2.3: Permittivity

Permittivity describes the effect of material in determining the electric field in response to electric charge. For example, one can observe from laboratory experiments that a particle having charge \(q\) gives rise to the electric field \[{\bf E} = \hat{\bf R} ~ q ~ \frac{1}{4\pi R^2} ~ \frac{1}{\epsilon} \nonumber \] where \(R\) is distance from the charge, \(\hat{\bf R}\) is a unit vector pointing away from the charge, and \(\epsilon\) is a constant that depends on the material. Note that \({\bf E}\) increases with \(q\), which makes sense since electric charge is the source of \({\bf E}\). Also note that \({\bf E}\) is inversely proportional to \(4\pi R^2\), indicating that \({\bf E}\) decreases in proportion to the area of a sphere surrounding the charge – a principle commonly known as the inverse square law. The remaining factor \(1/\epsilon\) is the constant of proportionality, which captures the effect of material. Given units of V/m for \({\bf E}\) and C for \(Q\), we find that \(\epsilon\) must have units of farads per meter (F/m). (To see this, note that 1 F \(=\) 1 C/V.)

Permittivity

Permittivity (\(\epsilon\), F/m) describes the effect of material in determining the electric field intensity in response to charge.

In free space (that is, a perfect vacuum), we find that \(\epsilon = \epsilon_0\) where: \[\epsilon_0 \cong 8.854 \times 10^{-12} ~\mbox{F/m} \nonumber \] The permittivity of air is only slightly greater, and usually can be assumed to be equal to that of free space. In most other materials, the permittivity is significantly greater; that is, the same charge results in a weaker electric field intensity.

It is common practice to describe the permittivity of materials relative to the permittivity of free space. This relative permittivity is given by: \[\epsilon_r \triangleq \frac{\epsilon}{\epsilon_0} \nonumber \] Values of \(\epsilon_r\) for a few representative materials is given in Appendix A1. Note that \(\epsilon_r\) ranges from 1 (corresponding to a perfect vacuum) to about 60 or so in common engineering applications. Also note that relative permittivity is sometimes referred to as dielectric constant. This term is a bit misleading, however, since permittivity is a meaningful concept for many materials that are not dielectrics.