Insulators — Question 6

Question 6

Breakdown field imposes a maximum dielectric energy density

A linear dielectric has permittivity $\varepsilon=4.00\times10^{-11}\,\mathrm{F/m}$ and idealized breakdown field $E_b=2.00\times10^7\,\mathrm{V/m}$. A slab thickness is $d=.200\,\mathrm{mm}$.

1. Find the breakdown voltage and maximum electrostatic energy density.

2. Find the maximum energy for dielectric volume $V_d=1.00\,\mathrm{cm^3}$.

Question 6 - Solution

Breakdown field imposes a maximum dielectric energy density

(a) Method and calculation

$$\boxed{V_b=E_bd=4.00\times10^3\,\mathrm V}.$$$$\boxed{u_{\max}=\tfrac12\varepsilon E_b^2=8.00\times10^3\,\mathrm{J/m^3}}.$$

(b) Interpretation and further calculation

$$U_{\max}=u_{\max}V_d,\quad1\,\mathrm{cm^3}=10^{-6}\,\mathrm{m^3}.$$$$\boxed{U_{\max}=8.00\times10^{-3}\,\mathrm J}.$$

Physical check

This ideal bulk-field limit omits defects and edge enhancement. It illustrates that thickness sets voltage capacity while energy per volume depends on permittivity and field.