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:

With dielectric: 𝐶=𝑄𝑉

𝑄=𝑐𝑜𝑛𝑠𝑡 because V<V0

C>C0

Κ := 𝐶𝐶0

=𝑉 0

𝑉 K>1: relative dielectric constant

d

What happens with the E-field in the presence of dielectric material𝑸=𝒄𝒐𝒏𝒔𝒕We know V<V0

E<E0 specifically Κ=𝑉 0

𝑉 =𝐸0

𝐸 𝐸=𝐸0

Κ

Recall:

𝐸0=𝜎𝜖0

𝐸=𝜎−𝜎 𝑖

𝜖0and

𝜎 𝑖=𝜎 (1− 1𝐾 ) 𝐸=𝜎𝐾 𝜖0

𝜖=𝐾 𝜖0 Definition of the permittivity

and

The surface charge (density) σ on conducting plates does not change butinduced charge σi of opposite sign

𝜎 𝑛𝑒𝑡reduced with dielectric material

DIELECTRICSExample: K1

K2

d/2d/2

+Q

-Q

E0E1E2

‖𝐸0‖=𝜎𝜖0

=𝑄𝜖0 𝐴

‖𝐸1‖=‖𝐸0‖𝐾1

= 𝑄𝜖0𝐴 𝐾1

‖𝐸2‖=‖𝐸0‖𝐾2

= 𝑄𝜖0 𝐴𝐾 2

V

𝐶=𝑄𝑉 = 𝑄

𝑄𝑑2𝜖0 𝐴

( 1𝐾 1+ 1𝐾 2

)=2𝜖0 𝐴𝐾 1𝐾 2

𝑑 (𝐾 1+𝐾2)

𝜎 1=𝜎 (1− 1𝐾1

) 𝜎 2=𝜎 (1− 1𝐾 2

)

==

24.4 DIELECTRICSDielectric breakdown or Dielectric strength

Cr2 O3

Ground GroundHigh Voltage

Air

GAUSS’S LAW IN DIELECTRICSRecall:

Conductor Dielectrics�⃗�=0 �⃗�≠0

𝜎−𝜎 𝑖

𝑄𝑒𝑛𝑐𝑙=(𝜎−𝜎 𝑖 ) 𝐴

∮𝐸 ∙𝑑𝐴=𝐸𝐴

AA

A

𝐸𝐴=(𝜎−𝜎 𝑖 ) 𝐴  

𝜖0

𝜎 𝑖=𝜎 (1− 1𝐾 )

𝐸𝐴=𝜎 𝐴𝐾 𝜖0

𝑄𝑒𝑛𝑐𝑙− 𝑓𝑟𝑒𝑒

𝜖0=∮𝐾 𝐸 ∙ �⃗�𝐴

GAUSS’S LAW IN DIELECTRICSExample:Capacitance of half filled spherical capacitor

Kra

rbr

𝑄𝑒𝑛𝑐𝑙− 𝑓𝑟𝑒𝑒

𝜖0=∮𝐾 𝐸 ∙ �⃗�𝐴

𝑄𝜖0

=∮ 𝐾 �⃗� ∙ �⃗�𝐴=𝐾 𝐸12𝜋𝑟2+𝐸22𝜋𝑟 2E1

E2

𝑄1

𝜖0𝑄2

𝜖0𝐸1=

𝑄1

2𝜖0𝐾 𝜋𝑟 2

𝐸2=𝑄2

2𝜖0 𝜋𝑟2

𝑉=∫𝑟 𝑎

𝑟 𝑏

𝐸1𝑑𝑟=𝑄1(𝑟𝑏−𝑟 𝑎)2𝜖0𝐾 𝜋 𝑟𝑎𝑟 𝑏

❑⇒𝑄1=¿

2𝜖0𝐾 𝜋𝑟 𝑎𝑟 𝑏𝑉(𝑟 𝑏−𝑟𝑎)

¿

𝑉=∫𝑟 𝑎

𝑟 𝑏

𝐸2𝑑𝑟=𝑄2(𝑟 𝑏−𝑟𝑎)2𝜖0 𝜋𝑟𝑎 𝑟𝑏

❑⇒𝑄2=

2𝜖0 𝜋𝑟𝑎𝑟𝑏𝑉(𝑟 𝑏− 𝑟𝑎)

𝑄=𝑄1+𝑄2=2𝜖0𝜋 𝑟𝑎𝑟 𝑏𝑉

(𝑟𝑏−𝑟 𝑎)(𝐾 +1)

𝐶=𝑄𝑉 =

2𝜖0𝜋𝑟 𝑎𝑟 𝑏(𝐾 +1)(𝑟 𝑏−𝑟𝑎)

Check: K->1 needs to reproduce empty =

MOLECULAR MODEL OF INDUCED CHARGE

EE

8

CLICKER QUESTIONA 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

0

0 0

1iEE K