Every electric charge generates an electric field. Like gravitational fields, electric fields are direction-dependent and are therefore described by vectors. If multiple charges are present, the electric fields at any point must be added vectorially.
When a charge is placed in an electric field, a force acts on it. If the charge is small enough not to influence the field itself, the electric field is defined as the ratio of force to electric charge:
$$
\vec{E} = \frac{\vec{F}}{q}
$$
The unit of charge \(q\) is the coulomb (C). The smallest naturally occurring unit of charge is the elementary charge:
$$
e = 1.602176634 \cdot 10^{-19}\,\mathrm{C}
$$
Its magnitude is identical to the charge of a proton or an electron. Thus, the electron has the charge \(q = -e\) and the proton \(q = +e\).
Field lines are used to visualize electric fields. They are drawn such that they point away from positive charges and toward negative charges. At every point, the field line indicates the direction of the electric field.
When constructing field line diagrams, one must remember that field lines never intersect. The density of field lines is a measure of the field strength at a given point.
In two-dimensional diagrams, the field of two charges of equal or opposite sign can be illustrated with field lines. The dashed lines represent so-called equipotential surfaces, on which the potential is constant. If a small test charge were moved along such a surface, no work would be performed—similar to moving an object parallel to the Earth’s surface.
Electric fields are often used to accelerate and deflect charges. If a free charge is located in an electric field, a force acts on it and causes it to accelerate. The acceleration is obtained by dividing the force by the mass \(m\) of the particle:
$$
\vec{a} = \frac{\vec{F}}{m} = \frac{q\vec{E}}{m}
$$
The mass of the electron has been determined experimentally as:
$$
m_e = 9.1093837015 \cdot 10^{-31}\,\mathrm{kg}
$$
Since the acceleration is constant, the motion can be described with the equation of motion:
$$
\vec{s}(t) = \tfrac{1}{2}\vec{a}t^2 = \tfrac{1}{2}\frac{q\vec{E}}{m}t^2
$$
In general, a charge in a homogeneous and constant electric field moves along a parabolic trajectory, analogous to an object in the Earth’s gravitational field. For the special case of motion perpendicular to the field lines, one obtains the following relationship between the vertical deflection \(y\) and the horizontal motion \(x\), analogous to projectile motion:
$$
y(x) = -\tfrac{1}{2}\frac{qE}{m}\frac{x^2}{v^2}
$$
This technique of deflecting charged particles played an important role in early oscillographs, where images were generated using an electron beam.
The electrons moved through an electric field that varied with the applied signal. When the electrons struck a fluorescent screen, the temporal course of the image corresponded directly to the signal being studied.