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UIUC 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede LECTURE NOTES 10

The Macroscopic Electric Inside a

When we discuss electric (and/or magnetic) fields, whether they are outside of/exterior to matter, or inside the matter itself, implicitly, we physically interpret these field quantities to be associated with macroscopic averages over (vast) numbers of electromagnetic quanta (i.e. virtual ), , , electric charges (both +ve and –ve) etc. The “true” E & B-fields inside of matter - at the atomic scale - are wildly varying from point to point (and also wildly varying in time, e.g. on short/atomic time-scales due to fluctuation(s) in thermal at finite temperature). For almost all applications that we are interested in, we are not concerned with these wild spatial (and temporal) fluctuations on the atomic scale; we are primarily concerned with the average / mean fields extant in these media, suitably averaged over large numbers of constituent particles involved. These (space and time-averaged) fluctuations die out as 1 N where N is the 23 number of constituents involved. If N 10 , then sinceσ N = N , then for random fluctuations −12 (i.e. Gaussian-distributed) the fractional fluctuations, σ N NNNN==13.210 × are extremely small – essentially negligible! Hence the macroscopic (i.e. microscopically averaged- over) E-field can be seen as being truly electrostatic, for so-called time-independent situations.

Suppose we want to calculate the macroscopic Er( ) at some point, r inside a dielectric sphere of radius, R as shown in the figure below. zˆ

Field point, P R r yˆ

O

Small imaginary sphere of radius δ centered on the field point, P @ xˆ | r | < R (for averaging purposes)

The macroscopic electric field at the field point P @ r inside the sphere consists of two parts: – A contribution from the average electric field Erout ( ) due to electric charges outside / external to a small imaginary sphere (of radius δ R ) centered on the point P, and: – A contribution from the average electric field Erin ( ) due to electric charges inside this small conceptual sphere. In other words, the macroscopic electric field at the field point P located at r (inside the dielectric sphere, i.e. rR< ), using the Principle of Linear Superposition is: Er()=+ Eout( r) E in ( r)

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 1 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede In Griffith’s problem 3.41(d), we learned that the electric field averaged over an imaginary sphere due to a single charge q outside of/exterior to the imaginary sphere was the same as the electric field due to the charge q, as observed at the center of that imaginary sphere. By the principle of superposition, this result then holds for any collection of exterior charges.

Thus, here for our dielectric sphere of radius R, Er( ) (with rR< ) is the electric field at out r due to the electric contained within the dielectric sphere of radius R that are outside of/exterior to the imaginary/conceptual sphere of radius δ centered on r .

Outside of/excluding the region of this small imaginary sphere of radius δ centered on the field point P @ | r | < R, the atomic/molecular electric dipoles are far enough away from the field point P that we may safely write the potential Vrout ( ) corresponding to Erout ( ) (with rR< ) as: 1 rˆ⋅Ρ(r′) Vr==−=− dτ ′, rr′′ , rr out () ∫ 2 rr 4πε o outside r where the is over the volume of the dielectric sphere, but excluding the small volume associated with the small imaginary sphere of radius δ centered on the field point P @ | r | < R.

The electric dipoles inside the small conceptual/imaginary sphere of radius δ centered on the field-point P @ r are too close to treat in this fashion.

However, in Griffith’s problem 3.41(a-c), we also learned that the average electric field inside a sphere of radius δ due to all of the contained within the sphere of radius δ (regardless of the details of the charge distribution within that sphere) is: 1 p E =− ave 4πεδ3 0 where p is the total electric moment of that sphere.

Thus, we know that we know that the average electric field @ r within the small conceptual / imaginary sphere of radius δ centered on the field-point P @ r must be: 1 p Erin ()=− 3 4πεδ0 where p()r is the total/net macroscopic associated with the (microscopic) electric dipoles contained within this conceptual/imaginary sphere centered on the field point P @ r : ⎛⎞Volume of conceptual / p rr=Ρ* =Ρ r44πδ33 = πδ Ρ r () ()⎜⎟4 3 ()33() ⎝⎠imaginary sphere, 3 πδ where Ρ()r = macroscopic electric = electric dipole moment per unit volume (@ r ). 4 3 Thus: pr()=Ρ( r)( 3 πδ ) 4 πδ3 Ρ r 11pr() ( 3 / ) () 1 And thus: Er=− =− =− Ρ r in () 3 3 ( ) 44πε00 δ/ πε/ δ 3ε 0

2 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede Thus we obtain: 1 Erin ()=− Ρ() r 3ε 0

Now because of the (infinitesimal) size of the conceptual/imaginary sphere of radius δ R centered on the field point P @ r , then the (macroscopic) electric polarization Ρ()r should not vary (on average) significantly over this small volume, thus the term/contribution that was left out of the integral for the outside potential : 1 rˆ⋅Ρ(r′) Vr==−=− dτ ′, rr′′ , rr out () ∫ 2 rr 4πε o outside r actually corresponds to that associated with the electric field at the center of a uniformly polarized dielectric sphere of radius δ , which is −Ρ13ε 0 (r ) !!! {see/read Griffiths Example 4.2 and/or Prof. S. Errede’s P435 Lect. Notes 9, p. 25-26}.

1 But this is precisely what the electric field Erin ()=− Ρ() r puts back in!!! 3ε 0 In other words, using the principle of superposition: VrVrVrToT( ) =+ out( ) in ( ) 1 and thus: Erin()=−∇ Vr in () =− Ρ() r 3ε 0 rˆ⋅Ρ(r′) Thus we see that Vr= dτ ′ works fine for the entire dielectric!!! Tot () ∫ 2 whole r volume v′

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 3 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede The Macroscopic Electric Field Due to Near Dipoles in a Polarized Dielectric

Consider a very large block of polarized dielectric (e.g. polarized by a uniform external E field, e.g. EEx= ext ˆ Imagine a small spherical volume of radius δ ~1 cm deep within the polarized ext o dielectric. The electric polarizationΡ inside the dielectric will then be uniform e.g. Ρ=Ρ xˆ and o int Eint inside the dielectric will also uniform, EExint= o ˆ

Imagine “excising” this small spherical volume from the polarized dielectric – but still having it precisely/magically retain all of its EM properties as they were when it was part of the polarized dielectric. By itself, it will appear as shown below:

Mathematically & physically, note that this situation here is equivalent to two overlapping spheres, 4 3 one with uniform volume ρ+ =+Q 3 πδ and another sphere with uniform volume 4 3 charge density ρ− =−Q 3 πδ whose centers are offset from each other by a distance d δ ()dm 1Å= 10 −10 . Thus equivalently, this sphere now has only a bound densityσ Bo()ξσ= cos () ξwhere the angle ξ is measured with respect to the +xˆ axis. Thus a uniformly polarized dielectric sphere of radius δ with uniform polarization Ρ=Ρo xˆ is equivalent to two uniformly oppositely charged spheres whose centroids are displaced from each other by a distance d δ . See figures on the immediately following page:

4 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede GREATLY EXAGGERATED PIX:

⎛⎞What is the E -field @ the center of this polarized dielectric sphere?

⎜⎟ ⎝⎠= E -field due to the near dipoles inside the polarized dielectric!!!

We know that for a single, uniformly electrically charged sphere (volume charge density ρ = constant), that the electric field inside such a single sphere is given (from Gauss’ Law) by 1 Qencl Erinside ()<=δ 2 rˆ 4πε 0 r where r is defined from center of that sphere.

4 3 But the charge enclosed by the of radius r (r < δ ) is QVrencl ==ρ *.ρπ3 Noting that the total charge contained in a single uniformly charged sphere is QV==ρρ*,4 πδ 3 or ρ = Q 4 πδ 3 , then we can rewrite Er< δ as: Tot11 Tot 3 Tot1 3 inside ( ) 4 3 11Q ρ π r 11QTot ⎛⎞r Er<=δ encl rˆ = 3 rrrˆˆ==ρ 1 r ˆ inside () 2 2 2 ⎜⎟ 4πε 0 r 4 π ε 0 r 34ε 00πε δ⎝⎠ δ

Radius δ of uniformly charged sphere

Gaussian surface of radius r.

