called the "line" (later called the maxwell) equal to 1 G-cm2. So for a given N pole face with a 1 Gauss field, there would be 1 line emitted normal per sguare cm. To my knowledge nobody attaches a standard value to a line anymore but the density of lines is still very useful as a visualization of field strength and they are sometimes calibrated on a drawing so one can determine flux by counting lines.
Obviously the exact placement of each line is ambiguous but their relative density and their direction is not.

Dale
Mentor
2020 Award
But the "conceptual" idea of field line at every point seems contradictory with the idea of saying that there are denser regions than others
Only if you are thinking of a countable number of lines. If you have an uncountable number then both properties are compatible.

hutchphd
Homework Helper
Dale can you elaborate...I am not really understanding your point. Are you saying that the normalization is not fixed?

Dale
Mentor
2020 Award
Dale can you elaborate...I am not really understanding your point. Are you saying that the normalization is not fixed?
Since there are an uncountable number of points in space then if you have a field line through each you must have an uncountable number of field lines.

With an uncountable number of lines you can have one for each point even if they diverge. For instance, consider the family of lines through the origin in the plane parameterized by ##y=mx ##. One and only one line passes through each point in the line segment ##x=1## and ##0<y<1##, each line corresponding to one ##m## in ##0<m<1##. They diverge such that those same lines pass through the larger segment ##x=2## and ##0<y<2##.

So even though the lines are less dense at ##x=2## there is still one through each point. This is what @AlanTuring thought was contradictory, but it is one of those weird things that can happen with uncountable sets like the continuum.

Since we can’t draw an infinite number of lines, we just draw a few. That means that our drawing can show the “lines diverge” idea but not the “line through every point” idea. But that is just a limitation in our drawing, the mathematical continuum of lines is not so limited.

Last edited:
hutchphd
Homework Helper
While I know what you say is true in general, the B field has zero divergence and so the lines cannot behave in the way you describe (?) Further the notion of a field "line" in this context just seems to me not useful for discussion. Why worry about it?

Dale
Mentor
2020 Award
the B field has zero divergence and so the lines cannot behave in the way you describe (?)
That is not correct. I am not sure why you think it is.

Edit: maybe you are thinking that if the field lines diverge it means the field has divergence? But that is not the case because the field lines are just integral curves of the field, not the field itself. The magnitude of the field can change along the integral curve to keep 0 divergence.

Further the notion of a field "line" in this context just seems to me not useful for discussion. Why worry about it?
I often use field lines to reason about electromagnetism.

hutchphd
Homework Helper
I don't spend much time in the transfinite realm. I understand your argument and I see it is correct but I will need think about the context a little more. Thanks (it is past my witching hour for today!)

• Dale
Ibix
2020 Award
Perhaps another (less rigorous) way to look at it is to pick some small area and trace the field lines running through the edges of it. Since field lines can't cross, any field lines crossing the initially chosen area must remain in the volume you've just defined. But field lines don't stay parallel, so the cross sectional area isn't constant and the field line density must be changing.

"Number of field lines" probably isn't really helpful because it reinforces the notion that there are a finite number of field lines. There aren't. The things you see when you drop iron filings in a magnetic field are a kind of self-organising behaviour of the filings in a magnetic field. I commented on it in an Insight article I wrote.

• Klystron, hutchphd and Dale
hutchphd
Homework Helper
I often use field lines to reason about electromagnetism
But for me there is a subtle distinction between "field" lines and lines of flux. When I represent a field by line pictures , either in my head or on paper, I choose a particular finite (countable) subset of the field lines defined to represent a particular unit of flux per line. This choice can be made to pass a line of flux through any one point but not, in general, any other arbitrary point, and only one choice is allowed per diagram.

"Number of field lines" probably isn't really helpful because it reinforces the notion that there are a finite number of field lines.
I guess I disagree as to their utility of flux lines . A diagram without the density feature would not be very useful as it would not provide any clues as to field strength without further annotation. I have designed some very useful magnets using flux lines and a piece of paper. Perhaps those days are long past.
I certainly do agree that the notion can be confusing and likely was the crux of this discussion! But I would just be careful to use the terms interchangeably: as is so often the case I believe the issue is mostly semantic.

Dale
Mentor
2020 Award
But for me there is a subtle distinction between "field" lines and lines of flux.
Indeed, the distinction is subtle. It is too subtle a distinction for me to spot.

• • weirdoguy and anorlunda
hutchphd
Homework Helper
I would point out that they carry different units.

Dale
Mentor
2020 Award
I didn’t know that the lines carried units at all. Are you just saying that the magnetic field and magnetic flux have different units?

