Direct Energy, 2018a

2 CAPACITORS AND PIEZOELECTRIC DEVICES 25
In Eq. 2.5, Q is the charge in coulombs, v is the voltage, and the constant of
proportionality is the capacitance C in farads. If we take the derivative with
respect to time, we get the more familiar expression relating the current
and voltage across the capacitor.
dQ
dt = i = C dv
dt
(2.6)
The capacitance of a capacitor is related to the permittivity of the
dielectric material between the conductors. Permittivity is a measure of the
amount of energy that can be stored by a dielectric material. As described
by Eq. 1.22, for a parallel plate capacitor this relationship is
C =
ɛA
d thick
(2.7)
where A is the area of the plates and d thick is the distance between the
plates. The energy E stored in a capacitor as a function of voltage applied
across it is given by
E = 1 2 Cv2 = 1 Qv. (2.8)
2
The capacitance of a vacuum-lled parallel plate capacitor is described by
Eq. 2.7 with permittivity ɛ = ɛ 0 , the permittivity of free space. As we
charge the capacitor, charges accumulate on the plates, and no change
occurs to the vacuum between the plates. If we replace the vacuum with
a dielectric with ɛ>ɛ 0 , the capacitance becomes larger. The dielectric
lled capacitor can store more energy, all else equal, because the dielectric
material changes as the capacitor charges. More specically, the material
polarizes. In an insulator, electrons are bound to their atoms, and current
cannot ow. Instead, the electrons in a dielectric move slightly with respect
to their nuclei while still staying bound to the atoms. Electrons are always
in motion for materials at temperatures above absolute zero, but when a
material polarizes, the net location of electrons with respect to the nuclei
changes. As the capacitor charges, the electrons are slightly displaced from
their atoms, balancing the charges on the plates, and more energy is stored
in the dielectric for a given voltage. We say that this process induces
electric dipoles. The larger the permittivity, ɛ, the more the materialcan
store energy by polarizing in this way. For this reason, capacitors are
often lled with dielectric materials like tantalum dioxide Ta 2 O 5 which has
ɛ =25ɛ 0 [18]. A materialwith ɛ =25ɛ 0 , for example, will be able to store
25 times the energy of an air lled capacitor of the same size with the same
applied voltage.