4 3 ρ = QTot 3 πδ 1

Now for two oppositely-charged spheres of uniform charge density ρ± whose centroids are laterally displaced from each other by an infinitesimal distance dm10−10 δ ~ 1 cm the net / total E -field at the center of the two overlapping spheres (by the principle of linear superposition) is: Tot ρρ+−11 EErErinside=+=+ inside() inside () ρ+ r+−−ρ r 33εε00 where ρ =±Q 4 πδ 3 and where the vectors r and r are defined in the figures shown below: ± Tot1 3 + −

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 5 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede

11 Define: ρ ≡=−ρρρ , then Thus: ErrdTot =−=−ρ ρ +− inside ()+− 33εε00 =−d Thus the E -field at center of two overlapping oppositely-but-uniformly charged spheres whose centroids are laterally displaced from each other by an infinitesimally small distance δδ∼10−10 mcm 1 and ddx= ˆ, is:

Tot 11 Eddxinside =−ρρ =− ˆ for ddx= ˆ. 33εε00

But ρ = Q 4 πδ 3 and p = Qd= total dipole moment of polarized dielectric sphere. Tot1 3 Tot1 Qd 11QTot 1 ( ToT1 ) EdxTot =−ρ ˆ =− 1 dx ˆˆˆ=− x but p = Q dx inside 4 33 ToT1 3ε 0 3ε 0 3 πδ4 πε0 δ Tot 1 p ∴ For ()rE<=−δ : inside 3 . 4πεδ0 We now drop the “Tot” superscript, since this simply referred to our (equivalent) model of the polarized dielectric sphere as the superposition of two uniformly-but-oppositely-electrically- charged spheres displaced by an infinitesimal distance δδ∼10−10 mcm 1 .

The electric polarization Ρ = electric dipole moment per unit volume = p 4 πδ 3 3 4433 ⇒=Ρp 33πδπδ = Ρ 11p 4 π δ 3 Ρ 1 1 ∴ For rE<=−=−δ : 3 = −Ρ ⇒ E =− Ρ ()inside 3 3 inside 4πε0 δ 4 π ε 0 δ 3ε 0 3ε 0

This is the macroscopically averaged E -field at the center of/inside an imaginary/conceptual small diameter sphere of radius δ (somewhere) deep inside of a uniformly polarized dielectric.

Note that this E -field arises solely from the contributions of the near dipoles in the dielectric within this sphere of radius δ . Note further that it explicitly does NOT include the externally applied electric field (that was used to polarize the block of dielectric in the first place).

This E -field DOES NOT include ANY contributions from electric dipoles (or anything else) EXTERIOR to this imaginary/conceptual sphere!

6 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede

THE MACROSCOPIC χe OF A DIELECTRIC

(Lossless) (in Eext) (uniform, no voids) (rotationally invariant → e.g. not a crystalline material) (i.e. amorphous) a.k.a. "Class A " Dielectric For an "ideal", linear, homogeneous & isotropic dielectric the electric polarization (a.k.a. the electric dipole moment per unit volume) Ρ is simply related to the internal electric field, Eint of the dielectric, by a simple proportionality constant, i.e.

Ρ=()rmEr int () Ρ(r ) m = slope of straight line

m = simple constant Eint (r ) (i.e. m = scalar quantity) n.b. This relation is ONLY true for CLASS A - i.e. ones which are linear, homogenous, ideal and isotropic. (We will discuss modifications to this relation shortly…)

Now: SI units of Ρ r : () meter2 SI units of Er : Newtons () Ρ()r Coulombs meter22 Coulombs ⇒ m has SI units of m == = 2 Define: m ≡ ε 0 χe Erint () Coulombs Newton-m 2 −12 Coulombs whereε 0 = (macroscopic) electric of free space () =×8.85 10 2 Newton-m = Farads/m and the (macroscopic) electric susceptibility of the dielectric material, χe is a pure number

(i.e. χe is a scalar quantity – it is dimensionless).

⎛⎞Coulombs ⎛⎞ Coulombs 2 ⎛⎞Newtons Then: Ρ=()rEr ⎜⎟2 εχ0 eint() ⎜⎟*⎜⎟ ⎝⎠ meter ⎜⎟Newton -m2 Coulomb ⎝⎠⎝⎠ = Coulombs m2 For class-A dielectrics: Ρ=()rErεχ0 eint( )

For free space (“empty” vacuum), the (macroscopic) electric susceptibilityχe = 0 because free space/vacuum has no MATTER in it.

The electric susceptibility χe and electric polarization Ρ(r ) explicitly refer to the dielectric properties of matter (and not the underlying/inter-penetrating vacuum). By the principle of linear superposition, the dielectric properties of matter and vacuum are additive to / independent of each other, thus we can define the (total) electric permittivity associated with a block of “Class-A” type dielectric as a scalar point function, defined at each point r in space as:

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 7 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede

ε =+=+=+εεχεχχε11 ⇐ ooeoe ( ) ( eo) SI Units same as total electric electric electric permittivity permittivity permittivity for ε o (Farads/m) of dielectric of vacuum of dielectric

In some dielectrics, under certain conditions χe →∞. In plasmas (i.e. ionized gases), χe < 0 .

In most typical/garden-variety dielectric materials, 0≤ χe ≤ 10.

We can also define a relative electric permittivity (a.k.a. dielectric “constant”) which is (obviously) dimensionless:

ε ()1+ χεeo ⎛⎞ε Ker ()=≡=""ε =+() 1χ e and/or: χee= K −=11⎜⎟ − εεoo ⎝⎠ε o

Consider a “real life” situation (i.e. an actual physics experiment): A Class-A dielectric block of -type material is inserted between two parallel plates, which have a potential difference ΔV across the parallel plates of the , as shown in the figure below:

σ ΔV We know that: Ex==free ˆˆ x () /m from the (empty) parallel plate capacitor ext ε abc++ 0 If the Class-A dielectric is in a uniform/constant Eext (i.e. the gap of the parallel-plate capacitor is small relative to size (length/width dimensions of the parallel plates), then the electric polarization Ρ=Ρ()rxo ˆ is must also be uniform/constant inside the gap of the parallel-plate capacitor, and thus no bound volume charge density exists inside the dielectric material:

ρBound ()rr= −∇i Ρ( ) = 0

However, on the RHS and LHS surfaces of the dielectric (see above figure, with nxnxˆˆˆˆ=+, =− ), that σ =Ρrni ˆ =+Ρ and σ = Ρ=−Ρrni ˆ , 12 Bound+ () 1 RHS o Bound− ( ) 2 LHS o surface surface respectively, or, expressing this more compactly: σ = Ρ=±Ρ(rn)i ˆ Bound± surface o

8 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede Thus, here again we see that we can replace the polarization Ρ of the dielectric altogether, here simply by the (equivalent) bound surface charge distributions σ (since ρ r = 0 inside Bound± Bound () the dielectric). Then we see that: Ρ=Ρoxˆˆ =σεχ BoundxE = 0 e int

What is Erint ()??? macroscopic Erint()= vector sum of Er ext( ) and E molecular ( r) dipoles using the principle of linear superposition!!!

macroscopic Thus: Erint()= Er ext( )+ E molecular ( r) dipoles

macroscopic What is: Ermolecular () ? dipoles

Note that the situation (here) with bound surface charges of σ = +Ρ on the RHS surface Bound+ o and σ =−Ρ on the LHS surface is analogous to that for the free surface charge densities Bound− o

σ free=+σ o (RHS) andσ free− =−σ o (LHS) associated with the parallel plate capacitor itself, except + macroscopic note the direction of Emolecular ()r relative to Eext (and hence note the sign change below)!!! We dipoles can thus easily see that:

macroscopic σ Bound σ free Ermolecular () =− xˆ c.f. with the external field of ||-plate capacitor: Exext =+ ˆ dipoles ε 0 ε 0

Thus we also see that: macroscopic σ free σ bound Erint()==+− Er ext ()+ E molecular () r xˆˆ x dipoles εε00 σ boundΡ o 1 macroscopic σ boundΡ o 1 But: xxˆˆ==Ρ, Thus: Ermolecular ()=− xxrˆˆ =− =− Ρ() ε 000εε dipoles εεε000

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 9 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede macroscopic 1 Therefore: Erint()==−Ρ Er ext ()+ E molecular () r Er ext () () r dipoles ε 0

Rearranging this relation: 1 Erext()=+Ρ Er int () () r But: Ρ=(rrEr) εχ0 eint( ) ( ) ε 0 1 ∴ Erext()=+ Er int () ε 0 χχχeEr int()=+ Er int () e Er int () =+ ( 1 e ) Er int () ε 0

Thus: Erext()=+ (1 χ e ) Er int () or: Erint( ) =+ Er ext( ) (1 χ e )

We see that the macroscopic/averaged-over internal electric field inside the dielectric Erint ( ) is reduced by a factor of 11()+ χe relative to the external/applied electric field Erext (), because the macroscopic electric field associated with the (now polarized) molecular dipoles, Ermolecular () opposes the dipoles external applied electric field! Using the dielectric constant, Keo ≡=+ε εχ(1 e) we see the same thing, namely that Erint()=+= Er ext ()(1 χ e ) ErK ext( ) e i.e. the internal electric field is “screened” / reduced from the Eext value by the dielectric constant K of the dielectric material.

