What is the number of electric lines of force originating from a charge of 1 Coulomb

Question

This is a question from SL Arora, so don't bash me. My initial thoughts were "Number of Field lines? well it should be infinity, You can draw how much ever field lines you want" However the author went on to say

Number of lines of force originating from a charge of 1 Coulomb = Electric flux through a closed surface enclosing the charge.

And so the author used the concept of flux to the number of field lines originating. The concept of flux is indeed related to the number of field lines passing through the surface.

But Flux is defined as "the total electric or magnetic field passing through a surface." It does NOT however tell us the number of field lines originating from the proton. At the end of the day, Flux is directly proportional to the number of field lines crossing through a surface, not equal to it. I think this is where Arora got it wrong

Interestingly, I remember stumbling upon a question from HC Verma telling

In some old texts, it is mentioned that 4$\pi$ lines of force originate from each unit positive charge. Comment on the statement in the view of the fact that 4$\pi$ is not an integar

However unlike SLA, HC Verma didn't bother giving the answer, so I can't tell what the answer is in his opinion

So what exactly is the consensus on the number of Line of force emerging from a unit positive charge

Edit1: I pulled out all dodgy citations and decided to simply compare what is given by the two books with regards to the line of force originating from a positive charge. Also, I will now use the terms "Line of force" and "Electric fieldlines interchangeably" unless someone objects

Edit2: I also went into whether flux can actually give you the number of fieldlines and came to conclusion, (1) NO IT CANT (2) most importantly the Question could be incorrect then

Answer 1

I really can't make any sense of the references you have given, especially not seeing them in context. However, it is important to realize that electric field lines are only a pictorial tool used to visualize and analyze electric fields qualitatively (not quantitatively) as opposed to a physical entity.

The primary use of this pictorial tool is to give you (1) information on the direction of the electric field at a location as indicated by the arrows on the lines and (2) qualitative information on the relative strength (and relative electric flux) of the electric field based on the density of the field lines in one area compared to another.

Aside from the above, the lines themselves have no physical meaning.

I think i need to ask, is the question the "number of field lines" relevant?

The actual number of field lines is irrelevant. Only the relative number of lines at one location vs another. That may provide information on the relative electric strength or the relative flux at the two locations.

just to re-clarify, we can never find the "number of electric lines of force originating from a charge of 1 Coulomb" right?" Right.

Right. The electric field from say a 1 Coulomb point charge exists everywhere, not just on the lines. Think of the field about the charge as a sphere that is darkest gray nearest the charge where the field is strongest and gradually gets lighter and lighter as you move radially outward as the field gets weaker and weaker. That would be a better representation of where the field exists and its relative strength than lines. But the lines are a better and easier representation of the direction of the field and its relative strength for more complex charge distributions.

Hope this helps.

Answer 2

No, the number of field lines cannot be determined by the electric flux as it cannot be quantified and is also physically meaningless. The number only has a meaning when measured relatively. A simple counterexample to this statement:

Number of lines of force originating from a charge of 1 Coulomb = Electric flux through a closed surface enclosing the charge.

Is by considering the definition of electric flux by Gauss's law for a closed surface: $$\phi_E = \frac{q_{in}}{\epsilon}$$ This shows that the electric flux has units of $\frac{N.m^2}{C}$, while a number is essentially dimensionless, so an equality between the two is very clearly incorrect. The statement is therefore both incorrect and misleading. What is correct is that the electric flux for an open surface is proportional to the number of field lines that pass through a surface, and for a closed surface is proportional to the net number of lines leaving the closed surface.

In some old texts, it is mentioned that 4π lines of force originate from each unit positive charge. Comment on the statement in the view of the fact that 4π is not an integar

These "old texts" are incorrect in both the number of field lines, and the fact that they could even quantify electric field lines. The only possible explanation I can think of, viewing it in reference to the fact that 4π is irrational, is that the author assumed that the number of field lines originating from a charge is a whole number and that 12.56637 field lines emerging from a charge is insensible.