Parallel-Plate Capacitor

The parallel-plate capacitor is one that includes two conductor plates, each connected to wires, separated from one another by a thin space.

Learning Objective

Calculate the maximum storable energy in a parallel-plate capacitor

Key Points

Assuming the plates extend uniformly over an area of A and hold ± Q charge, their charge density is ±, where ρ=Q/A.
Assuming that the dimensions of length and width for the plates are significantly greater than the distance (d) between them, $E=\frac {\rho}{\epsilon}$ can be used to calculate the electric field (E) near the center of the plates. In this equation, ε represents permittivity.
$V=\frac {\rho d}{\epsilon}=\frac {Qd}{\epsilon A}$ can be used to calculate the potential between the plates.
$C=\frac {\epsilon A}{d}$ can be found from the previous equation, adjusting the terms to solve for capacitance (C).
$U=\frac {CV^2}{2}=\frac {\epsilon A(U_dd)^2}{2d}=\frac {\epsilon AdU_d^2}{2}$ solves for the maximum storable energy in a parallel-plate capacitor (U) as a function of U_d, the dielectric strength per distance as well as capacitor's voltage (V) at its breakdown limit.

Terms

capacitor
An electronic component capable of storing an electric charge, especially one consisting of two conductors separated by a dielectric.
permittivity
A property of a dielectric medium that determines the forces that electric charges placed in the medium exert on each other.
dielectric
An electrically insulating or nonconducting material considered for its electric susceptibility (i.e., its property of polarization when exposed to an external electric field).

Full Text

One of the most commonly used capacitors in industry and in the academic setting is the parallel-plate capacitor . This is a capacitor that includes two conductor plates, each connected to wires, separated from one another by a thin space. Between them can be a vacuum or a dielectric material, but not a conductor.

Parallel-Plate Capacitor

In a capacitor, the opposite plates take on opposite charges. The dielectric ensures that the charges are separated and do not transfer from one plate to the other.

The purpose of a capacitor is to store charge, and in a parallel-plate capacitor one plate will take on an excess of positive charge while the other becomes more negative.

Assuming the plates extend uniformly over an area of A and hold ± Q charge, their charge density is ±, where ρ=Q/A. Assuming that the dimensions of length and width for the plates are significantly greater than the distance (d) between them, the electric field (E) near the center of the plates can be calculated by:

$E=\frac {\rho}{\epsilon}$

Potential (V) between the plates can be calculated from the line integral of the electric field (E):

$V=\int_0^d \! E \mathrm{d}z$

where z is the axis perpendicular to both plates. Through simplification and substitution, this integral can be changed to:

$V=\frac {\rho d}{\epsilon}=\frac {Qd}{\epsilon A}$

Given that capacitance is the quotient of charge and potential:

$C=\frac {\epsilon A}{d}$

Accordingly, capacitance is greatest in devices with high permittivity, large plate area, and minimal separation between the plates.

The maximum energy (U) a capacitor can store can be calculated as a function of U_d, the dielectric strength per distance, as well as capacitor's voltage (V) at its breakdown limit (the maximum voltage before the dielectric ionizes and no longer operates as an insulator):

$U=\frac {CV^2}{2}=\frac {\epsilon A(U_dd)^2}{2d}=\frac {\epsilon AdU_d^2}{2}$

[ edit ]

Prev Concept

Capacitors with Dielectrics

Combinations of Capacitors: Series and Parallel

Next Concept