Coulomb's Law and Gauss Law

1. Oct 30, 2011

-Castiel-

Why use k = 1/[4(pi)(epislon)] and epsilon = 8.8 * 10^(-12) and subsequently, k = 9 * 10^9

It could simply be k = 9 * 10^9, and different k for different medium instead of different permittivity for different medium.

What I mean is why does Pi, (I can handle the 4) comes into the equation, one reason I can think of is that it comes if you use Gauss Law to derive Coulomb's law but I am looking for something more convincing.

The Permittivity of free space is a constant experimentally derived. Why would one derive that in the first place? If the force between two charges was directly proportional to some constant K. One would experimentally see that it is 9 * 10^9. Where was the idea of breaking it down to 1/[4(pi)(epislon)] entertained?

2. Oct 30, 2011

grzz

Many problems in this area of physics will have a 4pi in the numerator. So a 4pi in the denomenator will simplify things. Perhaps that is the reason.

3. Oct 30, 2011

-Castiel-

That is the reason I thought of seconds after I clicked the Submit button, it does seem a likely reason but I will wait and see if someone knows something better.

4. Oct 30, 2011

Simon Bridge

Your epsilon is usually $\epsilon_o$ - the permitivity of free space. We can modify the basic equation for different media by including the relative permitivity which is different for different materials.

It is possible to adopt a system of units where you don't have these constants - however, doing so hides some of the physics. For instance, there is an equivalent number for magnetism $\mu_o$ which is the permiability of free space. Together they make the speed of light: $c^2=1/\epsilon_o\mu_o$

The idea is to expose the interaction of fundamental properties and the relationships between physical models.

The pi is more interesting - it comes from the symmetry of the situation the equation describes: spherical.

Last edited: Oct 30, 2011
5. Oct 30, 2011

grzz

I cannot resist posting this comment:

Is not $c^2=1/\epsilon_o\mu_o$

a beautiful formula.

It shows the permitivity on which the electric field depends and the permeability on which the magnetic field depends. And that is what light is - an electromagnetic wave. (At least in classical Physics)

6. Oct 30, 2011

-Castiel-

You can use K here too. epsilon(naught)/epsilon = K/K(naught)

No it doesn't really hide it.

k = 1/[4(pi)(epislon)]

Let l = mu/4(Pi) (I know l doesn't exist but bear with me.)

(I really need to learn how to type those formulae.)

then

epsilon = 1/[4(pi)(k)]

mu = 4(Pi)l

$\epsilon_o\mu_o = kl$

You can now put this back in your formula.

Like I said before. It is just a ratio. Whatever ratio epsilon is in, K is in inverse of that ratio. So relationships between models isn't affected.

7. Oct 30, 2011

-Castiel-

I completely agree. But my point is that permittivity and permeability are experimentally derived.

Why not use 9 * 10^9 and 10^(-7) directly. As I proved earlier they do not deform the equation in any major way.

8. Oct 31, 2011

-Castiel-

bump.

9. Oct 31, 2011

Born2bwire

Permittivity and permeability are not experimentally derived, they are defined constants. Permeability is defined as 4\pi e-7 and permittivity is defined using c and the permeability of freespace. The value for the permittivity and permeability is purely a choice of your unit system. As previously stated, we can use a Gaussian system where the permeability and permittivity of free space are unity. In addition, we can move the 4\pi from the factor k to Maxwell's equations (as we do in Gaussian units). Placing the 4\pi off of the Maxwell equations is called rationalized units.

We do not use a numerical value for k directly since k is dependent upon the permittivity of the background. It is not a constant across all possible problems.

10. Nov 1, 2011

Simon Bridge

When using the equations to do calculations, this is what you do.

When you want to derive general relations, discovering new physical laws, the math works out easier if you keep the geometric constants separate from the physical ones. It's also more rigorous philosophically, and exposes the symmetries of the laws you discover.

Much the same sort of reasoning is behind using radiens instead of degrees for angles.