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.