handbook of modern sensors

3.2 Capacitance 47
(A)
(B)
Fig. 3.6. Polarization of dielectric: (A) dipoles randomly oriented without an external electric
field; (B) dipoles aligned with an electric field.
with an external electric field. This process is called dielectric polarization. It is illustrated
in Fig. 3.6A which shows permanent dipoles before an external electric field
is applied to the capacitor, and in Fig. 3.6B, which shows permanent dipoles after
an external electric field is applied to the capacitor. In the former case, there is no
voltage between the capacitor plates, and all dipoles are randomly oriented. After the
capacitor is charged, the dipoles will align with the electric field lines; however, thermal
agitation will prevent a complete alignment. Each dipole forms its own electric
field which is predominantly oppositely directed with the external electric field, E 0 .
Due to a combined effect of a large number of dipoles (E ′ ), the electric field in the
capacitor becomes weaker (E = E 0 + E ′ ) when the field, E 0 , would be in the capacitor
without the dielectric.
Reduced electric field leads to a smaller voltage across the capacitor: V = V 0 /κ.
Substituting it into formula (3.19), we get an expression for the capacitor with a
dielectric:
C = κ q V 0
= κC 0 . (3.22)
For the parallel-plate capacitor, we thus have
C = κε 0A
d . (3.23)
In a more general form, the capacitance between two objects may be expressed through
a geometry factor, G:
C = ε 0 κG, (3.24)
G depends on the shape of the objects (plates) and their separation. Table A.5 of the
Appendix gives the dielectric constants, κ, for various materials.
Dielectric constants must be specified for test frequency and temperature. Some
dielectrics have a very uniform dielectric constant over a broad frequency range (e.g.,