Dielectric materials

Question 1

Electric dipole

Answer 1

pair of electric charges separated by some distance and having equal & opposite magnitudes.
p = ql

Question 2

What acts on a dipole?

Answer 2

Force: F = p grad(E)
Torque: T = p x E
Energy: E = -pE

Question 3

What is a dielectric material?

Answer 3

Electrical insulator that can be polarized by an applied E-field

Question 4

What is the polarization?

Answer 4

Microscopic charge displacement, charges assume a certain direction as response to an applied E-field.
Colleciton of those dipoles we call polarization

Question 5

Dielectric constant

Answer 5

Factor by which the E-field between two charges in a material is increased relative to vacuum.

Question 6

Macro vs. micro polarization

Answer 6

Macro: P = N<p>dipoles with density N and average dipole moment <p> (which is a micro quantity!!)

Micro: electronic, ionic and dipolar polarization

Question 7

Induced polarization

Answer 7

Emerging of a net dipole moment in the presence of an E-field called E_loc which can differ from the macroscopic E (which is more an average quantity)

Question 8

Consider an uniformly polarized infinite slab: describe in terms of polarization

Answer 8

The only uncompensated charges are the one at the very top/bottom of the slab –> equivalent to two opposite sheets of surface charges.

Question 9

Polarization outside and inside infinite parallel slab

Answer 9

Outside: E-field = 0
Inside: E_pol_in = -P/ε0

Question 10

Ouroboros of polarization P and E-field

Answer 10

P depends on E as P =εϗE
E depends on P as E = E_ext + E_pol
–> E_pol_in acts against E_ext

Question 10

What happens to the capacitance when a dielectric material is inserted in between the metal plates?

Answer 10

The capacitance is increased by a factor of C’ = C0εr.

Question 11

Dielectric capacitor: two cases

Answer 11

• For V constant: the free charges Q on the metal plate must increase to maintain the same E_field = E_ext + E_pol_in (the last acts against E_ext!)
• For Q constant: E decreases inside the capacitor because of screening provided by the charge density due to dielectric, thus also the V decreases.

Question 12

Connecting macro and micro polarization for gases

Answer 12

E = E_loc then we obtain ε = 1 + ϗ = 1+Nα/ε0

Question 13

Lorentz model for solids

Answer 13

E_loc = E - E_continuum_sphere + E_discrete_sphere –> E_loc = E+P/3ε0 = (εr+2)/3

We obtain the Clausius Mosotti eq: εr-1/εr+2 = Nα/3ε0 which links macroscopic measurable quantity permettivity ε wih microscopic quantity electrical, ionic and dipolar polarizability

Question 14

Dielectric breakdown

Answer 14

when current flow through an electrical insulator; the voltage as which the insulator becomes conductive is called breakdown voltage

Question 15

Examples of breakdown

Answer 15

• In gases: plasma
• In liquids: triggered by bubbles or impurities
• In solids: breakdown preceeded by long time partial discharge
• Arc discharge: breakdown on gas environment that produces an ongoing electrical discharge
• Corona discharge ≠ breakdown, it occurs in the vicinity of sharp corners and is characterized by current from electrode at high potential towards a neutral fluid –> formation of plasma

Question 16

How do we get that electronic polarizability αe is a complex function?

Answer 16

Oscillating E field as driving force –> put into equation of motion given by classical oscillator model –> we find α(ω) = -eR(ω)/E_0 –> α = α’ + α’’

Question 17

What tells us the frequency dependence of the relative permettivity?

Answer 17

We suppose that E = E_loc as in gases, then is εr = 1+Nα/ε but we know that α = α’ + α’’ –> also εr = εr’ + iεr’’
εr’ = in-phase response –> energy stored in the system
εr’’ = out-of-phase response –> absorption –> dissipation

Question 17

What is the resonance frequency?

Answer 17

Nearby this frequency, the relative permettivity varies a lot, this permit to polarize more a material.
For electronic polarization we speak about electronic resonance frequency, for ionic polarization is called “Debye frequency” = Phonon frequency

Debye freq &laquo_space;Electronic resonance freq.

Question 18

What is different from the dipolar polarization frequency dependence?

Answer 18

There is no characteristic energy ℏω at which the dipole would absorb energy more efficiently , i.e. there is no resonance behavior!
Once the E-field is removed, the dipoles take a characteristic time to change their orientation, which happens through scattering between dipoles themselves.