• Ibix
Ibix
2020 Award
I guess I disagree as to their utility of flux lines . A diagram without the density feature would not be very useful as it would not provide any clues as to field strength without further annotation. I have designed some very useful magnets using flux lines and a piece of paper.
You seem to be describing something slightly different from what I was considering. In the diagrams in my Insight, I merely drew a finite set of integral curves of the magnetic field, with no "strength" association. If I'm understanding you correctly, you pick a finite set of integral curves of the field and call these "flux lines". Then you assign the total magnetic flux through some area around each chosen curve (edit: presumably so that the total across your lines captures all flux) to that line - right?

Certainly with what I drew, "number of field lines" is a somewhat incomplete description. With what you seem to be describing, you don't want "number of flux lines" either - you need to weight by your strength value, I think. Unless you choose your curves so that the strength is equal for all, I suppose.

Assuming I've understood correctly then, like @Dale, I must say I've never seen this before.

• hutchphd and Dale
Mister T
Gold Member
As far as I know, there is no such thing as a line of flux. Certainly you determine the relative amount of flux by looking at the relative number of field lines passing through different surfaces.

A property of field lines is that the field is tangent to the line at any given point. But flux is a scalar quantity.

hutchphd
Homework Helper
I didn’t know that the lines carried units at all. Are you just saying that the magnetic field and magnetic flux have different units?
Yes the (derived) CGS unit of magnetic flux is the maxwell.

1 maxwell ≡ 1 gauss cm2t​

The archaic CGS unit for flux was the line which is equal to the maxwell. Obviously the flux density and field carry the same unit.
As I mentioned these distinctions are mostly semantic but the drawing of "fields lines" willy nilly without any further prescription about their density renders the diagram unrecognizable. I have always called this construction "lines of flux" to emphasize the (subtle) difference. I assumed it was common usage but perhaps not. (It should be!?)
Incidently when drawing lines of flux each field/flux line always attaches to a the same fixed amount of "charge" (electric or magnetic) at the sources. So you can draw the field pictures largely sans calculation.

@Ibix I apologize for not yet having read your insight article but look forward to doing so. I believe you understood what I was trying to say.

vanhees71
Gold Member
I think the two ways to visualize the magnetic field are both valid referring simply to different points of view, both of which are physically useful in different contexts.

The ##\vec{B}##-field (in modern terminology called the magnetic field) is operationally defined by its action on point charges. The force on a charge due to the magnetic field (in Heaviside-Lorentz or Gaussian units, because the SI is a nuissance when it comes to the physical understanding of electromagnetism though it's much more convenient for practical purposes in engineering) is
$$\vec{F}_{\text{mag}}=\frac{q}{c} \vec{v} \times \vec{B}(t,vec{x}).$$
Here the intensity and direction of ##\vec{B}## at the position ##\vec{x}## is important, giving the strength and direction of the force on the charge. I.e., here a picture with field vectors at each point in space (at a given time) is most intuitive.

Another important aspect is Faraday's Law of induction,
$$\vec{\nabla} \times \vec{E}=-\frac{1}{c} \partial_t \vec{B}.$$
Putting it in integral form (for simplicity for a surface ##A## and its boundary ##\partial A## assumed to be at rest in the computational reference frame) it reads, using Stokes's integral theorem
$$\int_{\partial A} \mathrm{d} \vec{x} \cdot \vec{E}=-\frac{1}{c} \mathrm{d}_t \int_A \mathrm{d}^2 \vec{f} \cdot \vec{B}.$$
Here at the right-hand side the flux of the magnetic field through the surface ##A## is of importance. Here it's more intuitive to imagine ##\vec{B}## as a flow field "streaming" through the surface. Here the corresponding "field-line picture" describes the situation best.

• • Klystron and hutchphd
Mr Turing I think to get an understanding of this you could do the following. Go back to the beginning if necessary:

1. Get a bit of a bit of an understanding of the concept of field lines, which you already have.
2. Get familiar with the magnitude and directions of the forces on current carrying conductors and moving charged particles in B fields.
3. Look (again) at the equations that quantify the forces and see that they make sense. In fact when you think about it they're predictable.
4. Ideally look at the experiments that can be done to verify the equations.
5. Get an understanding of the definition of the unit you use to measure magnetic flux density eg the Tesla (T)
6. Get an understanding of how magnetic flux is related mathematically to magnetic flux density. Get familiar with the unit of magnetic flux density eg the Weber (Wb).

In your original post the points you made were very valid so if we say, for example, that magnetic flux is proportional to concentration of field lines and magnetic flux is proportional to number of field lines the descriptions can be considered as being vague to the point of being meaningless. But, the concept of field lines can be useful, for example it can give an intuitive feeling what we mean by concepts such as "the strength of a magnetic field at a particular point". So use the concept of field lines if it helps. But remember that the concept is very flawed. Of course you know that already.

Dale
Mentor
2020 Award
So use the concept of field lines if it helps. But remember that the concept is very flawed
Why is it flawed? It is just another way of thinking about the field.

• vanhees71 and hutchphd