How Is the Number of Electric Field Lines Calculated in Gauss' Law?

[ananthu]

The electric flux around a charge is represented by electric lines of force. But it is said that the electric lines of force is purely an imaginary concept to visualize a field. In the Gauss' law, using the formula q/epsilon we calculate the total number of lines of force. For a charge of 1 coulomb, it works around 1.129x10^11 in free space. How do you explain this exact number they arrive at?

Is this number is a real one and if it is so, then how is it possible to count the number of some thing which itself is just an imaginary way of visualizing an abstract idea? Does it mean that actually so many number of lines are emerging from a point charge like fine hairs sticking out of a sphere in space?

 

[tiny-tim]

… In the Gauss' law, using the formula q/epsilon we calculate the total number of lines of force. For a charge of 1 coulomb, it works around 1.129x10^11 in free space. How do you explain this exact number they arrive at?

It makes a lot more sense if you look at the flux of the displacement D field, rather than the E field …

the electric displacement field D has lines of force coming perpendicularly out of a conductor, and (in equilibrium, where all the charge is on the surface) the number of lines equals the surface charge density: D = σ.

So D is measured in coulombs per square metre (C/m²).

The electric field E however is measured so that we can use it in the Lorentz force equation, F = qE, so E is measured in Newtons per coulomb (N/C).

To convert one to the other, we define (in the absence of polarisation) D = εE, where ε is the permittivity of the medium (in anisotropic media, ε is a tensor so the two fields needn't be parallel).

In vacuum, ε = ε₀, the permittivity of the vacuum.

It equals 1/µ₀c² where c is the speed of light and µ₀ is the permeability of the vacuum.

µ₀ is defined as 4π 10^-7 henrys per metre (H/m), and µ₀c² happens to be around 1.129 10¹¹ henry metres per second squared (Hm/s²), so ε₀ is around 8.85 10^-12 farads per metre (F/m).

(a henry times a farad equals a second squared: HF = s²)

(C/m² per N/C = C²/Nm per m = C²/J per m = F/m)

If it wasn't for that arbitrary 10^-7 in the definition of µ₀, then ε₀ would simply be 1/4πc² (in farads per metre). ​

Is this number is a real one and if it is so, then how is it possible to count the number of some thing which itself is just an imaginary way of visualizing an abstract idea? Does it mean that actually so many number of lines are emerging from a point charge like fine hairs sticking out of a sphere in space?

For the D field, the number of lines is the surface charge density, so it's as if there's one line sticking out of each charge.

 

[vector22]

hmm lines o force

I think there is no such thing as "lines" of force in either magnetism or electrostatics

Iron fillings will line up in what seems to be lines but that does not prove the lines exist.

 

[ananthu]

thanks for all the reply.

Regarding. tiny-tim's reply, the explanation is too technical to explain to a class of higher secondary school students.

When I say that there are millions of sand grains present in the handful of sand I hold, it is o.k.
No body will question us. But if I say, there are 15 lakh, 28 thousand and 233 sand grains in my hand, then I should justify the number I assert.

There is another difference in the above comparison also. At least, in the case of sand example. there is some possibility of counting them as they are visible.

 

[tiny-tim]

hi ananthu!

Regarding. tiny-tim's reply, the explanation is too technical to explain to a class of higher secondary school students.

what's technical about …

For the D field, the number of lines is the surface charge density, so it's as if there's one line sticking out of each charge.

… ?? ​

 

[diazona]

The electric flux around a charge is represented by electric lines of force. But it is said that the electric lines of force is purely an imaginary concept to visualize a field. In the Gauss' law, using the formula q/epsilon we calculate the total number of lines of force. For a charge of 1 coulomb, it works around 1.129x10^11 in free space. How do you explain this exact number they arrive at?

The "number" you get is actually not just a number, and it doesn't count anything. It's a value with units,
$$\frac{1\ \mathrm{C}}{\epsilon_0} = 1.129\times 10^{11}\ \mathrm{Vm}$$
(that's volt-meters). It represents the total amount of electric flux coming out of the charge.

Whenever you draw electric field lines, you have to choose an amount of flux for each line to represent. For example, you might choose to have one line per $1.41\times 10^{10}\ \mathrm{Vm}$ of flux, and in that case there will be 8 lines coming out of a 1-coulomb charge. Of course, usually you just pick a number of lines to draw and don't worry about exactly how much flux each one represents, but still, that choice is implicitly being made.