Confusion regarding field density

Question

In my query, I'll be discussing a test charge and I'd appreciate any responses

My doubts stem from this statement and might involve misinterpretation of it

"Since the magnitude of B is 10T, there is a uniform density of 10 field lines per m2 so that the flux through a 1m2 square that is perpendicular to the field is 10Wb if we take one field line as being 1Wb.

If we place a 10m2 surface perpendicular, there are 100 field lines in the surface, so 100Wb.

We could also say each field line is 10Wb and say there is 1 field line per m2 or say each field line is 100Wb and there is 1 field line per 10m2. It doesn't matter as long as the magnetic flux density (B) is the same"

Q1)When it's stated that there's magnetic flux density (B)/Field Intensity somewhere on an Area is 10T would saying intensity vectors of 6T and 4T pass through it be accurate? but the field intensity everywhere is on the area as a whole is said to be 10T. Is this an incorrect interpretation? i asked people about it but according to them my interpretation is wrong which is what brought me here

Q2) It's often said Density is proportional to Strength but I don't get the essence of it, There's an intensity vector everywhere at every point so why is the density around? discussing an area near a charge and farther from the charge would essentially still have a Field passing. then why are distant and crowded lines observed in fields? how does the intensity vary in the field throughout an area? simply saying density is proportional to intensity doesn't make sense to me.

simply put prove that strength on a area away and near a point is different or not , would we get a constant value considering how there's a diff value on each point on the area?

i understand that this might just sound like a stupid question but i urge you to give a response.

Answer 1

Magnetic field strength / Magnetic flux density:

The magnetic field $\vec{B}$ is a field that relates the force experienced by a charge by the relation

$$\vec{F} = q (\vec{V} × \vec{B})$$

The magnetic field intensity, is often referred to as the magnetic flux density.

The magnetic field is not a series of lines, there are no such thing as "n lines of magnetic flux density passing through an area". The magnetic field is a vector field, a vector field is a mathematical object that assigns a vector to every point in space, in any finite area there are infinitely many vectors, one at (0,0,0) one at (0,0,0.00000000....1) and so on.

Magnetic flux:

Magnetic flux is defined using a flux integral

$$\phi_{B} = \iint \vec{B} \cdot \vec{da}$$

In its simplest form, the flux integral can be thought of as splitting a specific area into infinitely many rectangular area pieces, and multiplying this area by the magnetic field strength at this location. Once you have done this for every infinitely small area peice in an area, you add them up and that is the magnetic flux across the area, measured in webers (Wb)

Definition of a field line

Although there is no such thing as a field line, you could in principle define a field line as a specific portion of the flux integral. In your first example, they have said that a field of 10T is incident a surface of area 1m^2.

In reality to calculate the flux we would say that the flux:

$$\phi_{B} = \iint 10 da = 10\iint da = 10 $$

Turning this into a finite sum instead for those that haven't learned the flux integral yet. We split the area into n chunks, each chunk having area (A/n)

The integral becomes $$\sum\limits_{i=0}^{n} B (A/n)$$

It is these elements B (A/n) that we define as one field line. If we say that are e.g 10 field lines over an area of 1m^2, with a B field strength of 10T we get that each field line has a value of 1wb. Field lines have units wb and not a measure of the magnetic field strength/flux density

B is a constant 10T in this case, the only degree of freedom we get is changing n the member of field lines we want to define. Changing n does not allow for a 6wb and 4 wb field line ( notice k said 6 wb not 6t, as before field lines are a measure of flux not field intensity)

If the field was not uniform, there could be a way of defining two field lines with the values of 6wb and 4wb. But not in the case you have mentioned