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©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 1
LECTURE NOTES 10
The Macroscopic Electric Field Inside a Dielectric
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
photons), atoms, molecules, 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 energies 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
number of constituents involved. If
23
N 10 ,then since σ N = N , then for random fluctuations
(i.e. Gaussian-distributed) the fractional fluctuations,
12
σ N N N N 1 N 3.2 10
− = = × 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 electric field E r ( )
G G
at some point, r
G
inside a
solid dielectric sphere of radius, R as shown in the figure below.
The macroscopic electric field at the field point P @ r
G
inside the sphere consists of two parts:
- A contribution from the average electric field Eout (^) ( r )
G G
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 (^) ( ) in E r
G G
due to electric charges inside this small
conceptual sphere.
In other words, the macroscopic electric field at the field point P located at r
G
(inside the dielectric sphere, i.e. r < R
K
), using the Principle of Linear Superposition is:
E r ( (^) ) = Eout (^) ( r (^) ) + Ein (^) ( r )
G G G G G G
x^ ˆ
y^ ˆ
z ˆ
R
O
r
G
Field point, P
Small imaginary sphere of radius δ
centered on the field point, P @
| r
G
| < R (for averaging purposes)
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2
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 , Eout ( r )
G G
(with r < R
K
) is the electric field at
r
G
due to the electric dipoles contained within the dielectric sphere of radius R that are outside
of/exterior to the imaginary/conceptual sphere of infinitesimal radius δ centered on r
G
Outside of/excluding the region of this small imaginary sphere of radius δ centered on the field
point P @ | r
G
| < R , the atomic/molecular electric dipoles are far enough away from the field point
P that we may safely write the potential Vout ( r )
G
corresponding to Eout ( r )
G G
(with r < R
K
) as:
2
out o outside
r V r d τ r r r r πε
= ′^ = − ′^ = − ′
G G
G r^ G G G (^) G G G r r r
where the integral 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
G
| < R.
The electric dipoles inside the small conceptual/imaginary sphere of radius δ centered on the
field-point P @ r
G
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 electric charge contained within the sphere of radius
δ (regardless of the details of the charge distribution within that sphere) is:
3 0
ave
p E
G
G
where p
G
is the total electric dipole moment of that sphere.
Thus, we know that we know that the average electric field @ r
G
within the small conceptual /
imaginary sphere of radius δ centered on the field-point P @ r
G
must be:
0
in
p E r
G
G G
where p ( r )
G G
is the total/net macroscopic electric dipole moment associated with the (microscopic)
electric dipoles contained within this conceptual/imaginary sphere centered on the field point P @
r
G
4 3 4 3 4 3 3 3 3
Volume of conceptual /
imaginary sphere,
p r r r πδ πδ r
G G G^ G G^ G G G
where Ρ( r )
G G
= macroscopic electric polarization = electric dipole moment per unit volume (@ r
G
Thus: ( ) ( )( )
4 3
p r = Ρ r 3 πδ
G G G G
And thus: ( )
4
3 0 0
in
p r E r πε δ πε
G G
G G
3 3
( π δ/ ) (^ )
3
r
δ
G G
0
r ε
G G
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 4
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
G
field,
e.g. ˆ
ext Eext = Eo x
G
Imagine a small spherical volume of radius δ ~ 1 cm deep within the polarized
dielectric. The electric polarization Ρ
G
inside the dielectric will then be uniform e.g. Ρ = Ρ o x ˆ
G
and
E int
G
inside the dielectric will also uniform, ˆ
int Eint = Eo x
G
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,
one with uniform volume charge density 4 3
ρ+ = + Q 3 πδ and another sphere with uniform volume
charge density 4 3
ρ− = − Q 3 πδ whose centers are offset from each other by a distance
d δ ( )
10 d 1Å 10 m
− =. Thus equivalently, this sphere now has only a bound surface charge
density σ (^) B ( ξ (^) ) = σ (^) o cos( ξ)where the angle ξ is measured with respect to the + x ˆaxis.
