The Electric Field

Coulomb's law, as stated in Section 1.3, appears to describe an "action at a distance": charge $q_1$ seems to exert a force on charge $q_2$ instantly, across empty space, with nothing physical passing between them. The concept of the electric field resolves this uncomfortable idea by proposing, instead, that a charge modifies the space surrounding it, and it is this modification -- the field -- that then acts locally on any other charge placed within it, rather than the source charge reaching across space directly.

Definition. Imagine placing a very small "test charge" $q_0$ (small enough that it does not itself disturb the positions of the source charges being investigated) at some point in space, and measuring the force $\vec{F}$ that acts on it there. The electric field $\vec{E}$ at that point is defined as the force per unit positive test charge:

The field is a vector quantity: at every point in space it has both a magnitude and a direction, the direction being the direction of the force that a small positive test charge would feel there. Its SI unit, from the definition, is newton per coulomb (N/C), which turns out to be exactly equivalent to volt per metre (V/m), the unit more commonly used once electric potential is introduced.

Field of a point charge. Applying this definition to Coulomb's law immediately gives the field due to an isolated point charge $q$: since the force on a test charge $q_0$ at distance $r$ from $q$ is $F=kqq_0/r^2$, dividing by $q_0$ gives

where $\hat{r}$ is the unit vector pointing away from $q$ toward the point where the field is measured. The field points radially outward from a positive source charge (since a positive test charge would be repelled) and radially inward toward a negative source charge (since a positive test charge would be attracted).

Superposition of fields. Just as forces from several charges add as vectors, the resultant field at any point due to several source charges is the vector sum of the fields each source charge would produce on its own:

$\vec{E} = \vec{E}_1+\vec{E}_2+\cdots+\vec{E}_n$ …