Electric Displacement Field - Example: Displacement Field in A Capacitor

Example: Displacement Field in A Capacitor

Consider an infinite parallel plate capacitor placed in space (or in a medium) with no free charges present except on the capacitor. In SI units, the charge density on the plates is equal to the value of the D field between the plates. This follows directly from Gauss's law, by integrating over a small rectangular pillbox straddling one plate of the capacitor:

On the sides of the pillbox, dA is perpendicular to the field, so that part of the integral is zero, leaving, for the space inside the capacitor where the fields of the two plates add,

,

where A is surface area of the top face of the small rectangular pillbox and Q / A is just the free surface charge density on the positive plate. Outside the capacitor, the fields of the two plates cancel each other and |D| = 0. If the space between the capacitor plates is filled with a linear homogeneous isotropic dielectric with permittivity ε the electric field between the plates is constant: |E| = Q / (ε A).

If the distance d between the plates of a finite parallel plate capacitor is much smaller than its lateral dimensions we can approximate it using the infinite case and obtain its capacitance as

.