Thus a uniformly polarized dielectric sphere of radius δ with uniform polarization Ρ = Ρ o x ˆ
G
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:
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 5
GREATLY EXAGGERATED PIX:
What is the -field @ the center of this polarized dielectric sphere?
= -field due to the near dipoles inside the polarized dielectric!!!
E
E
G
G
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
( ) (^2) 0
encl inside
Q
E r r r
G
where r is defined from center of that sphere.
But the charge enclosed by the Gaussian surface of radius r ( r < δ)is
4 3
Qencl = ρ * V =ρ 3 π r.
Noting that the total charge contained in a single uniformly charged sphere is
1 1
4 3 3
QTot = ρ* VTot = ρ πδ ,or
1
4 3 3 ρ= QTot πδ , then we can rewrite (^) ( ) inside
E r < δ
G
as:
( ) (^2) 0
encl inside
Q
E r r r
G
4
0
3 π^
3 r 2 r
1 2 0 0
Q Tot (^) r
r ρ rr r
Radius δ of uniformly charged sphere
Now for two oppositely-charged spheres of uniform charge density ρ± whose centroids are
laterally displaced from each other by an infinitesimal distance
10
d 10 m δ~ 1 cm
− the net /
total E
G
-field at the center of the two overlapping spheres (by the principle of linear superposition)
is:
( ) ( ) 0 0
Tot Einside Einside r Einside r r r
ρ ρ
G G G G G G G
where 1
4 3
ρ± = ± Q Tot 3 πδ and where the vectors r +
G
and r −
G
are defined in the figures shown below:
1
4 3
ρ= QTot 3 πδ
Gaussian surface
of radius r.
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 7
THE MACROSCOPIC ELECTRIC SUSCEPTIBILITY χ^ e OF A DIELECTRIC
(Lossless) (in Eext ) (uniform, no voids) (rotationally invariant → e.g. not a crystalline material)
(i.e. amorphous)
For an
a.k.a. "Class " Dielectric
"ideal", linear, homogeneous & isotropic dielectric
A
the electric polarization (a.k.a. the
electric dipole moment per unit volume) Ρ
G
is simply related to the internal electric field, int
E
G
of the
dielectric, by a simple proportionality constant, i.e.
Ρ ( r ) = m Eint ( r )
G G G G
Ρ ( r )
G G
m = slope of straight line
m = simple constant Eint (^) ( r )
G G
(i.e. m = scalar quantity)
n.b. This relation is ONLY true for CLASS A dielectrics - i.e. ones which are linear, homogenous,
ideal and isotropic. (We will discuss modifications to this relation shortly…)
Now: SI units of (^) ( ) : Coulombs 2 meter
Ρ r
G G
SI units of (^) ( ) Newtons : Coulomb
E r
G G
⇒ m has SI units of
( )
( )
2 2
2
Coulombs meter Coulombs
int Newton Coulombs^ Newton-m
r m E r
G G
G G Define: m ≡ ε 0 χ e
where ε 0 = (macroscopic) electric permittivity of free space (vacuum)
2 12 2
= Farads/m
Coulombs 8.85 10 Newton-m
− = ×
and the (macroscopic) electric susceptibility of the dielectric material, χ e is a pure number
(i.e. χ e is a scalar quantity – it is dimensionless).
Then: (^) ( ) ( )
2
2 0
Coulombs Coulombs
meter
r ε χ e Eint r
G G G G
Newton
2
Newtons
-m
⎝ ⎠ Coulomb
=Coulombs m^2
For class- A dielectrics: Ρ^ ( r^ ) =^ ε χ 0 e Eint^ ( r )
G G G G
For free space (“empty” vacuum), the (macroscopic) electric susceptibility χ e = 0 because free
space/vacuum has no MATTER in it.
