Intro Physics

11bLecture 7

DipolesDielectrics

What We Will Do Today

Dipoles

Dielectrics and Capacitors

Towards Computing!

DipolesThe total charge is not the only thing

that mattersThe distribution of charge also matters

Let us define a simple (primary) dipole as consisting of two opposite charges q separated by distance d

Total charge is 0But dipole is not electrically inert!Dipole moment p

Vector direction is from negative to positive charges

qdp =|| r

q

-q

pd

∫= dVxxp )(ρrr

Moment of inertiaDipole moment p

Vector direction is from negative to positive charges

(This equation is only meant to remind you of I from your mechanics class! We won’t actually use it)

Like moment of inertia

qdp =|| r

q

-q

pd

∫= dVxxp )(ρrrm

m

Id

Torque on DipolesEven though neutral, the dipole is

sensitive to electric fieldsLet’s put a third charge Q (positive)

along bisector+q feels outward push, -q feels inward

Net force is smallApproaches 0 as Q becomes far away compared to d

The tangential components, though, result in substantial torque!

Does not disappear as Q goes further away

Dipole would like to spin

q

-q

pQ

Torque on DipolesCompare dipole direction p to

local field direction E(xp)If field is constant over

small region near dipoleIf perpendicular to field

Torque as we just sawIf aligned with field

No total forceIf anti-aligned with field

No total force, but unstable

)( pxEprrr ×=τ

+-

Total force on dipoleIf field is constant

No net force on dipolePositive, negative charges pulled in opposite directions

But held together

+ -

+ -

If field not constantOne end of dipole may feel

stronger force than the other endDipole can be sucked into or

pushed out of region

Total Force on DipoleHow to measure change

of field across ends?Derivative

+ -

+ -

[ ]

xEp

xExq

xExxq

xExxxExEq

xExEqF

∂∂

=∂∂

∆=∂∂

−=

⎥⎦⎤

⎢⎣⎡

⎭⎬⎫

⎩⎨⎧

∂∂

−+−=

−=∆

−+

+−++

−+

)()(

)()()(

)()(

xx EpF ∇⋅=rr

Don’t worry about the details of this formula. It’s for cultural purposes only. The important point is to realize that the dipole is pulled towards regions of larger field

Induced DipolesA charge can induce a dipole in an

originally neutral, dipole-less objectAtoms

Negative charge can repel electrons, attract proton

Electrons re-arrange themselves around nucleus

0)( == ∫ dVxxp ρrr

-

∫= dVxxp )(ρrr

Result: net dipole momentField has gradient

Field of single charge Neutral atom attracted to single charge!

Will also work if single charge is neutral dipoleOrigin of van der Waals forces

(reminder formula, don’t memorize!)

Dipole ForceSuppose a molecule is “polarizable”

It develops dipole moment p that is proportional to E-field, with coefficient ε

What is the force between a single fixed charge Q and this molecule, as function of separation R?

+-

Q p=εE

R

Dielectric BreakdownIf field not constant

One end of dipole may feel stronger force than the other endDipole can be sucked into or pushed out of region

If field very strong and very rapidly varying

Tidal force may result Forces stronger than atomic bond between the charges“Dielectric breakdown”

aka “sparking”~10,000 V / cm in air

+ -

Potential EnergyIn constant E-field of

strength EWhat is potential energy

of dipole with strength p and angle φ, shown at right?

Careful with signss+

- φ

θτ sinpE=θτrr ddU ⋅= =⎟⎟

⎠

⎞⎜⎜⎝

⎛−−=−= ∫

φ

θθ0

sin dpEWU

)cos1( φ−pEp x E into page

θ vector out of page (RH rule)

DielectricsInsert molecule between

capacitor platesConstant E-field

If the molecule highly polarAlign themselves to field

If molecule highly polarizableInduced dipole moments throughout materialAtomic “cloud” of electrons distortedBut atoms themselves stay in place

Recall simple hydrogen atom in E-field of charge

∫= dVxxp )(ρrr

+-

)( pxEprrr ×=τ

MaterialsMaterials consist of many

moleculesLiquids: randomly arranged and oriented

Solids: arranged in arrays

+ -+-

+-

+-

+-

+-

+-

+-+-+

-

+-+-+

-

+-

+-

Dielectric MaterialsConsider now arrays of

molecules after aligning themselves

Rotatable dipoles (liquid)Centers fixed (solid)

Two things to noteDipole moments (induced or aligned) all point along field directionDipoles line up head-to-tail

+-+-+-+-+-

+-+-+-+-+-

+-+-+-+-+-

Head-to-Tail Dipole ChainsHead of one cancels tail of nextUltimately left only with head on

one end, tail on otherIf dipole has length d

Each dipole is qdOne head, one tail separated by ndDipole moment qndSame as adding up all the dipole moments n(qd)

Dipole moments arranged head to tail simply sum upActs simply as if charges are at

the ends of the dipole chainsSometimes the end charges are called “bound charge”

qd+-

qd+-

qd+-

qd+-

qd+-

qnd+-

Dielectric ConstantLarge dipole now induced in the original

field E0Induced electric field Einduced opposes the original, imposed field!

(E-field of dipole points opposite the dipole vector)Induced field is smaller than original, imposed field

Strength of induced dipole depends on material

Can in principle be calculated in quantum mechanics

Field E’ in internal region sum of Original field E0Induced field Eind

+-+-+-+-+-

+-+-+-+-+-

+-+-+-+-+-

KEE 0'r

r=

E0

Dielectric constant

+++++

-----E0

E0

Einduced

Dielectric Constant, Capacitance

Insert dielectric material (K>1) between capacitor plates

Induce dipole momentEquivalent to induced charge on faces of sheet

Head-to-tail chainsReduced E-field

E’=E/KReduced Voltage (work / charge)

∆V’=E’d=Ed/K=∆V/KSame charge Q

C’=Q’/∆V’=Q/(∆V/K)=KQ/∆V=KCCapacitance increased by factor K!

All actual capacitors built with a dielectric layerDefine permittivity Kε0=ε

+Q -QK=2

dA

dAKC εε == 0

Making ComputersA capacitor stores chargeYou could imagine

A capacitor with charge on it = stored value of 1A capacitor without charge = stored value of 0

This is basically how RAM is madeRecall Q=C∆V

For fixed ∆V, larger capacitance means the stored charge is larger

This means the stored signal is larger (and lasts longer)

For computer memory, would like large CC=Kε0A/d

Also want small ACompromise

Use small A, large K!Can find special materials with K>>1000

Lets you shrink your capacitor by 1000

Voltage, Potential, Work, Capacitance, and Fields

E-Field(N/Coulomb or Volts/m)

Electric Potential

(Volts; J/Coulomb)

Potential Energy(Joules or N.m or eV)

Force(Newtons)

Charges(Coulomb)

Integrate E.x over path

Gradient

Integrate F.x over path

GradientEqFrr

=VqU ∆=∆

iesch i

i rr

qE ˆ4arg

20

∑=πε

r

∑esch i

i

rq

arg 04πεGauss’ Law

Voltage(Volts; J / C)

BAABV Φ−Φ=

Q=C∆V

Coulomb’s Law

∫ =⋅=Φ0

)(εenclQadEflux rr

Dipole(Coulomb.m)

qdp =

Φ

U F

E

p

∆V q

SummaryDipoles

Dielectrics and CapacitorsHead-to-tail dipolesPermittivityDielectric Capacitors

Demo: Large variable capacitor

∑= pP rr

0εε K=

dA

dAKC εε == 0

qdp =|| r )( pxEprrr ×=τ xx EpF ∇⋅=

rr

)cos1( φ−pE