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Mathematics for Physics

36 Dielectrics

An insulating material that is used to weaken the net electric field inside a capacitor by becoming polarized, where an insulator is a material whose charges are not free to travel through the bulk.

Polarization of a linear dielectric. An external field induces a dipole moment in the molecules of a dielectric. Polarization is the dipole moment per unit volume. This principle is used to describe how the material responds to the applied field.

The polarization is

\[ \mathbf{P} = \epsilon_{0}\chi_{e}\mathbf{E} \]

where

  • \(\mathbf{P}\) is the polarization.
  • \(\epsilon_{0}\) is the permittivity of free space.
  • \(\chi_{e}\) is the electric susceptibility.
  • \(\mathbf{E}\) is the electric field in the material.

The dielectric constant. The net field inside a linear dielectric is the applied field reduced by the factor \(\kappa\). The dielectric constant is that reduction factor. This principle is used to compute the field and capacitance when a dielectric fills the gap.

The dielectric constant is

\[ \kappa = 1 + \chi_{e} \]

where

  • \(\kappa\) is the dielectric constant.
  • \(\chi_{e}\) is the electric susceptibility.

The electric displacement. The displacement field includes both the electric field and the polarization. This principle is used to write Gauss’s law in the presence of linear dielectrics.

The displacement field is

\[ \mathbf{D} = \epsilon_{0}\mathbf{E} + \mathbf{P} \]

where

  • \(\mathbf{D}\) is the electric displacement.
  • \(\mathbf{E}\) is the electric field.
  • \(\mathbf{P}\) is the polarization.
  • \(\epsilon_{0}\) is the permittivity of free space.

Note: Also called insulating materials in electrostatics.