The electric susceptibility χ (^) e and electric polarization Ρ( r )
G G
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
G
in space as:
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 8
N (^) N o (^) N o e (^) o (^1 e )^ (^1 e ) o total electric (^) electric electric permittivity (^) permittivity permittivity of dielectric (^) of vacuum of dielectric
In some dielectrics, under certain conditions χ e → ∞. In plasmas (i.e. ionized gases), χ e < 0.
In most typical/garden-variety dielectric materials, 0 ≤ χ e ≤10.
We can also define a relative electric permittivity (a.k.a. dielectric “constant”) which is
(obviously) dimensionless:
( )
( ) ( )
e o e r e o o
K
ε^ χ^ ε
= ≡ = = + and/or: (^) e e 1 1 o
K
Consider a “real life” situation (i.e. an actual physics experiment): A Class- A dielectric block of
insulator-type material is inserted between two parallel plates, which have a potential difference
Δ V across the parallel plates of the capacitor, as shown in the figure below:
We know that: (^) ( )
0
ˆ ˆ Volts/m
free ext
V
E x x a b c
G
from the (empty) parallel plate capacitor
If the Class- A dielectric is in a uniform/constant ext
E
G
(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 (^) ( r (^) ) = Ρ ox ˆ
G G
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:
ρ Bound (^) ( r (^) ) = −∇ Ρ (^) ( r ) = 0
G K^ G G
i
However, on the RHS and LHS surfaces of the dielectric (see above figure, with
n ˆ 1 (^) = + x ˆ , n ˆ 2 = − x ˆ), that (^) Bound ( ) ˆ 1 RHS o surface
σ r n
G G
i and (^) Bound ( ) ˆ 2 LHS o surface
σ r n
−
G G
i ,
respectively, or, expressing this more compactly: (^) Bound ( ) ˆ o surface
σ r n
±
G G
i
SI Units same as
for ε o (Farads/m)
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 10
Therefore: ( ) ( ) ( ) ( ) ( ) 0
macroscopic int ext molecular ext dipoles
E r E r E r E r r
G G G G G G G G G G
Rearranging this relation:
( ) ( ) ( ) 0
E ext r Eint r r
G G G G G G
But: Ρ (^) ( r (^) ) = ε χ 0 e ( r (^) ) Eint (^) ( r )
G G G G G
∴ (^) ( ) ( )
0
E ext r Eint r
G G G G
ε 0 χ e Eint (^) ( r (^) ) = Eint (^) ( r (^) ) + χ e Eint (^) ( r (^) ) = (^) ( 1 +χ e ) Eint (^) ( r )
G G G G G G G G
Thus: Eext^ ( r^ ) =^ ( 1 +^ χ e ) Eint^ ( r )
G G G G
or: Eint^ ( r^ ) =^ Eext^ ( r ) ( 1 +^ χ e )
G G G G
We see that the macroscopic/averaged-over internal electric field inside the dielectric Eint (^) ( r )
G G
is
reduced by a factor of (^1) ( 1 + χ e )relative to the external/applied electric field Eext (^) ( r )
G G
, because the
electric field associated with the (now polarized) molecular dipoles, (^) ( )
macroscopic molecular dipoles
E r
G G
opposes the
external applied electric field! Using the dielectric constant, Ke ≡ ε ε o = (^) ( 1 + χ e )we see the same
thing, namely that Eint (^) ( r (^) ) = Eext (^) ( r (^) ) ( 1 + χ e ) = Eext (^) ( r (^) ) Ke
G G G G G G
i.e. the internal electric field is
“screened” / reduced from the Eext
G
value by the dielectric constant K of the dielectric material.
