Relative Permittivity or Dielectric Constant

In stating Coulomb's law above, it was implicitly assumed that the two point charges sit in vacuum (or, to an excellent approximation, in air). When the same two charges are instead placed inside a material medium -- water, mica, paraffin-coated paper, and so on -- the medium itself modifies (reduces) the force between them.

In a medium of absolute permittivity $\epsilon$, the force between the charges becomes $F_{med}=\frac{1}{4\pi\epsilon}\frac{q_1q_2}{r^2},$ while the force between the SAME two charges at the SAME separation, but now in free space/vacuum, is $F_{vac}=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2},$ where $\epsilon_0=8.85\times10^{-12}\,\text{C}^2\text{N}^{-1}\text{m}^{-2}$ is the (universal) permittivity of free space. Dividing the vacuum expression by the medium expression, the $q_1q_2/r^2$ factors cancel exactly, leaving $\frac{F_{vac}}{F_{med}}=\frac{\epsilon}{\epsilon_0}.$

The ratio $\epsilon/\epsilon_0$ is called the RELATIVE PERMITTIVITY, or DIELECTRIC CONSTANT, of the medium, denoted $\epsilon_r$ or K: $K=\epsilon_r=\frac{\epsilon}{\epsilon_0}=\frac{F_{vac}}{F_{med}}.$ From this definition three things follow directly: (i) $\epsilon_r$ is the ratio of a medium's absolute permittivity to the permittivity of free space; (ii) equivalently, $\epsilon_r$ is the ratio of the force between two charges in free space to the force between the same two charges, at the same separation, when a given medium replaces the vacuum between them; (iii) $\epsilon_r$ is a purely dimensionless number (also occasionally called the specific inductive capacity of the medium). The force between the two charges when actually immersed in a medium of relative permittivity $\epsilon_r$ can then be written directly as $F=\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{q_1q_2}{r^2}.$ …