http://www.phys.uri.edu/~gerhard/PHY204/tsl105.pdf

Cylindrical Capacitor

For a cylindrical geometry like a coaxial cable, the capacitance is usually stated as a capacitance per unit length. The charge resides on the outer surface of the inner conductor and the inner wall of the outer conductor. The capacitance expression is

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Cylindrical Capacitor

The capacitance for cylindrical orspherical conductors can be obtained by evaluating the voltage difference between the conductors for a given charge on each. By applying Gauss' law to an infinite cylinder in a vacuum, the electric field outside a charged cylinder is found to be

The voltage between the cylinders can be found by integrating the electric field along a radial line:

From the definition of capacitance and including the case where the volume is filled by a dielectric of dielectric constant k, the capacitance per unit length is defined as

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Parallel Plate Capacitor

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The capacitance of flat, parallel metallic plates of area A and separation d is given by the expression above where:

= permittivity of space and

k = relative permittivity of the dielectric material between the plates.
k=1 for free space, k>1 for all media, approximately =1 for air.
The Farad, F, is the SI unit for capacitance, and from the definition of capacitance is seen to be equal to a Coulomb/Volt.
Any of the active parameters in the expression below can be calculated by clicking on it. Default values will be provided for any parameters left unspecified, but all parameters can be changed. After editing data, you must click on the desired parameter to calculate; values will not automatically be forced to be consistent.

Capacitance = k x ε₀ x area / separation

Capacitance of Parallel Plates

The electric field between two large parallel plates is given by

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The voltage difference between the two plates can be expressed in terms of the workdone on a positive test charge q when it moves from the positive to the negative plate.

It then follows from the definition of capacitance that

Calculation

Effect of dielectric

What is the dielectric constant?

Parallel Plate with Dielectric

When a dielectric is placed between charged plates, the polarization of the medium produces an electric field opposing the field of the charges on the plate. The dielectric constant k is defined to reflect the amount of reduction of effective electric field as shown below. The permittivity is a characteristic of space, and the relative permittivity or "dielectric constant" is a way to characterize the reduction in effective field because of the polarization of the dielectric. Thecapacitance of the parallel plate arrangement is increased by factor k.

Table of dielectric constants

Capacitance of two parallel plates

The most common capacitor consists of two parallel plates. The capacitance of a parallel plate capacitor depends on the area of the plates A and their separation d. According to Gauss's law, the electric field between the two plates is:

Since the capacitance is defined by

one can see that capacitance is:

Thus you get the most capacitance when the plates are large and close together. A large capacitance means that the capacitor stores a large amount of charge.

If a dielectric material is inserted between the plates, the microscopic dipole moments of the material will shield the charges on the plates and alter the relation. Materials have a permeability ewhich is given by the relative permeability k, e=ke₀. The capacitance is thus given by:

All materials have a relative permeability, k, greater than unity, so the capacitance can be increased by inserting a dielectric. Sometimes, k is referred to as the dielectric constant of the material. The electric field causes some fraction of the dipoles in the material to orient themselves along the E-field as opposed to the usual random orientation. This, effectively, appears as if negative charge is lined up against the positive plate, and positve charge against the negative plate. In the figure to the right, the blue material is the dielectric.

Spherical Capacitor

The capacitance for spherical or cylindrical conductors can be obtained by evaluating the voltage difference between the conductors for a given charge on each. By applying Gauss' law to an charged conducting sphere, the electric field outside it is found to be

The voltage between the spheres can be found by integrating the electric field along a radial line:

From the definition of capacitance, the capacitance is

Does an isolated charged sphere have capacitance?

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Isolated Sphere Capacitor?

An isolated charged conducting sphere has capacitance. Applications for such a capacitor may not be immediately evident, but it does illustrate that a charged sphere has stored some energy as a result of being charged. Taking the concentric sphere capacitance expression: