7
Fundamentals of Electric Capacitors
only to capacitors having two parallel planar electrodes. For cylindrical,
spherical, and isolate sphere capacitors, the capacitance expressions are different. For a long cylindrical coaxial capacitor of length L, the inner conducting cylinder has a radius of a and the outer cylinder has a radius of b. Its
capacitance can be expressed as
L
C = 2πε 0 b a
(1.13)
ln( / )
For a spherical capacitor with an inner sphere radius of a and an outer sphere
radius of b, its capacitance is
ab
C = 4πε 0 b a
(1.14)
−
For an isolated sphere capacitor that only has a single isolated spherical conductor with a radius of R, its capacitance can be expressed as
C = 4πε 0 R
(1.15)
Equations (1.12) through (1.15) demonstrate that capacitance is strongly
dependent on the dielectric constant ε 0 if the configuration of the capacitor
is fixed. This ε 0 is the dielectric constant of an empty vacuum. However,
for a non-vacuum dielectric, the material’s relative permittivity or relative
dielectric constant is defined as the relative dielectric constant (ε r = ε/ε 0 )
where ε is the dielectric constant of the material. Note that this equation
will be discussed further in Section 1.3.1. Every dielectric material has a
different dielectric constant resulting in a different capacitance. Table 1.1
lists the dielectric constants of common materials used in capacitors. All of
them have larger dielectric constants compared to those of air or a vacuum.
In this case, Equation (1.12) can be rewritten as
Q εA εε 0 A
C = =
=
r
V
d
d
(1.12a)
Larger dielectric constants can effectively reduce the magnitude of the electric field of a charged object, as seen in Equations (1.4) and (1.7). However,
larger dielectric constants have more energy storage because the energy
stored in a capacitor is proportional to its capacitance.
The intrinsic dielectric strength describes the maximum electric field the
capacitor can tolerate prior to the breakdown voltage, and is an important
consideration when selecting dielectric materials. For example, when an
applied voltage exceeds the dielectric strength of a material, the insulation of