We can also show that, since: Ρ=()rErεχ0 eint( ) Then: Erint( ) =Ρ( r) ε 0 χ e and Exext=()σε free 0 ˆ Ρ⋅()rnˆˆ =Ρ=σ Ρ=Ρ( r) x surface o Bound{ o } Ρ()r Ρoxˆ ⎛⎞σ Bound ⎛⎞11 ⎛⎞⎛⎞σ free Thus: Erint ()===⎜⎟ xˆ and: Erint()==⎜⎟ Er ext () ⎜⎟⎜⎟ xˆ εχ00ee εχ⎝⎠ εχ 0 e ⎝⎠1++ χχεeeo ⎝⎠⎝⎠1 Then we see that: ⎛⎞ ⎛⎞⎛⎞⎛⎞σ Bound 1 σ free ⎛⎞χe σχBound⎛⎞ e ⎜⎟⎜⎟⎜⎟= or: σ Bound= ⎜⎟σ free or: ⎜⎟= ⎜⎟ 1 ⎜⎟σ 1+ χ ⎝⎠⎝⎠⎝⎠εχ0 eeo1+ χ ε ⎝⎠+ χe ⎝⎠free⎝⎠ e ⎛⎞ε But: χχee=−KK1, or: 1 +== ee⎜⎟ ⎝⎠ε 0

⎛⎞Kee−−1 ⎛⎞εε0 ⎛⎞ χ ∴ σ Bound===⎜⎟σσσ free⎜⎟ free ⎜⎟ free ⎝⎠Kee⎝⎠εχ ⎝⎠1+

i.e. The bound surface charge densityσ Bound on the surface of a dielectric is directly related to the free surface charge density σ free on the surface of the conducting plates of the parallel plate capacitor!!!

IMPORTANT NOTE: This relation between bound surface charge densityσ Bound and surface charge density σ free is NOT a universal one!!! It is specific only to the case of the parallel-plate capacitor!!!

10 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede The potential difference ΔV between the two capacitor plates of the parallel plate capacitor is: Δ=−VEdaEbEcEi = + + ∫ ext int ext C If ac= (i.e. the air gaps in the parallel plate capacitor the same dimension) 1 ⎛⎞b Then: Δ=VaEbE2ext + int But: EEint= ext ∴ Δ=Va⎜⎟2 − Eext Ke ⎝⎠Ke Define: dab≡+()2 = total gap between parallel plates of capacitor.

σ free Now: Exext = ˆ ε 0

⎛⎞b σ free ∴ Δ=Va⎜⎟2 + ⎝⎠Ke ε 0 A = surface area of one of the Q (σ free A) of parallel plate capacitor: C ≡= plates of the ||-plate capacitor ΔΔVV

Capacitance of the ||-plate capacitor (including the dielectric):

σ A σ free A ε A C ==free = 0 ΔV σ ⎛⎞ ⎛⎞b free ⎜⎟2a + b ⎜⎟2a + ⎝⎠Ke ⎝⎠Ke ε 0

ε If there is no dielectric, then Ke ==1 ()εε=0 = vacuum and b = 0, d = 2a ε 0 ε A ε A Then: C ==00 no dielectric 2ad

If there are no air gaps, then ac==0 and db=

εε00AA⎛⎞ Then: CKKCdielectric==ee⎜⎟ = no dielectric d ⎝⎠d Ke

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 11 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede

THE MACROSCOPIC ELECTRIC FIELD INSIDE A DIELECTRIC BOUNDARY CONDITIONS ON E

Suppose we have a linear, homogeneous, isotropic (Class-A) dielectric. We concern ourselves with the macroscopic E -field inside the dielectric (the microscopic/atomic scale E -field is wildly fluctuating, both in space (position) and time due to thermal fluctuations). We note that the basic properties of macroscopic E -field in a dielectric material should not change wildly/dramatically from those of the vacuum (free/empty) space. i.e. imagine we adiabatically change ε 0 → ε dielectric no fundamental changes will occur during this process.

In particular, we must keep in mind that the macroscopic Er( ) is a conservative field, whether inside the dielectric or the vacuum (microscopically, virtual photons are all the same kind/type in a dielectric vs. the vacuum) and since the macroscopic (e.g. on a test charge, QT ) Fr()= QErT () is also conservative in a dielectric medium or vacuum ⇒ hence both Fr()and Er()are derivable from a , Vr( ) . ⇒∇×Er() =0 (always), since ∇×∇( Vr( )) =0 (always) for a . Equivalently: ∇×Er() =0 ⇒ Er ()i d = 0 (Stoke’s Theorem) and vice-versa. ∫c

Consider a parallel plate capacitor with a dielectric between the plates of the capacitor. The dielectric has a very thin hole drilled through it, parallel to the electric field(s), as shown in the figure below:

Now Er()i d = 0 . Take contour C as shown in picture above, but shrink the contour C down ∫C to justε (i.e. infinitesimally) inside/outside the hole drilled in the dielectric: =⊥0; Evac ε2 =⊥0; Ediel ε3 Er()i d=+ E i E i1 ε + E i 1 ε 11εε== 1st half of , 2 nd half of ∫C ()vac12() vac 2 2 ()diel 2 3 ()22223323 ++EEi i1 ε + E i 1 ε 11εε== 1st half of , 2 nd half of ()diel34() diel 2 4 ()vac 2 1 ()22441141 =⊥0; E ε =⊥0; Ediel ε4 vac 1 =+()()EEvaci12 diel i 34 = 0

12 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede But: 12=− 34 ≡ ⇒ ∴ ()EEvac−= die i 0 But: Evac || and Ediel || ∴ EEvac= die at the surface/boundary of the dielectric. tangent tangent Specifically: EEvac= diel @ the interface/boundary of the dielectric. More generally: The tangential components of E are equal @ a dielectric interface i.e. EE12tt= @ the interface of dielectric. The tangential component of E is continuous across a dielectric interface.

Note that this result is valid regardless of the orientation of cavity/hole, provided (if and only if) the dielectric is Class-A (i.e. linear, homogeneous isotropic) – it is not necessarily true otherwise.

SOME EXAMPLES OF DIELECTRICS

∃ all kinds of dielectric materials - some are gases, some are liquids and some are .

⎛⎞ε Dielectric “constant” K ≡=+⎜⎟()1 χe ⎝⎠ε 0 −12 ε 0 =×8.85 10 Farads/m = electric permittivity of free space/vacuum = macroscopic constant/scalar quantity = constant @ all (Lorentz invariant quantity)

ε = electric permittivity of dielectric χe = electric susceptibility of dielectric = macroscopic constant/scalar quantity = macroscopic constant/scalar quantity for Class-A dielectrics for Class-A dielectrics SI Units: Farads/m SI Units: Dimensionless

n.b. The macroscopic parametersε , χe (and thus Ke ) have/exhibit dependence because microscopically, the induced and/or permanent electric dipole moments in atoms/molecules in the dielectric (in general) are frequency dependent over the frequency range 0≤ f ≤∞ Hz !!!

Dielectric “Constants” of various materials at STP and f = 0 Hz.

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 13 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede THE MACROSCOPIC ELECTRIC DISPLACEMENT FIELD, D()r GAUSS’ LAW IN THE PRESENCE OF DIELECTRICS

We have seen that the effect of polarization of a dielectric is to produce bound surface and volume charge densities within and/or on the surface(s) of the dielectric: 3 Bound volume charge density: ρBound (r) =−∇Ρi ( r) ( Coulombs meter ) Bound surface charge density: σ (r) =Ρ( r)i nˆ Coulombs meter 2 Bound surface ( ) We have also shown that the E -field inside a dielectric medium due to the electric polarization, Ρ()r is simply (equivalently) due to the bound charge distributions ρBound (r ) and/or σ Bound (r ) .

Suppose now that this dielectric also had embedded in it free electric charges – e.g. either embedded or positive ions (e.g. by irradiating it with an e− beam or proton/ion beam).

