Biomedical Engineering Reference
In-Depth Information
energy, there is also a difference in electric potential between two locations. In the
equation form, the electric potential difference is
dW
dQ
ΔΦ =
E
ch
is determined by the nature of the reactants and electrolytes, not by the
size of the cell or amounts of material in it. As the charge builds up in the charging
process, each successive element of charge
dQ
ch
requires more work to force it onto
the positive plate. In other words, charge and potential difference are interdepend-
ent. Summing these continuously changing quantities requires an integral.
ΔΦ
Q
ch
∫
W
=ΔΦ
dQ
(3.29)
E
ch
0
Substituting (3.16) into the above equation and integrating gives
1
2
Q
2
C
2
W
=
Ch
or
W
=
m
ΔΦ
(3.30)
E
E
C
2
m
The only place a parallel plate capacitor could store energy is in the electric
field generated between the plates. This insight allows energy (rather the energy
density) calculation of an electric field. The electric field between the plates is ap-
proximately uniform and of magnitude,
E
, which is
s/e
0
,
where
Q
Ch
/A
) is the
surface charge density in C/cm
2
. The electric field elsewhere is approximately zero.
The potential difference between the plates is
σ
(
=
(3.31)
ΔΦ =
Ed
where
d
is the distance between the plates. The energy stored in the capacitor is
written as
C
ε
AE d
2
(3.32)
2
W
=
m
Φ=
0
E
2
2
Ad
is the volume of the field-filled region between the plates, so if the energy
is stored in the electric field then the energy per unit volume, or
energy density
, of
the field is given by
Ad
ε
W
E
2
(3.33)
w
==
0
2
E
