Electric Field (Definition)

Electric field at a point is the force per unit positive test charge. The limit ensures the test charge does not disturb the source distribution.

Motivation

Coulomb's law requires two charges. But we want to describe the influence of a charge distribution at every point in space, independently of what test charge we choose to place there.

Definition

Place a small positive test charge $q_0$ at a point P. Measure the force $\vec{F}$ on it. The electric field at P is:

$$\vec{E}(P) = \lim_{q_0 \to 0} \frac{\vec{F}}{q_0}$$

The limit $q_0 \to 0$ is crucial: a finite test charge disturbs the source distribution (induces charges, exerts back-reaction). Taking $q_0 \to 0$ ensures we measure the field of the source, not the modified field.

Physical meaning

$\vec{E}$ is a vector field — it assigns a vector to every point in space. Once $\vec{E}$ is known, the force on any charge $q$ placed at that point is:

$$\vec{F} = q\vec{E}$$

The field itself carries energy and momentum. It is not merely a computational device.

$$\boxed{\vec{E} = \lim_{q_0 \to 0} \frac{\vec{F}}{q_0}}$$