Field Theory: Electric Field I am a grade 12 student, and we are learning electric flux. We covered electrostatic forces and, now we are on electric flux. We defined electric flux as the total number of lines of force passing through a surface. 
My question focus on the bold line. Does that mean we can count field lines? If we do, wouldn't it make it discrete which is a collection of lines with some kind of thickness?  So do electric field lines have thickness? What are field lines? or What is a field? 
 A: 
What are field lines?

A field line just shows the direction a test-charge (usually considered positive) would take during movement under the influence of all the other charges considered in the system. That charge is usually taken to be very small so that it's own field doesn't have any effect on the system.

What is a field?

A field is a region which has a value assigned to it at each point in space. In case of charges, it's called the electric field because when a charge is introduced, the points in space have different values assigned to them for a quantity like electric field strength or potential.
After all, we may draw as many field lines as we want for a particular system but then the number of field lines we draw for other systems are determined and must be proportional to the strengths of the electric field at those points. Electric field lines just show the direction of movement of a small test charge while their density gives us a clue about the field strength at a point.
A: The electric field lines were invented by Faraday to visualize the direction and strength of electric field in a given space. While the electric field has a direction and strength at any point in space, the an electric field line can be arbitrarily chosen to pass through any given point of the field. The number of field lines per unit area they are crossing normally is a measure for the strength of the field. You can chose an arbitrary, convenient number of field lines for this visualization 
A: 
Lets just analyze a point charge to simplify this
Firstly  we need to understand how is electrical energy is transferred, so you're gonna have to stay with me for a second.
Electrical energy is transferred through photon--waves, so it is quantized. We can imagine one of these quanta to be the crest of a three--dimensional wave. Think of bubble expanding in size over time centered on our point charge (field lines are basically just the direction of propagation of this wave. This is our electrical energy, the amount of energy contained in this bubble is constant. This energy (E*) is a function of charge (Q) and the "vacuum permittivity constant "e":
    $$ E* = Q\cdot e_0$$
 This Electrical energy is distributed equally along the wavefront (the surface of the bubble), therefore the energy at a point(E) is dependent on the total energy (E*) and the surface area of the bubble (A =4*pi*r^2):
     $$E = Qe_0/(4\pi e^2)$$
 The constants in this equation are condensed into one constant K, and it is neatly rewritten as Coulomb's familiar law.
    $$ E = KQ/r^2.$$
Now That we (hopefully) understand this law let's discuss electric field lines.
Electric field lines are basically just arrows pointing in the direction of wave propagation, as drawn above. They are used to represent the direction of the energy (and subsequent force on a charged particle) that is emanating (in the form of photon wave bubbles) from the point charge. These lines are describing where a wave is going, thickness in these lines, then, would be describing how much it is going there you could  describe the magnitude of the electric field and a point with thickness of the lines at that point if you so desired, but this is not standard. When multiple charges are involved, their "energy waves" can interfere constructively and destructively, producing field lines that are no longer radial and slightly more difficult for me to draw. electric field lines are only a visualization technique, not a physical reality in nature. The amount of field lines is really only dependent on how detailed a human "artist" wanted to get with their visualization of an electric field. My drawing has 8 field lines; this, however, means only that I was a being lazy but it can tell you nothing about the system which I have described. If we were to try to visualize all POSSIBLE field lines in a system, we would end up trying to trace the velocities of all photons present in a wave front; this is not possible, and even if it were, I would advise against trying it.