Within the dielectric, since the electric charge density distributions (obviously) obey the principle of linear superposition (i.e. due to charge conservation!), then the TOTAL volume electric charge density can be written as: ρρTot(rrr) =+ Bound( ) ρ free ( )

Then Gauss’ Law (in differential form) becomes: ερρρ0∇==iErTot() Tot () r Bound () r + free ( r) where: ErTot()=+ total electric field = " E bound( r) " " E free ( r) and ρBound (rr)()=−∇Ρi

We can rearrange Gauss’ Law Law (in differential form) as follows (dropping the “Tot” subscript on the E-field – but please keep this in mind!!!): ερ∇−iiEr() () r =∇+Ρ= ε Er () () r ρ() r 00Bound() free ≡= Dr() Electric Displacement The (macroscopic) Electric Displacement Field: Dr( ) ≡+Ρε 0 Er( ) ( r) 2 SI units of Dr()are the same as that for Ρ(r ) (same as that for σ Bound & σ free !!): Coulombs m

Then we realize that Gauss’ Law (for dielectrics) becomes: ∇=iDr( ) ρ free ( r) i.e. the divergence of the (macroscopic) D -field at the point (r ) is due to (i.e. equal to) the free volume charge density, ρ free that is present at the point (r ) !

Dr′′i dA= Qencl In integral form, Gauss’ Law (for dielectrics) becomes: ∫ ( ) free S′ Gauss’ Law for D physically tells us that the electric displacement field, Dr( ) is sensitive to the free charge that is present in a given situation, whereas Gauss’ Law for E tells us that the electric field Er()is sensitive to the total charge that is present in this same situation. Gauss’ Law for Ρ tells us that Ρ()r is sensitive to the bound charge that is present in this same situation.

14 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede Thusfar, we have obtained several useful relations for Class-A dielectrics, summarized here:

Dr()≡+Ρε 0 Er () () r and Ρ=()rErεχ0 e ( ) 2 SI units of Dr()are the same as that for Ρ(r ) (same as that for σ Bound & σ free !!): Coulombs m

∇=iErρ r ε Er′′i dAQ= encl ε ρρrrr=+ ρ ()Tot () 0 and ∫ () Tot o with Tot( ) Bound( )() free S′ ∇=iDr ρ r Dr′′i dA= Qencl QQencl=+ encl Q encl ()free () and ∫ () free with Tot Bound free S′ ∇Ρi ()rr =−ρ () and Ρ=−()rdAQ′′i encl and σ (rrn) =Ρ( )i ˆ Bound ∫ Bound Bound surface S′

⎛⎞ε and: Kee≡=+⎜⎟()1 χ ⎝⎠ε 0 The tangential components of E are continuous across a dielectric interface i.e. EE12tt= @ the interface of a dielectric.

For Class-A dielectric materials (i.e. linear, ideal, homogeneous isotropic materials): Dr()=+Ρε 0 Er () () r, but Ρ=()rErεχ0 e ( ) inside the dielectric ∴ Dr()=+εχεεχ000 Er ()ee Er () =+ (1 ) Er( ) but ε =+ε 0 ()1 χe and Keo≡=+ε ε ()1 χ e

∴ Dr()==+=εεχε Er ()00 (1 e ) Er () Er( ) +Ρ( r) in a Class-A dielectric material.

Griffiths Example 4.4:

A (very) long, straight conducting wire carries a uniform, free line electric charge λ which is surrounded by rubber insulation out to radius, a. Find the electric displacement Dr().

L λ Coulombs/meter s a free line charge zˆ

Take a cylindrical Gaussian surface of radius, and length, L: Dr( ′)i dAQ= enclosed s ∫S′ free From the intrinsic of this problem, we realize that Dr( ) will be radial (n.b. The E -field associated with the free line charge λ (alone) is radial)

The only contribution to surface integral is from the cylindrical portion of the Gaussian-surface, i.e. Dr()|| rˆ , and the end caps of the Gaussian surface (|| zˆ) do not contribute since Dr()⊥ zˆ .

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 15 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede In Cylindrical Coordinates: enclosed DL()2π s = λ L = Qfree sssss==rr ˆ = rˆ === rr λ Thus: Dr()= rˆ (Coulombs/m2) 2π r

Note that this formula holds inside the rubber dielectric (ra< ) as well as outside the rubber dielectric ()ra> , i.e. this formula is valid for any r. However, since Ρ>()ra =0 (i.e. no rubber dielectric for ra> ) 11λ ⎛⎞ Then: Er()== Dr() ⎜⎟ rˆ for ra> επε002 ⎝⎠r

Inside the rubber dielectric ()ra< , since we do not explicitly know the analytic form of Ρ<()rathen we do not know Er()< a. Note also that (here) neither ρBound (ra< ) nor

σ Bound ()ra= have been specified.

CAUTIONARY STATEMENTS ABOUT THE ELECTRIC DISPLACEMENT Dr( ) AND THE ELECTRIC POLARIZATION Ρ(r ) Inside Class-A dielectric materials, the so-called constitutive (a.k.a. auxiliary) relation between the three fields Dr()≡+Ρε 0 Er () () r holds/is true/valid.

Coulomb’s Law is true for ErTot (), because Er( ) is a conservative field, i.e. it is derivable from a scalar potential ()Er()=−∇ Vr (), and the∇×=Er( ) 0 (always) in problems: 1 ρ ()r′ Er()= Tot ˆ dτ ′, with ρρ(rrr) =+( ) ρ( ) ∫v′ 2 r Tot Bound free 4πε 0 r 1 σ ()r′ or: Er()= Tot ˆ dA′,with QQencl=+ encl Q encl ∫S′ 2 r Tot Bound free 4πε 0 r 1 λ ()r′ or: Er()= ˆ d′ ∫C′ 2 r 4πε 0 r The same/analogous thing is not true for the electric displacement, Dr( ) nor is it true for the electric polarization, Ρ()r , because neither Dr( ) nor Ρ(r ) are conservative, and neither is derivable from (the negative of) a scalar potential. As consequences of these facts: 1 ρ free ()r 1 ρ (r ) ′ Bound ′ Dr()≠ 2 rˆ dτ and Ρ≠()rd2 rˆ τ 4π ∫v′ r 4π ∫v′ r 1 σ free ()r 1 σ (r ) ′ Bound ′ Dr()≠ 2 rˆ dA and Ρ≠()rdA2 rˆ 4π ∫S′ r 4π ∫S′ r 1 λ free ()r′ 1 λBound (r′) Dr()≠ 2 rˆ d and Ρ≠()rdrˆ 4π ∫C′ r 4π ∫C′ r 2

16 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede Er()is a fundamental field. Er()is a conservative field. Dr()and Ρ()r are not fundamental fields. Dr( ) and Ρ(r ) are not conservative fields. Dr()and Ρ()r are auxiliary fields.

While Dr()=+Ρ⇒∇=∇+∇Ρεε00 Er () () r iii Dr( ) Er( ) ( r) holds/is true/valid for Class-A dielectrics, the divergence of a on its own is insufficient to uniquely determine/fully- specify the nature of a vector field.

Both ∇iAr() and ∇×Ar() must be specified in order to uniquely determine the Ar()-field.

Now ∇×Er() =0 always ()Er() ()and FE () r are conservative But ∃ many situations where ⎧⎫⎪⎪∇×Dr() ≠0 ⎛⎞has permanent electric polarization ⎨⎬ eg. . a bar ⎜⎟ ⎩⎭⎪⎪∇×Ρ()r ≠ 0 ⎝⎠ - analogous to bar !!!

Dr()and Ρ()r are auxiliary fields associated with matter – dielectric materials in particular.