We can also show that, since: Ρ (^) ( r (^) ) = ε χ 0 e Eint (^) ( r )
G G G G
Then: Eint (^) ( r (^) ) = Ρ( r ) ε 0 χ e
G G G G
and Eext =( σ (^) free ε 0 ) x ˆ
G
( ) ˆ^ o Bound { ( ) o ˆ} surface
Ρ r ⋅ n = Ρ = σ Ρ r = Ρ x
G G G G
Thus: (^) ( )
( )
0 0 0
o Bound int e e e
r (^) x E r x
G G
G G
and: (^) ( ) ( )
free int ext e e o
E r E r x
⎝ +^ ⎠ ⎝ + ⎠ ⎝ ⎠
G G G G
Then we see that:
0
Bound^ free
e e o
σ^ σ
or: 1
e Bound free e
or: 1
Bound e
free e
⎝ ⎠ ⎝^ ⎠
But:
0
e K^ e 1,^ or: 1 e Ke
e e Bound free free free e e
K
K
⎝ ⎠ ⎝^ ⎠ ⎝ + ⎠
i.e. The bound surface charge density σ 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 density σ Bound and surface
charge density σ free is NOT a universal one!!! It is specific only to the case of the parallel-plate
capacitor!!!
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 11
The potential difference Δ V between the two capacitor plates of the parallel plate capacitor is:
ext int ext C
Δ V = − E d = aE + bE + cE ∫
G G
i A
If a = c (i.e. the air gaps in the parallel plate capacitor the same dimension)
Then: Δ V = 2 aEext + bEint But:
int ext e
E E
K
= ∴ (^2) ext e
b V a E K
Define: d ≡ (^) ( 2 a + b )= total gap between parallel plates of capacitor.
Now: 0
free Eext x
G
0
free
e
b V a K
Capacitance of parallel plate capacitor:
Q (^ free A ) C V V
Capacitance of the ||-plate capacitor (including the dielectric):
free A^ free C V
σ^ σ
free
e
A
b a K
⎛ ⎞^ σ
0
0
e
A
a b K
If there is no dielectric, then (^) ( ) 0 0
1 = vacuum e
K
= = = and b = 0, d = 2 a
Then:
0 0 no dielectric 2
A A
C
a d
If there are no air gaps, then a = c = 0 and d = b
Then:
0 0 dielectric e e no dielectric
e
A A
C K K C
d (^) d K
ε ⎛ ε ⎞ = = = ⎜ ⎟ ⎝ ⎠
A = surface area of one of the
plates of the ||-plate capacitor
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 13
But: 12 = − 34 ≡
G G G
A A A ⇒ ∴ (^) ( Evac − Edie A) = 0
G G G
iA But: Evac ||
G G
A and Ediel ||
G G
A
∴ Evac = Edie A
G G
at the surface/boundary of the dielectric.
Specifically:
tangent tangent Evac = Ediel
G G
@ the interface/boundary of the dielectric.
More generally:
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 solids.
Dielectric “constant” (^) ( ) 0
K (^1) e
ε χ ε
12
− = × Farads/m
= electric permittivity of free space/vacuum
= macroscopic constant/scalar quantity
= constant @ all frequencies (Lorentz invariant quantity)
ε = electric permittivity of dielectric χ e = 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 parameters ε , χ e (and thus K e ) have/exhibit frequency 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 !!!
The tangential components of E
G
are equal @ a dielectric interface
i.e. E 1 (^) t = E 2 t @ the interface of dielectric.
The tangential component of E
G
is continuous across a dielectric interface.
Dielectric “Constants”
of various materials at
STP and f = 0 Hz.
