How did the expression for electric field? How did the expression for electric field come to be defined as F/q?
 A: Hi and welcome to physics stackexchange. Imagine you want to quantify how much force will a (test) charge feel if you place it somewhere near another charge, without considering the first (test) charge's properties (i.e. charge magnitude). So you want to define a quantity that is independent of the test particle's charge, but it is related to the second charged particle's properties (i.e. charge magnitude again). Well, the most natural way to do that is to divide by the magnitude of the test charge. The resulting quantity is what came to be known as the electric field, caused by the (non-test) charged particle to all the points in space.
It has the properties of a field, i.e. a given value (or several values if we are talking about a vector field) for each point in space and it also inherits the properties of the (non-test) charged particle, while at the same time neglects properties a test particle might have If we were placing it anywhere in space, where it would be influenced by the field caused by the non-test charge.
I know it doesn't sound like a lot, but I think this is simply a matter of definition. And the definition makes sense, because fields are essentially space-dependent functions (or vectors, whose components are functions of space). I hope this helps...
