Exploring the Origins of 4π in Coulomb's Constant

[yaseen shah]

i does not understand that in columbs constant where 4pi comes from

 

[cortiver]

I guess you're referring to Coulomb's law:

$$F = \frac{1}{4\pi\epsilon_0} \frac{q_1q_2}{r^2}$$

The factor of 4π is there just because of the way the constant $$\epsilon_0$$ (the "permittivity of free space") is defined. It's defined this way to avoid having extra factors of 4π in http://en.wikipedia.org/wiki/Maxwell%27s_equations" , which are used much more often than Coulomb's law in practice.(This is all in SI units; in the Gaussian units system, still widely used, the factors of 4π in Maxwell's equations are retained, and Coulomb's law doesn't have the 4π in it.)

 

[Dadface]

The constant depends on the medium and we could write it as a straightforward constant(k) or ,what you are referring to,as 1/4pi*epsilon.The two constants are the same and in many problems it is easier to use the straightforward k but if we did there would be other problems where,because of spherical symmety,4 pi would appear in the final answer.Using the second form of the constant makes the 4pis cancel and for the majority of problems the second form is more convenient.

Hello yaseen
and hello cortiver .you just beat me to it

 

[Filip Larsen]

Good question. I have always assumed it corresponds to the unit sphere area of 4 pi steradians, but I could be wrong.

 

[arildno]

This is what you could say part of mathematical aesthetics, rather expressive of any more deep insights.
We have the mathematical freedom to define a "constant" as the product (or, for that matter, the sum) of two other constants, and we choose the one definition that gives the most "elegant" answers.

 

[Tomsk]

Good question. I have always assumed it corresponds to the unit sphere area of 4 pi steradians, but I could be wrong.

It does. In fact you have the area of a sphere there, 4pi r^2. The constant epsilon0 is chosen so that the factor 4pi r^2 can be understood geometrically. There is nothing wrong with getting rid of the 4pi though, but the geometrical meaning is slightly less obvious.

The geometrical meaning comes from Gauss's Law, which says that the flux of the electric field out of a closed surface surrounding a charge is proportional to the charge inside that surface. The proportionality constant is determined by how you measure charge (i.e what system of units you're using), 1/epsilon0 is most common, but you can chose it so that the 4pi cancels. In this context flux is basically the number of electric field lines crossing a surface. So it depends not just on the strength of the electric field, but also on the area of the surface. So for a sphere you have $Flux = 4\pi r^2 E \propto Q$.