Electrostatic Constant Written Funny

[Pupil]

Main Question or Discussion Point

So the electrostatic constant k = 9*10^9, but I've also seen it written in my Fundamentals of Phys book (Resnick) and Wikipedia as k = 1/(4*pi*e0). Why is this? Thanks.

 

[Pupil]

No one knows?

 

[Doc Al]

So the electrostatic constant k = 9*10^9, but I've also seen it written in my Fundamentals of Phys book (Resnick) and Wikipedia as k = 1/(4*pi*e0). Why is this? Thanks.

The ε₀ is a constant called the electric permittivity of free space. Read about it here: http://hyperphysics.phy-astr.gsu.edu/HBASE/electric/elefie.html#c3"

 

[Bob S]

The capacitance of a parallel plate air capacitor of area A and plate separation d is

C = e₀ A/d

where the permittivity of free space e₀ = 1/(u₀ c²) = 8.85 x 10^-12 Farads per meter,
and the permeability of free space is
u₀ = 4 pi x 10^-7 Henrys per meter

 

[jtbell]

It's partly for historical reasons, and partly a matter of minimizing the number of equations that have factors of $4 \pi$.

Using $\epsilon_0$ Coulomb's Law is more complicated:

$$F_{elec} = \frac {1} {4 \pi \epsilon_0} \frac {q_1 q_2} {r^2}$$

but other equations like Gauss's Law and the parallel-plate capacitor equation are simple:

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

$$C = \frac {\epsilon_0 A}{d}$$

Whereas using k, Coulomb's Law is simpler:

$$F_{elec} = k \frac {q_1 q_2} {r^2}$$

but you have to insert factors of $4 \pi$ into other equations:

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

$$C = \frac {A}{4 \pi k d}$$

 

[madmike159]

Well 1/4pi*e0 is where 9x10^9 came from.