The electric field is a vector field that describes electric effects, and is defined as the field that correctly predicts the Lorentz force on a particle with electric charge :
This definition implies that the direction of is from positive to negative charges, since opposite charges attracts and like charges repel.
If two opposite point charges with magnitude are observed from far away, they can be treated as a single object called a dipole, which has an electric dipole moment defined like so, where is the vector going from the negative to the positive charge (opposite direction of ):
Alternatively, for consistency with magnetic fields, can be defined from the aligning torque experienced by the dipole when placed in an -field. In other words, satisfies:
Where has units of . The polarization density is defined from , and roughly speaking represents the moments per unit volume:
If has the same magnitude and direction throughout the body, then this becomes , where is the volume. Therefore, has units of .
A nonzero complicates things, since it contributes to the field and hence modifies . We thus define the “free” displacement field from the “bound” field and the “net” field :
Where the electric permittivity of free space is a known constant. It is important to point out some inconsistencies here: and contain a factor of , and therefore measure flux density, while does not contain , and thus measures field intensity. Note that this convention is the opposite of the magnetic analogues , and , and that has the opposite sign of .
The polarization is a function of . In addition to the inherent polarity of the material (zero in most cases), there is a (possibly nonlinear) response to the applied -field:
Where the are the electric susceptibilities of the medium. For simplicity, we often assume that only the term is nonzero, which is the linear response to . In that case, we define the absolute permittivity so that:
I.e. , where is the dielectric function or relative permittivity, whose calculation is of great interest in physics.
In reality, a material cannot respond instantly to , meaning that is a function of time, and that is the convolution of and :
Note that this definition requires for in order to ensure causality, which leads to the Kramers-Kronig relations.