# 24.4 Dielectrics ```Dielectrics
Experiment: Place dielectrics between
plates of capacitor at
Q=const condition
Observation: potential difference decreases to smaller value with
dielectric material relative to air
Without dielectric: 𝐶0 =
With dielectric:
𝐶=
𝑄
𝑉0
𝑄
𝑉
Κ: =
𝑄 = 𝑐𝑜𝑛𝑠𝑡
𝐶
𝑉0
=
𝐶0 𝑉
because V&lt;V0
C&gt;C0
K&gt;1: relative dielectric constant
What happens with the E-field in the presence of dielectric material
𝑸 = 𝒄𝒐𝒏𝒔𝒕
𝐸0
𝑉0 𝐸0
We know V&lt;V0
E&lt;E0 specifically
𝐸=
Κ=
=
Κ
𝑉
𝐸
d
Recall: 𝐸 =
𝜎𝑛𝑒𝑡
𝜖0
𝜎𝑛𝑒𝑡 reduced with dielectric
material
The surface charge (density) σ on
conducting plates does not change but
induced charge σi of opposite sign
𝜎
𝐸0 =
𝜖0
and 𝐸 =
𝜎 − 𝜎𝑖
𝜖0
𝜎
𝐾𝜖0
1
𝜎𝑖 = 𝜎(1 − )
𝐾
𝐸=
𝜖 = 𝐾𝜖0
Definition of the
permittivity
𝐴
1
𝐶 = 𝜖 𝑑 and 𝑢 = 2 𝜖𝐸 2
DIELECTRICS
+Q
Example:
E0
E1
E2
K1
K2
d/2
d/2
V
-Q
𝐸0
𝐸1
𝐸2
𝜎
𝑄
=
=
𝜖0 𝜖0 𝐴
𝐸0
𝑄
=
=
𝐾1
𝜖0 𝐴𝐾1
𝐸0
𝑄
=
=
𝐾2
𝜖0 𝐴𝐾2
𝜎1 = 𝜎(1 −
1
)
𝐾1
𝑄2 𝑑
𝑈=
= 4𝜖 𝐴
0
𝜖0
1
= 4 (𝐸2 𝐾2 )2 𝐾 +
1
1
𝑄𝑉
2
𝑑
𝑑
+ 𝐸2
2
2
𝑄
𝑄
𝑑
𝑄𝑑 1
1
=
+
=
( + )
𝜖0 𝐴𝐾1 𝜖0 𝐴𝐾2 2 2𝜖0 𝐴 𝐾1 𝐾2
V = 𝐸1
𝑄
𝑄
2𝜖0 𝐴𝐾1 𝐾2
𝐶= =
=
𝑄𝑑 1
1
𝑉
( + ) 𝑑(𝐾1 + 𝐾2 )
2𝜖0 𝐴 𝐾1 𝐾2
1
𝜎2 = 𝜎(1 − )
𝐾2
1
𝐾1
1
𝐾2
1
𝑈
1
𝑄
+ 𝐾 𝑎𝑛𝑑 𝑢 = 𝐴𝑑 = 4𝜖 (𝐴 )2
2
0
1
𝐾1
1
𝜖
+ 𝐾 = 40 (𝐸1 𝐾1 )2
2
1
𝐾1
1
+𝐾
2
24.4 DIELECTRICS
Dielectric breakdown or Dielectric strength
High Voltage
Ground
Ground
Air
Cr2O3
GAUSS’S LAW IN DIELECTRICS
Recall:
𝑄𝑒𝑛𝑐𝑙
𝜖0
=
𝐸 ∙ 𝑑𝐴
𝐸≠0
𝐸=0
Dielectrics
Conductor
𝑄𝑒𝑛𝑐𝑙 = 𝜎 − 𝜎𝑖 𝐴
𝐸 ∙ 𝑑𝐴 = 𝐸𝐴
𝐸𝐴 =
𝜎 −𝜎𝑖
𝜎 − 𝜎𝑖 𝐴
𝜖0
1
𝜎𝑖 = 𝜎(1 − )
𝐾
𝐸𝐴 =
𝜎𝐴
𝐾𝜖0
𝑄𝑒𝑛𝑐𝑙−𝑓𝑟𝑒𝑒
=
𝜖0
A
A
A
𝐾𝐸 ∙ 𝑑𝐴
GAUSS’S LAW IN DIELECTRICS
𝑄𝑒𝑛𝑐𝑙−𝑓𝑟𝑒𝑒
=
𝜖0
Example:
K
E1
ra
r
rb
E2
𝑉=
𝑟𝑏
𝑟𝑎
𝑉=
𝑟𝑏
𝑟𝑎
𝑄1 (𝑟𝑏 − 𝑟𝑎 )
𝐸1 𝑑𝑟 =
2𝜖0 𝐾𝜋𝑟𝑎 𝑟𝑏
𝐸2 𝑑𝑟 =
𝑄2 (𝑟𝑏 − 𝑟𝑎 )
2𝜖0 𝜋𝑟𝑎 𝑟𝑏
𝐾𝐸 ∙ 𝑑𝐴
𝑄
=
𝜖0
𝐾𝐸 ∙ 𝑑𝐴 = 𝐾𝐸1 2𝜋𝑟 2 + 𝐸2 2𝜋𝑟 2
𝑄2
𝑄1
𝜖0
𝜖0
𝑄1
𝐸1 =
2𝜖0 𝐾𝜋𝑟 2
𝑄2
𝐸2 =
2𝜖0 𝜋𝑟 2
2𝜖0 𝐾𝜋𝑟𝑎 𝑟𝑏 𝑉
𝑄1 =
(𝑟𝑏 − 𝑟𝑎 )
𝑄2 =
2𝜖0 𝜋𝑟𝑎 𝑟𝑏 𝑉
(𝑟𝑏 − 𝑟𝑎 )
2𝜖0 𝜋𝑟𝑎 𝑟𝑏 𝑉
𝑄 = 𝑄1 + 𝑄2 =
(𝐾 + 1)
𝑟𝑏 − 𝑟𝑎
𝐶=
𝑄 2𝜖0 𝜋𝑟𝑎 𝑟𝑏 (𝐾 + 1)
=
𝑉
(𝑟𝑏 − 𝑟𝑎 )
MOLECULAR MODEL OF INDUCED CHARGE
CLICKER QUESTION
A conductor is an extreme case of a dielectric, since if an electric field
is applied to a conductor, charges are free to move within the
conductor to set up “induced charges”. What is the dielectric constant
of a perfect conductor?
A. K = 0
B. K = 
C. A value depends on the material of the conductor
E
E0

0 i
0

1
K
8
```