Why is electric field strength indicated by the density of field lines?

Question

It is well known that the magnitude of the electric field is indicated by the density of field lines. However, is it a physical law or additional rule that helps us to draw informative diagrams?

In other words, in a problem of electrostatic equilibrium, if we somehow deduce that $A$ gets more field lines than $B$, does it mean the absolute value of the charge of $A$ is greater than $B$? (By Gauss's law, the electric flux is proportional to the net charge enclosed by the Gausiian surface. By the relationship between field line and field strength, the number of field lines pass through a surface is proportional to the electric flux. Hence, the number of field lines is proportional to the charge enclosed. However, is there a physical law that says the number of field lines passing through a surface is proportional to the electric flux? As I understand it, it's just an additional rule for drawing graphs. This is why I find where I find this argument dodgy.)

If you want to see a configuration that enable us to deduce who gets more field lines and want to see how, here is a specific example. This is only to provide context. It is not necessary to read the rest if you can already answer the question.

Problem: At electrostatic equilibrium, a positive point charge is placed near a neutral conductor. The absolute value of the induced negative charge is less or equal to that of the point charge.

The following is a solution I read.

First of all, the induced positive charge need to emit field lines. It cannot go to the induced negative charge as that would violate the fact that surface of the conductor is equipotential. Therefore, it goes to infinity.

Secondly, consider the induced negative charge. It has three possible places of departure:

a) other parts of the conductor ( it has been ruled out in the last paragraph)

b) infinity (the last paragraph indicates the potential at infinity is more negative than that of the conductor, so this option is impossible as well)

c) point charge (this is the one)

The field lines from the point charge can go to the induced negative charge or infinity. Whether it actually goes to infinity or not depends on the configuration of our model.

Therefore, the number of field lines coming from the point charge is greater or equal to that coming into the induced negative charge. The absolute value of the postive point charge is greater or equal to that of induced negative charge.

The context is finished.

In addition, can we convert it into a problem on differential geometry: Consider two closed surface in a smooth vector field, the one has greater 'flux' has the greater number of integral curves?

Answer 1

Field lines are only for illustration. They are not physical.

Answer 2

My question is in the last paragraph of this solution. Is there a physical law that says the number of field lines passing through a surface is proportional to the electric flux?

However, the question already contains the answer to itself:

In other words, in a problem of electrostatic equilibrium, if we can deduce that A gets more field lines than B, does it mean the absolute value of the charge of A is greater than B? (By Gauss's law, the electric flux is proportional to the net charge enclosed by the Gausiian surface. By the relationship between field line and field strength, the number of field lines pass through a surface is proportional to the electric flux)

Indeed, electric field lines are just a graphical representation of the Gauss law. What might lead a bit astray is that the electric field lines are usually drawn in two dimensions, whereas Gauss law is valid in three dimensions. Thus, any two-dimensional representation of the electric field lines of a point charge is necessarily an approximation. However, in problems where the charge distributions (and hence also the fields) do not vary in z direction, $\partial_z\rho=0, \partial_zE_x=\partial_zE_y=\partial_zE_z=0$ (e.g., problems dealing with homogeneously charges cylinders or sheets), the representation is exact graphical representation. That is, one can calculate electric flux as the product of the number of field lines times the length of the closed curve (corresponding to Gaussian surface in 2D.)

In this case, how frequently one decides to draw the lines (density of the lines) fixes the effective units of the electric field. The discreteness of lines does not allow showing continuously changing field magnitude, but choosing the units (i.e., the density per unit flux) allows to achieve any required precision.

Wikipedia article on Field lines covers a more general aspect of the field lines, not specifically for electrostatics:

An individual field line shows the direction of the vector field but not the magnitude. In order to also depict the magnitude of the field, field line diagrams are often drawn so that each line represents the same quantity of flux. Then the density of field lines (number of field lines per unit perpendicular area) at any location is proportional to the magnitude of the vector field at that point. Areas in which neighboring field lines are converging (getting closer together) indicates that the field is getting stronger in that direction.

Related:
Why does the density of electric field lines make sense, if there is a field line through every point?
Electric field line density : Theory vs Reality