# Gauss theorem and the nature of the surface

According to Gauss’ law the total number of lines of force over a closed surface is equal to 1/ε times the net charge enclosed within the closed surface. Why should it be a closed surface but not an open surface too? I am unable to find a convincing explanation for it. Since we take into account only the number of lines starting or reaching a charge, the total number of lines is not going to vary whether we take a closed surface or open surface near the charge. Then why should it be specifically a closed one? Can anyone giving a convincing explanation?

mfb
Mentor
If the surface is open, you can get any arbitrary value. It is important that you enclose the whole mass - without leaks, so to speak.
There is no "number of lines of force".

If the surface is open, you can get any arbitrary value. It is important that you enclose the whole mass - without leaks, so to speak.
There is no "number of lines of force".
Thank you for the reply. If there is no lines of force, then how the calculated value of 1.129 x 1011 lines of force from 1 C of charge placed in air or vacuum arrived? What does that number exactly stand for? The value has been obtained from the formula 1/ε for air.

I am by means an expert but, what Gaus law does is to collect all that flows through a surface. If you dont totally enclose the charge there will be flow going out that you are not calculating.

Because only a closed surface can enclose a charge.

How do you define "enclosed charge" when your surface is open?

mfb
Mentor
If there is no lines of force, then how the calculated value of 1.129 x 1011 lines of force from 1 C of charge placed in air or vacuum arrived?
I have no idea who did that, but it is wrong.
What does that number exactly stand for?
Nothing.
$$\frac{1C}{\epsilon_0}=1.13\times 10^{11} Vm$$
The numerical value depends on the units you use - if you convert this to imperial units, you get a different number, for example. This alone shows that the numerical value itself cannot have a physical meaning (like some number of "lines").

I have no idea who did that, but it is wrong.

This value has been given in the XII std physics text book published by Tamil Nadu (India) government.

One of the properties of the electric lines of force given in the book is: "each unit positive charge placed in free space gives rise to 1/ ε lines of force, which works out to be equal to the quoted value taking ε for air as 8.854x10-12.

Though I could understand that the lines of force is merely an imaginary concept to visualize the electric flux, I find it difficult to convincingly explain how such an exact number could be specified when no one can count the number of lines of force as such, coming out of a closed area since such an idea is purely an abstract one. I will be happy if you could come out with a more convincing reply.

mfb
Mentor
It is pointless to talk about a "number of lines of force". Such a thing does not exist.
I think "line of force" itself is a problematic concept, but at least it has some clear meaning (=field lines). Let's look at the points 1m away from a charge: Every point is on its own "line of force", and the number of points in a distance of 1m is infinite. There is no way to get any finite number for "lines of force". If you want to draw them, you have to restrict yourself to a finite number, but that is not an exact drawing of the field.