BOUNDARY CONDITIONS ON DE , and Ρ for DIELECTRIC MATERIALS

BOUNDARY CONDITIONS ON THE ELECTRIC DISPLACEMENT, D AT AN INTERFACE

Suppose we concern ourselves with what happens at the boundary/interface of two dielectric materials, e.g. (air and water) or (glass and plastic) Gaussian pillbox centered SIDE VIEW: D1 nˆ1 on dielectric interface. Shrink height h of pillbox

Dielectric S1′ to zero/infinitesimally small. Material # 1: ee ε111,,K χ Free charge surface density,

Boundary/ σ free exists on interface Interface h ΔS

σ free Dielectric

Material # 2: S3′ ee ε 222,,K χ nˆ3

S2′

nˆ 2 D2

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 17 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede

Gaussian Surface SSSS′′′′=++123 ΔS = area of disk at interface boundary D() ri dA== Qenclosed σσ () r dS′ =Δ S ∫∫SS′′free free free =+DrndS1112()iiˆˆ′′ DrndS () 22333 + DrndS () i ˆ ′= σ freeΔS ∫∫SS ∫ S 11 1 =0

Now nnˆˆ12=− Shrink Gaussian Pillbox to zero height @ interface/ nˆ boundary of the two dielectrics 1 D1

Dielectric θ1 Interface Medium #1 Dielectric θ2 D2 Medium #2

nˆ2

Normal component of Dr1 () Dn11i ˆ =− D 1() rcosθ 1 ≡− D 1n ( r) = interface interface interface evaluated at/on the interface ⇒=−ΔDrndSi ˆ ′ D S 11() 1n imterface ∫S1 Normal component of Dr() Dni ˆ =+ D() rcosθ ≡+ D( r) = 2 22interface 2 2interface 2n interface evaluated at/on the interface ⇒=−ΔDrndS22()i ˆ ′ D 2n S ∫S interface 2

But: dS12′′==Δ dS S ∫∫SS 12

∴ ⎡⎤−+Dr() Dr () Δ S =Δσ S ⎣⎦12nninterface free or: ⎡⎤Dr()−= Dr () σ If σ == 0 at interface, then Dr( ) Dr() ⎣⎦21nninterface freefree12 ninterface n interface

The normal component of Dr() is discontinuous across a dielectric interface when σ free is

present, by an amount σ free

18 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede BOUNDARY CONDITIONS ON THE ELECTRIC FIELD, E AT AN INTERFACE

We have already shown (see pages 12-13 of these lecture notes) that taking the contour integral Er()i d = 0 across an interface between two dielectrics told us that the tangential components ∫C EE= of E are continuous across a dielectric interface: 12ttinterface interface

Er() 1 1 θ Shrink height h of Medium 1 1 4 2 contour C to 0,

Just above & θ2 Medium 2 Contour C below interface. Er2 ( ) 3

Er()i d =+ E i E i ++EEi i = 0 where = −= ∫C 11 2 2 33 44 13 with: EE11i=−=−tt and E 3 i E 2 interfaceinterface interface interface The tangential components of E are continuous across Thus: EE= or: EEsinθθ= sin 12ttinterface interface 11interface 2 2 interface a dielectric interface

If e.g. medium #1 is a conductor, then E = 0 inside the conductor. 1 If E1 = 0 inside the conductor, then DE11011= εε=+= EP0 ⇒=DP110 and = 0 inside conductor

∴ For conductor-dielectric interface: Material #1 is conductor and material #2 dielectric medium, then: DEP111===0 and D2nfree= σ and E2t = 0

Note that the potential Vr physically must be continuous at an interface between two ()interface materials, whether they are dielectrics or otherwise!

Qenclosed Also: From Gauss’ Law for E : Er()i dA′ = Tot ∫S′ ε 0

At a dielectric interface, as drawn on page 17 above, we see that: σ σ +σ EE−==Tot bound free The normal components []21nninterface εε00 of E are discontinuous across a dielectric interface

by the amount σTot ε 0

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 19 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede BOUNDARY CONDITIONS ON THE ELECTRIC POLARIZATION Ρ AT AN INTERFACE

Qenclosed QQenclosed+ enclosed From Gauss’ Law for E : Er()i dA′ ==Tot free Bound ∫S′ εε00 11 Now: Dr()≡+Ρε 0 Er () () r so: Er()=−Ρ Dr() () r εε00 ⎛⎞11 1 ∴ Er()ii dA′′=−Ρ=+ Dr() () r dA Qenclosed Q enclosed ∫∫SS′′⎜⎟()free Bound ⎝⎠εε00 ε 0

or: Dr()ii dA′′−Ρ () r dA = Qenclosed + Q enclosed ∫∫SS′′free Bound

But we already know that: Dr()i dA′ = Qenclosed and Ρ=−(rdAQ)i ′ enclosed ∫S′ free ∫S′ Bound

Take a (shrunken) Gaussian pillbox centered on the interface as shown in figure below:

=Ρ12nn =Ρ So: Ρ=−()rdAQi ′ enclosed Get: []Ρ+ΡΔ=−ΔiinnSˆˆ σ S But: nnˆˆ= − ∫S′ Bound 11 2 2 Bound 21 Thus: ⎡⎤Ρ−Ρ()rr () =−σ ⎣⎦21nninterface BoundThe normal components of Ρ(r ) are discontinuous

at an interface by the amount −σ bound

Since: Dr()=+Ρε 0 Er () () r we can also write this out for normal and tangential components as: Dr=+Ρε Er r and Dr=+Ρε Er r nnniii()0 () () tttiii() 0 ( ) ( ) Both of these component relations are valid on each side of interface, i.e. for the ith media, i = 1, 2.

⎧⎫DD21n−= nσσ free and PP 21 n −=− n bound ⎪⎪ Then: ⎨⎬11 at the interface of two dielectrics ⎪⎪EE21n−= nσσσ ToT =() free + bound ⎩⎭εε00

The tangential relations for fields at the interface are: DDPP2121tttt− =− ⇐ Not necessarily = 0! and: EE21tt−=0 ALWAYS (for electrostatics)!!!

20 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede RELATIONSHIPS BETWEEN ρ free (r )AND ρBound (r ) FREE & BOUND VOLUME CHARGE DENSITIES

Since: Dr()=+Ρε 0 Er () () r then: Ρ=(rDrEr) ( ) −ε 0 ( ) 1 However, for Class-A dielectrics: Dr( ) = ε Er( ) or: E ()rDr= () ε εεε00 ⎛⎞− Thus: Ρ=()rDrErDrDr () −ε 0 () = () −() =⎜⎟ Dr() but: ε = Keε 0 εε⎝⎠ ⎛⎞K −1 ∴ Ρ=()rDr⎜⎟e () ⎝⎠Ke ⎛⎞K −1 ⎛⎞1 Now: ∇Ρii()rDrDr =⎜⎟e ∇() = ⎜⎟1 − ∇ i() ⎝⎠KKee ⎝⎠ But: ∇Ρii()rrDrrErrrr =−ρρερρρBound () and ∇ () =+ free( ) , also o ∇ i( ) = Tot( )() = free + Bound ( )

⎛⎞1 ∴ −=−ρρBound()rr⎜⎟1 free () ⇐ NOTE: ρbound (r ) is opposite charge sign to ρ free ()r !!! ⎝⎠K ⎛⎞11 Then: ρρρTot()rr=+ free () Bound () rr =−−=+ ρ free () ⎜⎟1 ρ free() r ρ free () r ⎝⎠KKee 1 Thus: ρρTot()rr=+ free () Ke

The total volume charge density is reduced by the amount 1 Ke inside a dielectric. ⇒ Dielectric material screens out charge!!!

NOTE: if Kre →∞ then ρTot () → 0 (perfect screening!!!)

Ke →∞ also implies χe →∞ (infinite electric susceptibility) because Kee=+ 1 χ

Ke →∞ also implies ε →∞ (infinite electric permittivity) because Keo= ε ε Thus, we see (again) that ρBound ()r {partially} cancels out ρ free (r )

Since −∇ii Ρ()rr =ρρbound (), can only get ∇ Ρ( r) ≠ 0 if free ( r) ≠ 0!!!

IMPORTANT NOTE:

There is NO universal relationship between σ free & σ bound .

Sometimes, but not always, a relationship does exist between σ bound & σ free , but it is not universal (i.e. valid for any/all situations).

It is NOT necessary to have σ free ≠ 0 in order to have a non-zero σ bound present on a dielectric.

Example: Bound surface charge density, σ bound on a bar electret (permanently polarized material).

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 21 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede We have previously discussed (above, p. 10 of these lecture notes) the example of free and bound surface charge densities σ free andσ bound at a dielectric-conductor interface, e.g. with the parallel- plate capacitor:

On LHS Plate: σ =Ρ()rni ˆˆˆˆ where Ρ( r) =Ρ x , and n =− x Bound LHSinterface o LHS ⎛⎞Coulombs ˆˆ ∴ σ Bound=−Ρ o ⎜⎟2 since xni LHS = −1 ⎝⎠meter Now: Ρ=()rDrEr () −ε 0 () σεBound=Ρ()rniiˆˆ LHS = Drn() LHS − 0 Ern( ) i ˆ LHS interface interface interface ⎛⎞Ke −1 ⎛⎞1 σ Bound=−Ρ o =−Dr() =−1 − Dr() but: Dr( ) = σ (here) ⎜⎟interface ⎜⎟ interface interface free ⎝⎠KKee ⎝⎠

⎛⎞1 ∴ σ bound=−⎜⎟1 − σ free (here) Note also that σ Bound has opposite charge sign to σ free !!! ⎝⎠Ke

Again we remind the reader that:

There is NO universal relationship between σ free & σ bound .

