The polarization vector \(\mathbf{P}\) and the resultant electric field \(\mathbf{E}\) in a dielectric are related through the dielectric's response to the applied electric field. Here's a brief overview of the relationship:
1. **Electric Field and Polarization**: In a dielectric material, the polarization vector \(\mathbf{P}\) describes how the material's dipole moments align with an external electric field. The polarization is given by:
\[
\mathbf{P} = \epsilon_0 \chi_e \mathbf{E}
\]
where \(\epsilon_0\) is the permittivity of free space, \(\chi_e\) is the electric susceptibility of the dielectric, and \(\mathbf{E}\) is the electric field applied to the dielectric.
2. **Electric Displacement Field**: The electric displacement field \(\mathbf{D}\) is related to the electric field \(\mathbf{E}\) and the polarization \(\mathbf{P}\) by:
\[
\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}
\]
This relation shows how the electric field in the dielectric material is influenced by the polarization.
3. **Permittivity**: The dielectric constant \(\epsilon_r\) (relative permittivity) of the material is related to the susceptibility \(\chi_e\) by:
\[
\epsilon_r = 1 + \chi_e
\]
The permittivity of the dielectric \(\epsilon\) is then:
\[
\epsilon = \epsilon_0 \epsilon_r
\]
4. **Resultant Electric Field**: The relationship between the electric field \(\mathbf{E}\) and the polarization \(\mathbf{P}\) can also be expressed through the dielectric constant:
\[
\mathbf{E} = \frac{\mathbf{D}}{\epsilon_0 \epsilon_r} = \frac{\mathbf{D}}{\epsilon}
\]
In summary, the polarization vector \(\mathbf{P}\) is proportional to the electric field \(\mathbf{E}\), with the proportionality constant being \(\epsilon_0 \chi_e\). The resultant electric field within the dielectric material is affected by both the applied electric field and the induced polarization.