11.1 Basic Concepts and Formulae
463
ε 0 ϕ E = q
(11.29)
ε 0
E · ds = q
(Gauss’ law)
(11.30)
Capacitors in series
1
c
=
n
1
c n
(11.31)
Capacitors in parallel
C =
n
C n
(11.32)
Energy of a charged capacitor
W =
1
2
qV =
1
2
q 2
C
=
1
2
C V
2
(11.33)
Parallel Plate Capacitor with Dielectric
If a dielectric slab of thickness t and dielectric constant K is introduced in a parallel
plate air capacitor, whose plates have area A and are separated by a distance d, the
capacitance becomes
C =
ε 0 A
d − t
1 −
1
K
(11.34)
If a metal of thickness t is introduced in the air capacitor, the effective distance
between the plates is reduced and the capacitance becomes
C =
ε 0 A
d − t
(11.35)
Energy Loss in the Combined System of Capacitors
If the positive end of a capacitor of capacitance C charged to potential difference
V is connected in parallel with the positive end of the capacitor of capacitance C
charged to potential difference V , then common potential difference will be
V =
C 1 V 1 + C 2 V 2
C 1 + C 2
(11.36)
and the energy loss will be
W =
1
2
C 1 C 2
C 1 + C 2
(V 1 − V 2 )
2
(11.37)