Sometimes, but not always, a relationship does exist between σ bound & σ free (as we just showed), but it is not universal (i.e. valid for any/all situations).

22 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede Griffiths Example 4.5

A conducting metal sphere of radius a carries a free charge Q and is surrounded by a Class-A dielectric sphere of radius b > a as shown in the figure below:

b a Q •

Find the potential Vr()at the center of sphere. Q From Gauss’ Law for D : Dr′′i dA= Qenclosed gives: Dr>= afree rˆ for r > a ∫ () free () 2 S′ 4π r 1 Now: Dr()= ε Er () so: Er()= Dr() ε QQ1 Q ⇒ for arb<<: Ea()<< r b =free rˆˆ = free r and for rb> : Er()>= bfree rˆ 44πεrK22 πε r 4πε r 2 eo 0 and for ra< : Era()()<= Dra <=Ρ<=( ra) 0!!!

The potential at the center of sphere is therefore: 0 b ⎛⎞QQa ⎛⎞0 Q ⎛⎞111 Vr()==−00 Er ()i d =−free dr − free dr −() dr=+− ∫∫∞∞⎜⎟22∫b ⎜⎟ ∫a ⎜⎟ ⎝⎠44πε0rr⎝⎠ πε 4π ⎝⎠εεε0bab QQ⎛⎞⎛⎞111 1111⎛⎞ Vr()==0⎜⎟⎜⎟ + − = + ⎜⎟ − 44πε⎝⎠⎝⎠00bab ε ε πε bKabe ⎝⎠

If Er()is known, then Ρ()r is also known, because Ρ=(rEr) εχ0 e ( ) εχQQ1 ⎛⎞χ 0 e freeˆˆe free Thus, for arb≤≤: Ρ=()rErrεχ0 e () =22 =⎜⎟ r (i.e. inside the dielectric) 441πεrr π⎝⎠+ χe 11⎛⎞χ ∂ ⎛⎞Q ∴ ρ rr=−∇Ρi =− e r2 free = 0 !!! Bound () () ⎜⎟2 ⎜⎟2 41πχ⎝⎠+∂e rr⎝⎠r ⎧ 1 ⎛⎞χe Qfree ⎪+==+⎜⎟2 at rb () n.b. nˆˆrb= r ⎪⎝41πχ+ e ⎠b and: σ Bound =Ρ()rni ˆ =⎨ interface 1 ⎛⎞χ Q ⎪−==−e free at ra n.b. nˆˆ r ⎪ ⎜⎟2 ()ra= ⎩ 41πχ⎝⎠+ e a n.b. By convention, nˆ is the outward pointing unit vector from the surface(s) of the dielectric.

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 23 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede When all space is filled with a Class-A dielectric material, the E -field inside the dielectric is reduced by factor of 1 Ke from its free-space value.

For example: A point (free) electric charge q is embedded at the center of a solid Class-A dielectric sphere of radius R as shown in the figure below:

The E -field inside the dielectric sphere, due to the point free charge at the center of the sphere is (using Gauss’ Law for D and then using the relation Er( ) = Dr( ) ε ): 111qq⎛⎞ Er()<= R22 rˆˆ =⎜⎟ r 44πεrK⎝⎠eo πε r

1 q However!! Note that for rR> : Er()>= R2 rˆ 4πε o r

Outside the dielectric ( rR> ) the E -field is the same as if the dielectric sphere wasn’t there at all! This is a consequence of Gauss’ Law for E : Qenclosed Er()i dA′ = Tot where QQQenclosed=+ enclosed enclosed ∫S′ ε Tot Bound free 0 i.e. we get E -field contributions from all enclosed charges:

1) +q at the center of sphere

2) −σ Bound at the inner cavity surface, radius δ << R

3) +σ bound at r = R Note that 2) and 3) cancel each other for r > R!!! (They don’t cancel for r < R!! obviously)

Can you show that QrBound ()==−δ qand QrRqBound ( = ) =+ ??

24 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede

THE ELECTRIC SUSCEPTIBILITY χe OF NON-CLASS A DIELECTRICS

A crystalline dielectric material (e.g. salt, diamond, etc.) has preferred internal axes so that the electric polarization, Ρ()r is different along the different internal axes in such materials.

For crystalline materials, Ρ()r is related to Er( ) by the relation: Ρ=(rEr)()εχ0 e ⎛⎞χχχ eeexx xy xz ⎜⎟ Where χ = Susceptibility Tensor: χχχχe≡ ⎜⎟ eee e ⎜⎟yx yy yz ⎜⎟χχχ ⎝⎠eeezx zy zz Ρ=εχEEE + χ + χ x 0 ()exxy ey xy ez xz ⎛⎞χχχ ⎛⎞Ρx eeexx xy xz ⎛⎞Ex ⎜⎟⎜⎟ ⎜⎟ i.e. Ρ=εχEEE + χ + χ or: Ρ=εχχχ⎜⎟E yexeyez0 ()yx yy yz ⎜⎟yeeey0 yx yy yz ⎜⎟ ⎜⎟⎜⎟ ⎜⎟ ⎝⎠Ρzz⎜⎟χχχ ⎝⎠E Ρ=εχEEE + χ + χ ⎝⎠eeezx zy zz zexeyez0 ()zx zy zz

3 or: Ρ=rErεχ and noting that χχ= ⇒ χ has only six independent components. iej()0 ∑ ij () eeij ji e j=1

Again, if we carefully choose the coordinate axes to coincide with the internal symmetry axes of the crystalline dielectric material, then in reality there are only 3 independent components of χe - the off-diagonal elements vanish; only the diagonal elements χ ,χχ and are non-zero. eexx yy e zz

For crystalline dielectric materials: ∇=iDr()ρεfree () r is valid, Dr () =0 Er( ) +Ρ( r) is also valid. ∴ ∇=∇+Ρ=∇+∇Ρ==−iiDr()()εε00 Er () () r ii Er() () r ρρρfree Tot Bound is valid

However, in a crystalline dielectric material, generally speaking Dr( ), Er( )()& Ρ r are NOT all pointing in the same direction!!!

i.e. a tensor relation also exists between Dr( ) & Er( ) : Dr( ) =εε Er( )()= oe KEr where Keo≡=−ε εχ()1 e for a linear, anisotropic dielectric material and whereε = electric permittivity tensor, thus Ke = relative electric permittivity tensor (a.k.a. dielectric “constant” tensor).

⎛⎞ 33 ⎛⎞DE⎛⎞εεε KKKeee ⎛⎞ x xx xy xz ⎜⎟xx xy xz x Dr==εε E r KE r ⎜⎟⎜⎟ ⎜⎟ iijjej()∑∑ ()0 ij () DKKKE==εεε ε⎜⎟ jj==11 ⎜⎟yyxyyyzeeey⎜⎟0 yx yy yz ⎜⎟ ⎜⎟ ⎜⎟⎜⎟εεε ⎜⎟ εε== and KK , etc. ⎝⎠DEz ⎝⎠zx yz zz ⎜⎟KKK ⎝⎠z ij ji ij ji ⎝⎠eeezx zy zz

Again, if choose the symmetry axes of the crystal for coordinate axes, then the off-diagonal elements of ε and Ke vanish. ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 25 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede For extremely high externally applied Eext -fields, the electric polarizationΡ()r becomes increasingly non-linearly related to Eext :

33 3 Ρ=i()raErbErErcErErEr∑∑ ij j () + ijk j () k () + ∑ ijkl j () k () l () +… jjkjkl==1,1,,1 = linear quadratic cubic response response response

Linear Ρi Non-Linear Regime Regime j = i

j ≠ i

E j

26 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede Example: Gauss’ Law for Parallel Plate Capacitor with “Class-A” Dielectric Between Plates:

Take Gaussian pillbox centered on LHS conducting plate: Dr()i ndSˆ ′′== Qσσ dS =Δ S ∫∫SS′′free free free Because of the parallel-plate geometry:ED , , Ρ are constant fields between the plates of the capacitor (we neglect the fringe-field/edge region(s) of the parallel plate capacitor, since dAw ()=× ) ∴ Dn11iiˆˆΔ+ S Dn 2 2 Δ=+ Sσ free Δ S or: Dn11iiˆˆ+= Dn 2 2 σ free But: nnˆˆ21=− , thus: Dn11iiˆˆ−= Dn 21 σ free or: DD12nnfree− = σ ⇒ Normal components of D discontinuous across dielectric interface, by amount σ free . Now DE101==ε 0 and P01 = since LHS of Gaussian pillbox ends inside the LHS plate of ||-plate capacitor, also the conducting metal is not a dielectric, so it has no electric polarization.

