## PHYS208 Fundamentals of Physics II

### Evaluation of Parallel-Plate Capacitance

**1. Consider the system of two conductors,
placing a charge ***+Q* on one conductor and *-Q* on the other.

Here we are considering two parallel plates of area *A*, not necessarily of rectangular shape,
separated by a distance *d*.

**2. Evaluate the electric field everywhere along some path joining the two conductors.**

The plates are considered to be large enough in area and close enough in distance that the
electric field is uniform well away from the edges (fringing fields at the edges are ignored).
In addition, the excess charge deposited on the conductors is assumed to spread out uniformly
so that

Application of Gauss's law can be made to determine the dependence of the electric field
to the surface charge density present; the symmetry of the charge sheet suggests a
gaussian can enclosing one surface. The cross section of the appropriate gaussian can
is shown below.

Considering a gaussian can with base area of *a*, the only contribution to the
flux will be that of the end cap between the plates. The portion of the gaussian surface
immersed in the conducting material will have no flux through it since the electric field
is zero there. The tube wall outside the conductor is constructed so that there is no
flux through it either; *i.e.* the surface vector at all points on the cylindrical
surface is perpendicular to **E**.

**3. Evaluate the potential difference along that same path using the relationship
between** *V* **and ****E**.

Ideally you may select a path over which the line integral will be the simplest to evaluate;
*i.e.* one in which the line integral reduces to a simple integral.
In this case, choose the shortest possible path, a straight line perpendicular to the plates,
starting from the negatively charged plate (lower potential) and moving toward the positively
charged plate.
The direction vector of the line integral will then always be antiparallel to the electric
field and the dot product in the integrand will reduce the line integral to a simple integral
as shown.

**4. The capacitance is the ratio of the charge deposited on one conductor
to the potential difference between the two.**

"http://www.physics.udel.edu/~watson/phys208/parallel-plate.html"

Last updated March 9, 1998.

Copyright George Watson, Univ. of Delaware, 1998.