I just wanted to make sure I understood the difference between the two.
Coulombs law is the force between two charges. Two charges that are physical(maybe not the right word) located on some axes. We can use the equation
F=k(q1 * q2 * r(hat))/r^2
r^2=|r| = |x-x'|
where r hat equal= r/|r|= (x-x')/|x-x')|
if we only have one charge and we want to know what the force would be at some random point q. We know that F=qE(x). So, F/q=E(x). which would give us
They both calculate the force, coulombs is the force between two charges and the electric field is the force a some point q. (q would be a field point.)
As the distance grows on both the force gets weaker and weaker.