Electric Field - Quantitative Definition

Quantitative Definition

Consider a point charge q with position (x,y,z). Now suppose the charge is subject to a force due to other charges. Since this force varies with the position of the charge and by Coloumb's Law it is defined at all points in space, is a continuous function of the charge's position (x,y,z). This suggests that there is some property of the space that causes the force which is exerted on the charge q. This property is called the electric field and it is defined by

Notice that the magnitude of the electric field has units of Force/Charge. Mathematically, the E field can be thought of as a function that associates a vector with every point in space. Each such vector's magnitude is proportional to how much force a charge at that point would "feel" if it were present and this force would have the same direction as the electric field vector at that point. It is also important to note that the electric field defined above is caused by a configuration of other electric charges. This means that the charge q in the equation above is not the charge that is creating the electric field, but rather, being acted upon by it. This definition does not give a means of computing the electric field caused by a group of charges.

From the definition, the direction of the electric field is the same as the direction of the force it would exert on a positively charged particle, and opposite the direction of the force on a negatively charged particle. Since like charges repel and opposites attract, the electric field is directed away from positive charges and towards negative charges.