Definition of Eo (Coulomb's law)

[maxbashi]

F=$$\frac{q1q2}{4\pi\epsilon_{0}}$$

If epsilon 0 was defined as the proportionality constant for this equation, why was 4 pi not included in Eo? Why does there have to be a 4pi in the equation, instead of just having Eo equal its current value times 4pi?

 

[K^2]

Because in differential form, the equation is written as:

$$\nabla \cdot E = \frac{\rho}{\epsilon_0}$$

If you apply Gauss Theorem to it, you get the 1/4pi term.

More fundamentally, it's because epsilon works as polarization coefficient for vacuum.

It also works neatly with the wave equation. $\epsilon_0\mu_0 = \frac{1}{c^2}$

 

[jtbell]

The $4\pi$ and the $\epsilon_0$ appear where they do because the equations assume SI units. In other systems of units, they appear in different places, or not at all. For example, in Gaussian cgs units, $\epsilon_0 = 1$ and we have

$$F = \frac {q_1 q_2}{r^2}$$

$$\nabla \cdot E = 4 \pi \rho$$

See http://en.wikipedia.org/wiki/Gaussian_units

 

[AJ Bentley]

Epsilon 0 has that silly form because it came to the party last and Mu 0 got in first.

The Ampere was defined in terms of the force between two parallel conductors one meter apart. The force is caused by the magnetic field, which as you know is circular around the wire.
That's where the 4 pi came from (actually it's 2 *2 pi because there are two wires)

The clincher is the fact that Epsilon 0 * Mu 0 = 1/c²

Once Mu has that pi in it, Epsilon has no choice.

If Epsilon had got in first, Mu 0 would be stuck with the pi instead.

 

[sardar]

Coulomb's law is a fundamental equation in electromagnetism that describes the force between two charged particles. It states that the force (F) between two point charges (q1 and q2) is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. This is represented by the equation F = (q1q2)/4πε0.

The constant 4π in the equation is known as the permittivity of free space and has a value of approximately 8.85 x 10^-12 C^2/Nm^2. It is included in the equation to account for the distribution of electric field lines in three-dimensional space. Without this constant, the equation would not accurately describe the force between charged particles at different distances.

The reason why the constant is not included in the definition of ε0 is because it is a fundamental constant in physics, similar to the speed of light or the gravitational constant. It is not something that can be changed or derived from other quantities. Therefore, it is given its own symbol and value.

In summary, the inclusion of 4π in Coulomb's law is necessary to accurately describe the force between two charged particles at different distances, and it is a fundamental constant in physics with a specific value.