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 14
THE MACROSCOPIC ELECTRIC DISPLACEMENT FIELD, D^ (^ r )
G (^) G
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:
Bound volume charge density: (^) ( ) ( ) ( )
3
ρ Bound r = −∇ Ρ r Coulombs meter
G K^ G G
i
Bound surface charge density: (^) ( ) ( ) ( )
2 ˆ
σ Bound r = Ρ r n surface Coulombs meter
G G G
i
We have also shown that the E
G
-field inside a dielectric medium due to the electric polarization,
Ρ ( r )
G G
is simply (equivalently) due to the bound charge distributions ρ Bound (^) ( r )
G
and/or σ (^) Bound ( r )
G
Suppose now that this dielectric also had embedded in it free electric charges – e.g. either
embedded electrons 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:
ρ Tot ( r (^) ) = ρ Bound ( r (^) ) +ρ free ( r )
G G G
Then Gauss’ Law (in differential form) becomes:
ε 0 ∇ ETot (^) ( r (^) ) = ρ Tot ( r (^) ) = ρ Bound ( r (^) ) +ρ free ( r )
K G G G G G
i
where: ETot (^) ( r (^) ) = total electric field = " Ebound (^) ( r (^) ) " +" E (^) free ( r )
G G G G G G
and ρ (^) Bound ( r (^) ) = −∇ Ρ( r )
G G G G
i
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!!!):
( ) ( ) ( ( ) ( ))
( )
0 0 (^ )
Electric Displacement
Bound free
D r
ε E r ρ r ε E r r ρ r
≡ =
G (^) G
G G G G G G G G G G
i i
The (macroscopic) Electric Displacement Field: D r (^ ) ≡^ ε 0 E r (^ ) + Ρ( r )
G G G G G G
SI units of D r ( )
G G
are the same as that for Ρ( r )
G G
(same as that for σ Bound & σ free !!):
2 Coulombs m
Then we realize that Gauss’ Law (for dielectrics) becomes: ∇ D r ( (^) ) = ρ free ( r )
G G G G
i
i.e. the divergence of the (macroscopic) D
G
-field at the point (^) ( r )
G
is due to (i.e. equal to) the
free volume charge density, free ρ that is present at the point (^) ( r )
G
In integral form, Gauss’ Law (for dielectrics) becomes: ( )
encl free S
D r dA Q
′
∫
G G G
i v
Gauss’ Law for D
G
physically tells us that the electric displacement field, D r ( )
G G
is sensitive to the
free charge that is present in a given situation, whereas Gauss’ Law for (^) E
G
tells us that the electric
field intensity E r ( )
G G
is sensitive to the total charge that is present in this same situation. Gauss’ Law
for Ρ
G
tells us that Ρ( r )
G G
is sensitive to the bound charge that is present in this same situation.
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 16
In Cylindrical Coordinates:
D (^) ( 2 π s L ) = λ L
enclosed = Qfree = r = r ˆ= r ˆ = = r = r
K G G G
s s s s s
Thus: ( ) ˆ 2
D r r r
G G
(Coulombs/m
2 )
Note that this formula holds inside the rubber dielectric (^) ( r < a )as well as outside the rubber
dielectric (^) ( r > a ), i.e. this formula is valid for any r.
However, since Ρ (^) ( r > a ) = 0
G
(i.e. no rubber dielectric for r > a )
Then: (^) ( ) ( ) 0 0
E r D r r r
G G G G
for r > a
Inside the rubber dielectric (^) ( r < a ), since we do not explicitly know the analytic form of
Ρ (^) ( r < a )
G
then we do not know E r ( < a )
G
. Note also that (here) neither (^) ( ) Bound
ρ r < a nor
σ (^) Bound ( r = a )have been specified.
CAUTIONARY STATEMENTS ABOUT THE ELECTRIC DISPLACEMENT D r ( )
G G
AND THE ELECTRIC POLARIZATION Ρ( r )
G G
Inside Class- A dielectric materials, the so-called constitutive ( a. k. a. auxiliary) relation between the
three fields D r ( (^) ) ≡ ε 0 E r ( (^) ) + Ρ( r )
G G G G G G
holds/is true/valid.