∴ DDDD22nfreen===⊥σσ or free ( 2 n to surface of plates) But: DE= ε for Class- A dielectric. ∴ DE= ε = σ or: EK= σ εσ= ε free free free e o

And: E =ΔVd for ||-plate capacitor, σ free= QA free and: QCVfree = Δ σ Q Thus: EVd=Δ =free = free KKAeeε 00ε Q KAε ε A ε A So: CK==e 00 = But now recall that for no dielectric, CK==0 , 1 DielΔVd e d 00d

Cdiel ∴ = Ke C0

⇒ Capacitance of parallel-plate capacitor with dielectric increased by factor of Ke over vacuum.

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 27 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede EXAMPLE: Gauss’ Law for “Class-A” Dielectric Sphere with Point Charge Q at its Center:

Use Gauss’ Law to obtain D -field inside the dielectric and thus obtainE , Ρ inside the dielectric: enclosed enclosed D() ri ndSˆ ′ = Q but: QQfree = and (from above): nrˆˆout = + , and: nrˆˆin =− . ∫S′ free rR= r=δ Note that Dr()is radial (i.e. no θ, ϕ dependence) due to rotational symmetry/invariance of problem. Q ⇒ Dr()= rˆ for δ ≤≤rR. inside 4π r 2 For “Class-A” dielectrics But: Drinside()=+Ρ==ε 000 Er inside () inside () rErKErεε inside( ) e inside( ) with: εε = K e QQ11 ⇒ Erinside()==22 rˆˆ rDr = inside() = Dr inside () 44πεrKr πeo ε ε K e ε 0 ε Ρ=()rD () r −ε E () rD = () r −0 D() r inside inside0 inside insideε inside ∴ ⎛⎞εε−−−⎛⎞KK11 ⎛⎞Q 0 eeˆ ===⎜⎟Drinside() ⎜⎟ Dr inside () ⎜⎟2 r ⎝⎠επ⎝⎠KKree ⎝⎠4

⎛⎞K −1 Q e ˆ ⇒ Ρ=inside ()rr⎜⎟2 ⎝⎠Kre 4π

At the outer surface of the dielectric sphere the bound surface charge density is:

⎛⎞K −1 Q σ ≡Ρ()rniiˆˆ =Ρ() rr =Ρ() r = e BoundrR= inside outrR== inside rR inside rR= ⎜⎟2 ⎝⎠KRe 4π

⎛⎞K −1 Q ⎛⎞K −1 σ = e QRQ==σπ4 2 e Bound rR= ⎜⎟2 and thus: BoundrR== Bound rR ⎜⎟ on the outer surface. ⎝⎠KRe 4π ⎝⎠Ke

28 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede

At the inner surface of the dielectric sphere the bound surface charge density is:

⎛⎞K −1 Q σ ≡Ρ()rniiˆˆ =Ρ()( r − r ) =−Ρ() r =− e BoundrR= inside inrr==δδ inside inside r=δ ⎜⎟2 ⎝⎠Ke 4πδ

⎛⎞K −1 Q ⎛⎞K −1 σ =− e QQ==−σπδ4 2 e Bound r=δ ⎜⎟2 and thus: Boundrr==δδ Bound ⎜⎟ on the inner surface. ⎝⎠Ke 4πδ ⎝⎠Ke

⎛⎞⎛⎞KK−−11 QQ=− =−ee Q =− Qenclosed Thus, we explicitly see that: BoundrrR==δ Bound⎜⎟⎜⎟ free ⎝⎠⎝⎠KKee

⎛⎞Ke −1 Q Now from above, we found that: Ρ=inside ()rr⎜⎟2 ˆ ⎝⎠Kre 4π Then inside the dielectric sphere: ρBound(rr) =−∇Ρi inside ( ) for δ < rR< . 1 ∂ Now∇ in spherical coordinates: ∇=rr2 ˆ +……θˆ + ϕˆ rr2 ∂ these terms not important here, since Ρ=Ρrˆ only ⎡⎤⎡⎤ 11∂∂22⎛⎞Ke −1 Q 1 ∂ ⎛⎞Ke −1 Q Then: ρBound()rrrr=− Ρ inside () =− ⎢⎥⎢⎥⎜⎟ = −=⎜⎟ 0 rr22∂∂ rr K 4π r 2 rr2 ∂ K 4π ⎣⎦⎣⎦⎝⎠e ⎝⎠e =constant, ≠ fcn of r

⇒ ρBound ()r = 0 for δ <

Using Gauss’ Law for E we also see that the total charge as seen by an observer inside the dielectric (i.e. for δ <

enclosed enclosed enclosed ⎛⎞⎛KKKeee−−+11 ⎞1 We see that: QQQQTot=+=−= free Bound ⎜⎟⎜ Q ⎟ QQ = forδ < rR< . ⎝⎠⎝KKKeee ⎠ ⇒ Forδ <

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 29 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede Outside the dielectric sphere (i.e. for rR> ) again using Gauss’ Law for E we also see that the total charge as seen by an observer outside the dielectric is: Qenclosed QQenclosed+ enclosed E() ri ndSˆ ′ ==Tot free Bound . ∫S′ εεoo ⎛⎞⎛⎞KK−−11 QQenclosed = QQenclosed = +=−+ Qee Q Q =0 Since free for rR> and Bound BoundrrR==δ Bound ⎜⎟⎜⎟ ⎝⎠⎝⎠KKee enclosed enclosed enclosed Thus we see that: QQQQQTot=+=+= free Bound 0 for rR> !!! ⇒ For rR> the total/net charge “seen” by the E -field (e.g. using a test charge Q in the region T rR> ) is not “screened” by the dielectric – the E -field outside the dielectric is the same as the E - field associated with a “bare” point charge, Q located at the origin! The bound surface charge density σ (located at the outer radius rR= of the dielectric) precisely cancels the effect(s) associated Bound rR= withσ (located at the inner radius r = δ of the dielectric)!!! Bound r=δ

Outside the dielectric sphere (i.e. for r > R): enclosed Q 1 Gauss’ Law for D : D() ri ndSˆ = Q ⇒ Droutside ()= rˆ for rR> . ∫S free 4π r 2 Q 1 Gauss’ Law for E : E() ri ndSˆ = Qenclosed ε ⇒ Er()= rˆ for rR> . ∫S Tot o outside 2 4πε o r Then: DrEroutside()= ε 0 outside () for rR> . Obviously, Ρoutside (r ) = 0 for rR> !!!

30 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede

THE BAR ELECTRET: The Electrostatic Analog of A Bar Magnet An electret is a polar dielectric which has permanent polarization, Ρ . can be made e.g. by heating a polar dielectric material (i.e. a dielectric material which has permanent molecular dipole moments), heating it in the presence of a (very) strong uniform external electric field. The electret is then cooled e.g. to ambient/room temperature in the presence of the external E -field. It is then removed from the external E -field, still retaining a net, permanent polarization!

Eext + − + − + − + − + − + − P + − + − + − + − + −

+ −σ b +σ b −

+σ f −σ f Heat Source Heat polar dielectric material, then cool to room temperature.

Afterwards: Electret retains uniform, permanent electric polarization: Ρ=Ρo zpˆ = Volume = Electric dipole moment per unit volume.

Ρ=Ρo zˆ zˆ

−σ b +σ b

Note that since ρρBound()rr=−∇Ρi () =0 ⇒ free = 0 too!!

∴ ∃ no free charge, σ free on surface (or ρ free within volume) of electret.

