Linear Dielectrics
In many substances, the polarization \(\vec{P}\) is proportional to the electric field. These are called linear dielectrics. While the microscopic physics can be complex, the macroscopic behavior is often simple and elegant.
Susceptibility and Permittivity
For a linear dielectric, the polarization is proportional to the total macroscopic field \(\vec{E}\). We define this relationship using the electric susceptibility \(\chi_e\):
This simple scalar relation assumes the material is isotropic, meaning its properties are identical in all directions (e.g., gases, liquids, glass).
In anisotropic materials (like crystals), the polarizability depends on the direction of the field. In such cases, \(\vec{P}\) is not necessarily parallel to \(\vec{E}\), and \(\chi_e\) becomes a tensor rather than a scalar number.
Substituting this into the definition of Displacement (\(\vec{D} = \epsilon_0 \vec{E} + \vec{P}\)), we find that \(\vec{D}\) is also proportional to \(\vec{E}\):
- Electric Susceptibility (\(\chi_e\)) A dimensionless measure of how easily a material polarizes. For vacuum, \(\chi_e = 0\).
- Permittivity (\(\epsilon\)) Defined as \(\epsilon = \epsilon_0 (1 + \chi_e)\). This determines the field strength inside the material.
- Dielectric Constant (\(\epsilon_r\)) The relative permittivity: \(\epsilon_r = 1 + \chi_e = \epsilon / \epsilon_0\). This is the factor by which the field is reduced.
Energy in Dielectric Systems
To polarize a dielectric, work must be done to displace the charges against their internal forces. In a linear dielectric, this work is stored as electrostatic potential energy.
The total energy required to assemble a charge distribution in the presence of a linear dielectric is given by a volume integral over all space:
This expression generalizes the vacuum case (\(W = \frac{\epsilon_0}{2} \int E^2 \, d\tau\)). Notice that the presence of the dielectric, with its large permittivity \(\epsilon\), typically increases the amount of energy a system can store for a given field strength.
Physical Interpretation: Part of this energy is stored in the field itself, and part is stored as potential energy in the distorted molecules of the dielectric.