Coulomb’s Law is true for ETot (^) ( r )
G G
, because E r ( )
G G
is a conservative field, i.e. it is derivable from a
scalar potential (^) ( E r ( (^) ) = −∇ V (^) ( r ))
G G G
, and the ∇ × E r ( ) = 0
G G G
(always) in electrostatics problems:
( )
( ) 2 0
Tot v
r E r d
∫
G
G G
r r
with ρ Tot ( r (^) ) = ρ Bound ( r (^) ) +ρ free ( r )
G G G
or: (^) ( )
( ) 2 0
Tot S
r E r dA
∫
G
G G
r r
with
encl encl encl QTot = QBound + Qfree
or: (^) ( )
( ) 2 0
4 C
r E r d
∫
G
G G
r A r
The same/analogous thing is not true for the electric displacement, D r ( )
K K
nor is it true for the
electric polarization, Ρ( r )
G G
, because neither D r ( )
K K
nor Ρ( r )
G G
are conservative, and neither is
derivable from (the negative gradient of) a scalar potential. As consequences of these facts:
( )
( )
2
free v
r D r d
π ′^
∫
G
G G
r
r
and ( )
( ) 2
Bound v
r r d
π ′^
∫
G
G G
r
r
( )
( )
2
free S
r D r dA
π ′^
∫
G
G G
r
r
and ( )
( ) 2
Bound S
r r dA
π ′^
∫
G
G G
r
r
( )
( ) 2
free
C
r D r d
∫
G
G G
r A
r
and ( )
( ) 2
Bound C
r r d
∫
G
G G
r A
r
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 17
E r ( )
G G
is a fundamental field. E r ( )
G G
is a conservative field.
D r ( )
G G
and Ρ( r )
G G
are not fundamental fields. D r ( )
G G
and Ρ( r )
G G
are not conservative fields.
D r ( )
G G
and Ρ( r )
G G
are auxiliary fields.
While D r ( (^) ) = ε 0 E r ( (^) ) + Ρ (^) ( r (^) ) ⇒ ∇ D r ( (^) ) = ε 0 ∇ E r ( (^) ) + ∇ Ρ( r )
G G G G G G G G G G G G G G G
i i i holds/is true/valid for Class- A
dielectrics, the divergence of a vector field on its own is insufficient to uniquely determine/fully-
specify the nature of a vector field.
Both ∇ A r ( )
G G G
i and ∇ × A r ( )
G G G
must be specified in order to uniquely determine the A r ( )
G G
-field.
Now ∇ × E r ( ) = 0
G G G
always (^) ( E r ( (^) ) (^) ( and FE (^) ( r ))are conservative)
G G G G
But ∃ many situations where
( )
( )
(^0) has permanent electric polarization
.. a bar electret 0 - analogous to bar magnet!!!
D r e g r
⎧ ∇ × ≠ ⎫ ⎛ ⎞
⎪∇ × Ρ ≠^ ⎪ ⎝^ ⎠
G G G
G G G
D r ( )
G G
and Ρ( r )
G G
are auxiliary fields associated with matter – dielectric materials in particular.
BOUNDARY CONDITIONS ON THE ELECTRIC DISPLACEMENT, D
G
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: D 1
G
n ˆ 1 on dielectric interface.
Shrink height h of pillbox
Dielectric S 1 ′^ to zero/infinitesimally small.
Material # 1:
1 ,^1 , 1
e e
ε K χ Free charge surface density,
Boundary/ σ free exists on interface
Interface h Δ S
σ free
Dielectric
Material # 2: S 3 ′
2 2 2
e e
ε K χ
3 n ˆ
2
S ′
n ˆ 2
D 2
G
BOUNDARY CONDITIONS ON D , E andΡ
G G G
for DIELECTRIC MATERIALS
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 19
BOUNDARY CONDITIONS ON THE ELECTRIC FIELD, E
G
AT AN INTERFACE
We have already shown (see pages 12-13 of these lecture notes) that taking the contour integral
( ) 0 C
E r d = ∫
G G G
i A v
across an interface between two dielectrics told us that the tangential components
of E
G
are continuous across a dielectric interface: (^1) t 2 t interface interface
E = E
( ) (^1 1 2 ) C
E r d = E + E ∫
G G G G G G G
i A iA iA v 3 3 4 4
+ E + E
G G G G
iA iA = 0 where 1 = − 3 =
G G G
A A A
with: (^1 1) t and 3 2 t interface interface interface interface
E = E − E = − E
G G G G G G
iA iA
Thus: (^1) t 2 t interface interface
E = E or: 1 sin 1 2 sin 2 interface interface
E θ = E θ
If e.g. medium #1 is a conductor, then E 1 (^) = 0
G
inside the conductor.
