AcademyCapacitors and Dielectrics
Academy
Gauss's Law in Materials
Level 1 - Physics topic page in Capacitors and Dielectrics.
Principle
The displacement field separates free charge from bound polarization charge in dielectric problems.
Notation
\(\vec D\)
electric displacement field
\(\mathrm{C\,m^{-2}}\)
\(\vec E\)
electric field
\(\mathrm{V\,m^{-1}}\)
\(\vec P\)
polarization
\(\mathrm{C\,m^{-2}}\)
\(q_{\mathrm f}\)
free charge enclosed
\(\mathrm{C}\)
\(\epsilon\)
material permittivity
\(\mathrm{F\,m^{-1}}\)
\(\kappa\)
dielectric constant
1
Method
Derivation 1: Define the displacement field
Polarization accounts for bound charge. The displacement field groups the vacuum field and polarization so Gauss's law can count free charge only.
Displacement definition
\[\vec D=\epsilon_0\vec E+\vec P\]
Material Gauss law
\[\oint \vec D\cdot d\vec A=q_{\mathrm f}\]
Derivation 2: Linear dielectric form
For a simple linear dielectric, polarization is proportional to the electric field, so \(\vec D\) is also proportional to \(\vec E\).
Linear relation
\[\vec D=\epsilon\vec E\]
Permittivity
\[\epsilon=\kappa\epsilon_0\]
Electric field
\[\vec E=\frac{\vec D}{\epsilon}\]
Derivation 3: Filled parallel-plate capacitor
For plates with free surface charge density \(\sigma_\{\mathrm f\}\), a pillbox enclosing one plate gives the displacement field between the plates.
Free charge
\[q_{\mathrm f}=\sigma_{\mathrm f}A\]
Displacement flux
\[DA=q_{\mathrm f}\]
Displacement field
\[D=\sigma_{\mathrm f}\]
Dielectric field
\[E=\frac{\sigma_{\mathrm f}}{\kappa\epsilon_0}\]
Rules
These are the material-field relations.
Displacement field
\[\vec D=\epsilon_0\vec E+\vec P\]
Material Gauss law
\[\oint \vec D\cdot d\vec A=q_{\mathrm f}\]
Linear dielectric
\[\vec D=\epsilon\vec E\]
Permittivity
\[\epsilon=\kappa\epsilon_0\]
Plate field
\[E=\frac{\sigma_{\mathrm f}}{\kappa\epsilon_0}\]
Examples
Question
A dielectric-filled parallel-plate capacitor has free surface charge density
\[3.0\times10^{-6}\,\mathrm{C\,m^{-2}}\]
Find \(D\).Answer
Between the plates,
\[D=\sigma_{\mathrm f}=3.0\times10^{-6}\,\mathrm{C\,m^{-2}}\]
Checks
- \(\vec D\) flux counts free charge, not total free-plus-bound charge.
- \(\vec E\) is the field that acts on charges and appears in potential differences.
- In a linear dielectric, larger \(\kappa\) gives smaller \(E\) for the same free charge.
- At a material boundary, normal \(\vec D\) changes only if free surface charge is present.