4
Electrochemical Supercapacitors for Energy Storage and Delivery
where F represents the magnitude of electrostatic force present between
the two charges; q + and q – are the magnitudes of the positive and negative
charges; r is the distance between these two charges; ε 0 is the dielectric constant of the an empty vacuum; and the negative sign represents the force
as attractive rather than repelling. The dielectric constant is an important
parameter for a capacitor and will be discussed in detail in Section 1.3.1. Note
that the charge magnitude in Equation (1.1) can be quantized by n values of
the elementary charge of an electron, allowing charge q to be written as
q = ne
(1.2)
where n = ±1, ±2, ±3, etc., and e is the elementary charge constant equal to
1.602 × 10 –19 C.
1.2.2 Electric Field and Potential
A charge, for example, the positive charge (q + ) in Figure  1.2, can emit an
electric flux on the surrounding area to form an electric field. Figure 1.2 is
helpful for visualizing the electric field as lines of force that radiate outward
(positive charge) or inward (negative charge). The strength of this field can
be felt by the negative charge, and is expressed as
F
E =
q −
(1.3)
where E is the electric field strength of the positive charge q + . Combining
Equations (1.1) and (1.3), the electric field strength can be expressed alternatively as
q q
1
q
E = −
+ −
2
=
+
4πε 0 r (−q − ) 4πε 0 r
2
(1.4)
The corresponding electric potential (V q+ ) at the position of the negative charge
can be treated as the work done by moving the negative charge from its position through a distance r toward the positive charge and is expressed as
V q+ = Er
(1.5)
By combining Equations (1.4) and (1.5), the electric potential induced by the
positive charge can be expressed as
q
V =
+
q+
4πε 0 r
(1.6)