What is the derivation of Coulomb's constant?
Coulomb's constant 'k' in the equation F=kQQ/r2 is derived from
Gauss's law. Gauss's law stated that the charge enclosed by a
theoretical surface is equal to the permittivity constant,
represented by the Greek letter epsilon (because I can't use an
epsilon, I will use an X) times the electric flux through the
surface. Flux is equal to the closed integral of electric field
vector dot the vector dA (infinitesimal change in surface area) of
the surface. Becasue the surface surrounding one point charge is a
perfect sphere, the dot product can be ignored (The surface is
uniform and every change in area is normal to the electric field),
and the Electric field is constant so it can be brought out of the
integral leaving integral dA. When the integral is solved, the
resulting equation is XEA=Q. A equals the surface area of the
sphere so XE(4*pi*r2)=Q and E=Q/(4*pi*X*r2) and because F=EQ,
F=QQ/(4*pi*X*r2). This is probably looking pretty familiar. All we
have to do is make k=1/(4*pi*X) to make this equation equal to good
old Coulomb's law. X, the permittivity constant equals 8.854*10-12
Farads per meter, or coulombs squared seconds squared per kilograms
meters cubed. If you substitute this constant into the equation
k=1/(4*pi*X), you obtain Coulmb's constant.
