CAPACITANCE, DIELECTRICS AND ELECTRIC ENERGY STORAGE
CAPACITANCE
A capacitor is a device that can store electric charge and usually
consists two conducting objects.
It’s principle is so simple that even these two ugly charges
form a capacitor.
PARALLEL PLATE CAPACITOR
When a capacitor is charged, its plates have equal but opposite sign of
charges, +Q, -Q.
Q = C.V
Capacitance
C: How much charge must be put on the plates
to produce potential difference of 1V
Q
C=
V
Only depends on the
geometry of the capacitor
The SI unit of capacitance: FARAD
1 Farad = 1F = 1 Coulomb per Volt
Charging a Capacitor
One way is to place it in an electric circuit with a battery.
Battery: supplies a certain potential difference between Its terminals
Schematic diagram
Battery maintains potential
difference V.
CALCULATING CAPACITANCE
Q = C .V
1. Assume charge Q on the plates
2. Calculate Electric Field ( We can use Gauss’ Law for simple
geometries)
3. Calculate Potential Difference V
r r
∆V = Vb − Va = − ∫ E.dl
b
a
PARALLEL PLATE CAPACITOR
E-field from Gauss’ Law:
Electric Potential
r r
∆V = Vb − Va = − ∫ E.dl
b
a
CYLINDRICAL CAPACITOR
r r
∆V = Vb − Va = − ∫ E.dl
b
a
A SPHERICAL CAPACITOR
r r
∆V = Vb − Va = − ∫ E.dl
b
a
AN ISOLATED SPHERE
CAPACITORS IN PARALLEL
Q = Ceq .V
Each plate acquires charge:
Total charge Q from the battery:
Q1=C1V, Q2=C2V, Q3=C3V
Q=Q1 +Q2+Q3=C1V+C2V+C3V
=(C1+C2+C3)V
Ceq=C1+C2+C3
CAPACITORS IN SERIES
Q = Ceq .V
V is divided into three Parts :
Q
V1 =
C1
1
1
1
V = V1 + V2 + V3 = Q ( +
+ )
C1 C 2 C3
Q
V=
C eq
Q
V2 =
C2
Q
V3 =
C3
1
1
1
1
=
+
+
Ceq C1 C 2 C3
COMBINATION
ELECTRIC ENERGY STORAGE
• Battery must do work to charge or capacitor.
• That work is equal to the energy stored in the
capacitor
Assume charge q has been transferred to the capacitor then potential V
is formed between the plates
q
V =
C
If an extra increment of charge dq is transferred
Then battery does dW extra work to move the charge dq
dW = dqV
q
dW = dq
C
q
V =
C
dW = dqV
W = ∫ dW
Total work the battery perform to charge
Q
1Q
W = ∫ dW = ∫ qdq =
2 C
0
2
1
C
Q2
U=
2C
as
The potential energy stored in the capacitor
Q = CV
1
2
U = CV
2
Q =C V
2
2
2
1 C 2V 2
U=
2 C
Another form of U
Energy Density
In a parallel plate capacitor energy density:
U
u=
volume
U
CV 2
u=
=
Ad 2 Ad
As we found
V
E=
d
A
C = eo
d
1
2
u = eoE
2
1 V 2
u = eo ( )
2
d
Energy/Volume
Dielectrics
dielectrics
An insulating sheet of
material between the plates
• They break down less (higher voltage can be applied)
• Plates can be more close (Higher C values)
• Dielectric increases capacitance by a factor of K.
C = K .C air
Dielectric constant
Parallel plate capacitor
A
C = Ke o
d
In general
In air
1 Q
E=
4pe o r 2
1 Q
E=
4pKe o r 2
e = K .e o
Permittivity
• Dielectric weakens the electric field
In dielectric
AN ATOMIC VIEW OF DIELECTRICS
• Some materials have permanent dipole moments which
line up with Electric field.
• In some materials dipole moments induced by extend
electric field.
In both cases, dipole moments line up with E-field and weaken it.