Outside the electret: Drout()= ε 0 Er out ( ) (Ρout (r ) ≡ 0) Inside the electret: Drin()=+ε 0 Er in () Pr in ( ) (Ρin (rz) =Ρ0 ˆ)

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 31 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede Boundary Conditions on the Surfaces of the Electret: No free charge anywhere on/within electret. If Ρ=Ρin()rz o ˆ , there is no bound surface charge density on the cylindrical portion of the electret. Ρin(rz) =Ρ o ˆ

At Endcaps of Electret: ()nzˆ =±ˆ s′ zˆ

Take Gaussian Pillbox on, e.g. LHS end cap: −σ b +σ b D() ri ndSˆ ==⇒−= Qencl 0 D⊥⊥ D 0 ∫S′ free out in ⊥⊥ ⇒ DDout= in Normal component of D(⊥ to endcaps) is continuous across endcaps.

Stoke’s Law: Take line integral on LHS Endcap: E ()rdi=⇒0 E = E Tangential components of E continuous across endcaps. ∫C out in Also, Gauss’ Law for Ρ : enclosed ⊥ ⊥ ⊥ Ρ=−⇒Ρ()ri ndSˆ Q −Ρ =−σ or: Ρin= σ b ∫S′ bound out in b

On Cylindrical Surface of Electret: ()nˆ = ρˆ radial direction ⊥⊥ Gauss’ Law for D : ⇒ DDout= in Stoke’s Law: Er()i d = 0 ⇒ EE= ∫C out in ⊥ Gauss’ Law for Ρ : ⇒Ρ= in 0 since uniform polarizationΡin (rz) =Ρ0 ˆ

Outside Electret: Ρ=out ()r 0 , Drout()= ε o Er out ( )

Very Important Note: For the bar electret (or anything else with permanent electric polarization), cannot get conditions on ⊥ ⊥ EE) (in or out) from DD , except via explicit use of Dr( ) =+Ρε 0 Er( )() r. The reason for this is that the relations Drin( ) = ε Er in ( ) and Ρ=(rEr) εχ0 e ( ) are not valid here for permanently polarized / electret materials!!!

The relations DE= ε and Ρ=ε 0 χe E are valid only for “Class-A”/linear dielectrics!!!

32 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede Lines of E for the Bar Electret: (n.b. E terminates on (any) charges (free or bound)!!!)

Important note: Lines of E terminate on any charges – free or bound. Lines of D terminate only on free charges (n.b. none here in the electret!!!). Lines of P terminate only on bound charges.

+σ b In situations where the electret is a thin polarized sheet: Drin()=+Ρ=ε 0 Er in () in () r 0 −σ b i.e. Drin()=⇒0 Er in () =−Ρ in () rε 0 Ein Ρin

Inside e.g. a long bar electret: Ρ≥in()rEr ε 0 in ( ) Din points in direction of Ρin but Ein points in opposite direction of Ρin

Inside e.g. a thin polarized sheet: Ρin()rEr= ε 0 in ( ) Din = 0 and Ein points in opposite direction of Ρin

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 33 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede More Pix of the Bar Electret: Note direction of electric polarization here is opposite to above!

34 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede VACUUM POLARIZATION, CHARGE RENORMALIZATION, RENORMALIZATION

Suppose we use the previous example of the spherical dielectric with a (free) at its center in an attempt to understand what happens to the (physical) vacuum at very small distances from fundamental (i.e. point-like) electrically charged particles, such as the .

The physical vacuum is by no means “empty” – it is actually a (strange) form of dielectric medium (because microscopically, fundamentally it is entirely quantum mechanical in nature – seething with (very briefly appearing & disappearing) -antiparticle pairs of all possible kinds/types).

Nevertheless, the vacuum has two macroscopic (i.e. microscopically-averaged over) parameters associated with it: −12 – The electric permittivity of free space (the vacuum): ε o =×8.85 10 Farads/meter −7 – The magnetic permeability of free space (the vacuum): μπo =×410Henries/meter These two macroscopic properties of the vacuum are not independent of one another, because they are linked via a third macroscopic parameter associated with the vacuum, namely the “speed of ”, c =×3108 m/s (which is a misnomer, since c is the (maximum) speed for which any fundamental force (E&M, strong, weak, gravity) can propagate!) 1 1 These three quantities are related to each other by: c2 = or: c = ε ooμ ε ooμ “Empty” space also has a fourth macroscopic property associated with it – again not independent:

μo – The (the vacuum): o = 376.8 Ω ε o

For E&M, at the microscopic/quantum level only electrically -antiparticle pairs contribute to the macroscopic parameters ε o and μo – e.g. all of the charged fermion-antifermion pairs (in the context of the Standard Model of Electroweak Interactions – these would be the 3 generations of charged : e±±± , μ , τ and the 3 generations of charged up and down quarks (u, d, s, c, b, t)) and also the charged W ± bosons – the electrically charged carrier/mediator of the weak force.

1 −e Now the classical E&M, macroscopic E -field for a point charged particle is Er()= 2 rˆ 4πε 0 r 1 −e The corresponding potential is: Vr()= since Er( ) =−∇ Vr( ) 4πε 0 r Near r 0 , or rr≤=classical electron radius = 2.8 × 10−13 cm = 2.8 fm the electric field of the e 20 electron becomes extremely high, Er()=× re ~1.84 10 Volts/m !!! Note that this is a significantly larger field strength than those e.g. typical of atomic-scale fields, Er( =1Å) ~ 1011 Volts/m.

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 35 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede If we had a true macroscopic spherical dielectric, with an infinitesimally small spherical cavity of −13 radial size δ rcme =×2.8 10 with an electron at the center of this small spherical cavity, it might look something like that shown in the figure below:

By Gauss’ Law the effective electric charge is reduced from that of the bare charge (e) by an amount

QQKeff= bare e Where Ke = dielectric constant (of the physical vacuum, here).

In the region where the electric field of the charged particle is extremely high, the local field density there is sufficient e.g. to produce (virtual) ee+ − pairs in this region of space. These virtual ee+−pairs “live” only for an extremely short period of time – as allowed by the Heisenberg uncertainty principle ()ΔΔ≤Et . Because opposite (like) charges attract (repel), the e+ (e−) from the ee+ − pair that is “pulled” out of the vacuum, tends to be (on average) closer to (further from) the “bare” e−, respectively. Thus “vacuum polarization” results. The net effect, to an observer (who also lives in the same physical “dielectric” medium – the vacuum!!) is that the observed electric charge is reduced from the “bare” charge of the electron!!! The observed/physical value of e is in fact the one we know −19 and love! eobs =+×1.6021892 0.0000046 10 Coulombs.

Note the following “amusing” thought: suppose we were able to get “outside” our universe (i.e. outside of the dielectric of our physical vacuum (“empty” space)!!). Then Gauss’ Law for outside says that we should “see” the full bare charge, ebare and not the reduced charge, eobserved!!!

However, since we cannot get “outside” of the dielectric medium we live in we can never hope to directly observe the pure/bare charge, ebare, except perhaps via extremely high energy collisions between charged particles.

36 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 10 Prof. Steven Errede Using the theoretical formalism of Quantum Electrodynamics (QED), one finds (to first order in ⎛⎞e2 1 the so-called fine structure “constant”, α ) ⎜⎟α == that: ⎝⎠4πε oc 137.036...

n.b. formally divergent! (but only logarithmically…) ⎛⎞α ⎛⎞Λ ee22 1−Λ= log cutoff parameter, e.g. 10 mc2 observed bare ⎜⎟⎜⎟2 e ⎝⎠3π ⎝⎠mce 22 2 For: Λ= 10mce : e observed 0.9992 e bare ==Kedielectric const. Or: eeee=⇒=0.9996 1.00039 small effect!!! observed bare bare observed

Another effect of vacuum polarization in the region of space immediately surrounding a bare electron is that the apparent mass of the electron is increased, due to the observer mis-interpreting the “cloud” of ee+−, etc. pairs surrounding the (bare) electron as being part of the electron itself.

Experiments on a free electron measure m , e.g. the mass used in the Law, obs F =−eE − ev × B = m a . However, the observed mass, m is the sum of the bare mass, m , e− ext e ext e e obs bare plus the inertia of the electron’s “self-field”. Even from classical electrodynamics, we obtain the following mass relation:

⎛⎞4 Λ mmobs≈+ bare ⎜⎟1 α 2 where Λ = “cutoff parameter” ⎝⎠3π mce

2 2 Again, this relation is formally divergent, but if Λ 10mce ( mce = 0.511 MeV ) then mmobs 1.029 bare Or: mmbare 0.97 observed

Thus, we see that (to first order in α ) that mass renormalization is a larger effect than charge renormalization is.

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 37 2005 - 2008. All rights reserved.