If E 1 (^) = 0
G
inside the conductor, then D 1 = ε E 1 = ε 0 E 1 + P 1 = 0
G G G G
⇒ D 1 (^) = 0 and P 1 = 0
G G
inside conductor
∴ For conductor-dielectric interface:
Material #1 is conductor and material #2 dielectric medium, then:
D 1 = E 1 = P 1 = 0
G G G
and D 2 n = σ free and E 2 t = 0
Note that the potential (^) ( ) interface
V r
G
physically must be continuous at an interface between two
materials, whether they are dielectrics or otherwise!
Also: From Gauss’ Law for E
G
: (^) ( ) 0
enclosed Tot S
Q
E r dA
∫
G G G
i v
At a dielectric interface, as drawn on page 17 above, we see that:
[ 2 1 ] 0 0
Tot^ bound^ free E (^) n En (^) interface
σ^ σ^ σ
Shrink height h of
contour C to 0,
Just above &
below interface.
Medium 1
Medium 2
E 1 (^) ( r )
G G
E 2 (^) ( r )
G G
Contour C
θ 1
θ 2
The tangential components
of E
G
are continuous across
a dielectric interface
The normal components
of E
G
are discontinuous
across a dielectric interface
by the amount σ Tot ε 0
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 20
BOUNDARY CONDITIONS ON THE ELECTRIC POLARIZATION Ρ
G
AT AN INTERFACE
From Gauss’ Law for E :
G
( ) 0 0
enclosed^ enclosed^ enclosed Tot free^ Bound S
Q Q^ Q
E r dA
∫
G G G
i v
Now: D r ( (^) ) ≡ ε 0 E r ( (^) ) + Ρ( r )
G G G G G G
so: (^) ( ) ( ) ( ) 0 0
E r D r r
G G G G G G
∴ (^) ( ) ( ) ( ) ( )
0 0 0
(^1 1 1) enclosed enclosed
S E r^ dA^ S D r^ r^ dA^ Q^ free^ QBound
∫ ∫
G G G G G G G G
i i v v
or: (^) ( ) ( )
enclosed enclosed free Bound S S
D r dA r dA Q Q ′ ′
∫ ∫
G G G G G G
i i v v
But we already know that: (^) ( )
enclosed S ′ D r^ dA^ Qfree
∫
G G G
i v and (^) ( )
enclosed S ′ r^ dA^ QBound
∫
G G G
i v
Take a (shrunken) Gaussian pillbox centered on the interface as shown in figure below:
So: (^) ( )
enclosed S ′ r^ dA^ QBound
∫
G G G
i v Get:
P^1 P^2
1 1 2 2
[ ˆ^ ˆ]
n n
n n S σ Bound S
=Ρ =Ρ
G G
i i But: n ˆ 2 (^) = − n ˆ 1
Thus: (^2) n ( ) (^1) n ( ) Bound interface
⎡ Ρ r − Ρ r ⎤ = − σ
G G
Since: D r ( (^) ) = ε 0 E r ( (^) ) + Ρ( r )
G G G G G G
we can also write this out for normal and tangential components as:
( ) 0 ( ) ( ) ni ni ni
D r = ε E r + Ρ r
G G G
and ( ) 0 ( ) ( ) ti ti ti
D r = ε E r + Ρ r
G G G
Both of these component relations are valid on each side of interface, i.e. for the i
th media, i = 1, 2.
Then: ( )
2 1 2 1
2 1 0 0
and
1 1 at the interface of two dielectrics
n n free n n bound
n n ToT free bound
D D P P
E E
⎧ −^ =^ −^ = − ⎫
The tangential relations for fields at the interface are: D 2 (^) t − D 1 (^) t = P 2 (^) t − P 1 t ⇐ Not necessarily = 0!
and: E 2^ t −^ E 1 t =^0 ALWAYS (for electrostatics)!!!
The normal components of Ρ( r )
G G
are discontinuous
at an interface by the amount − σ bound
