dr. hugh blanton entc 3331. electrostatics in media
TRANSCRIPT
Dr. Hugh Blanton
ENTC 3331
Electrostatics in Media
Dr. Blanton - ENTC 3331 - Energy & Potential 3
• A medium (air, water, copper, sapphire, etc.) is characterized by its relative permittivity, (r).
Medium rvacuum 1
air 1.0006
conductors 1
glass 4.5 - 10
Dr. Blanton - ENTC 3331 - Energy & Potential 4
• Media can be grouped in two classes:conductors dielectrics
(insulators, semiconductors,
etc.)
free charges no free charges
charges will move until the conductor is field free
charges in the material are polarized by external fields
everywhere (this assumes we are dealing with an electrostatic problem with
0E
PED
oelectric flux density
field strength
polarization field
0 t
xE
Dr. Blanton - ENTC 3331 - Energy & Potential 5
+
+
+ +
+
+
+
+
xE
xE
xE
E
oP
D
0S
0S
Orientation of dipoles inside medium
Dr. Blanton - ENTC 3331 - Energy & Potential 6
• and are defined to be parallel.• A dielectric with field has positive and
negative surface charges on opposite sites.
E
P
++++
dielectric
Dr. Blanton - ENTC 3331 - Energy & Potential 7
• The polarization field is antiparallel to the polarization .
• The field inside the medium is smaller than the external field.
xE
P
oP
PE
mediumx EE
Dr. Blanton - ENTC 3331 - Energy & Potential 8
Microscopic Reasons for Induced PolarizationMicroscopic Reasons for Induced Polarization
• Deformation polarization in non-polar materials such as glass:
+
atom
+
polarized
atom
xE
Dr. Blanton - ENTC 3331 - Energy & Potential 9
• Orientation polarization in polar materials.
O
HH
O
H
H
xE
before dipoles line up
+ +
xE
after dipoles line up
Dr. Blanton - ENTC 3331 - Energy & Potential 10
• Note:• Isotropic implies that the , , and
fields are in the same direction. • Anisotropic implies that the , , and
fields may have different directions.• We limit the media to those that are linear,
isotropic, and homogeneous.• For such media, the polarization field is:
DE
P
DE
P
EP
eo
Electric susceptibility
Dr. Blanton - ENTC 3331 - Energy & Potential 11
• Since
• It follows that
• Materials with large permittivity also have a large susceptibility!
PED
o
EEED
eooeoo
ED
oe 1
r
Dr. Blanton - ENTC 3331 - Energy & Potential 12
Boundaries Between DielectricsBoundaries Between Dielectrics
• Maxwell’s equations are of general validity• In particular
dielectric1
dielectric2
Different amounts of surface charge at the
boundary.
What fields are at the boundary?
0ˆ0 C
drot lEE
Dr. Blanton - ENTC 3331 - Energy & Potential 13
• Construct a suitable path, C, about the boundary.
• and split the field into normal (n) and tangential (t) components.
ld
ld
a b
cd
2h
2h
medium1
medium2
0ˆˆˆˆˆ c
b
d
c
a
d
b
aC
ddddd lElElElElE
t
2E
n
2E
2E
t
1E
n
1E
1E
1
2
Dr. Blanton - ENTC 3331 - Energy & Potential 14
• Now make h smaller and smaller• This implies
• and
• Which implies
0ˆ c
bdlE
0ˆ
a
ddlE
0ˆˆ12
d
c
b
add lElE
Below boundary Above boundary
Dr. Blanton - ENTC 3331 - Energy & Potential 15
• Now, make l smaller and smaller, but not zero
tttt
12120ˆˆ EElElE
Dr. Blanton - ENTC 3331 - Energy & Potential 16
• Boundary conditions for the tangential components of the fields.• Across the boundary between any
media, the tangential component of is unchanged
• in all cases
tt EE12
E
tt DE111
tt DE 222
Dr. Blanton - ENTC 3331 - Energy & Potential 17
• However
• because
tt DD 22
11
ED
Dr. Blanton - ENTC 3331 - Energy & Potential 18
• Now use
• Construct suitable volume, V
QdVDdivsdDDdivS V
ˆGauss’s Law
3h
h32
medium1
medium2
n1D
upn
downn
S
S2
1
n2D
dVdS V
s sD ˆ
The only charge inside V is the surface charge on the boundary area S
Dr. Blanton - ENTC 3331 - Energy & Potential 19
S
Sdown
S
up dsdsdsd nDnDsD ˆˆˆ 21
3h
h32
medium1
medium2
n1D
upn
downn
S
S2
1
n2D
Let h go to zero,
Now, make the Gaussian surface smaller and smaller, but not zero
sincedsdsds Snn 21 DD
downup nn ˆˆ
Dr. Blanton - ENTC 3331 - Energy & Potential 20
• This implies
Snn 21 DD
Dr. Blanton - ENTC 3331 - Energy & Potential 21
• Boundary conditions for the normal component of the fields across the boundary between any two media.
• which implies
Snn 21 DD
Snn 2211 EE
Dr. Blanton - ENTC 3331 - Energy & Potential 22
Application of Boundary ConditionsApplication of Boundary Conditions
• Given that the x-y plane is a charge-free boundary separating two dielectric media with permittivities 1 and 2.• If the electric field in medium 1 is
• Find• The electric field in medium 2, and• The angles 1 and 2.
zyx 1111 ˆˆˆ EzEyExE
2E
Dr. Blanton - ENTC 3331 - Energy & Potential 23
• What are the angles between 1 and 2 between and , as well as between and .• For any two media:
• With no charges (charge free) on the boundary plane
1E
z2E
z
1E
t1E
n1E
t2E
2E
n2E
x-y plane
z
tt21 EE
nnS
nn22112211 0 EEEE
Dr. Blanton - ENTC 3331 - Energy & Potential 24
• It follows that:
• since the z-component of the field is the normal component of the field.
zyxzyx EzEyExEzEyExE 12
1112222 ˆˆˆˆˆˆ
Dr. Blanton - ENTC 3331 - Energy & Potential 25
• The tangential components for and are:
• Then
1E
2E
21
211 yx
t EE E
22
222 yx
t EE E
and
z
yx
E
EE
1
21
21
1tan
z
yx
E
EE
2
21
21
2tan
and
Dr. Blanton - ENTC 3331 - Energy & Potential 26
z
yx
E
EE
1
21
21
1tan
z
yx
z
yx
E
EE
E
EE
12
1
21
21
12
1
22
22
2tan
and
Dr. Blanton - ENTC 3331 - Energy & Potential 27
• The relation looks very similar to Snell’s law of Refraction
1
2
12
1
21
21
1
22
22
1
2
tan
tan
z
yx
z
yx
E
EE
E
EE
2
1
22
11
2
1
sin
sin
n
n
Dr. Blanton - ENTC 3331 - Energy & Potential 28
Dielectric with Conductor BoundaryDielectric with Conductor Boundary
• Very important practically:• Capacitor• Coaxial shielded cable
• External field cannot penetrate inside the shield.
shield
Dr. Blanton - ENTC 3331 - Energy & Potential 29
• Boundary conditions:
• Since a conduct is free field
tt21 EE
Snn 21 DD
conductor
dielectric
02 tE
02 nD
02 tE
Dr. Blanton - ENTC 3331 - Energy & Potential 30
• Field lines at a conductor surface have no tangential component.• They are always perpendicular to the
conductor surface!
• In addition
• The surface charge on the conductor defines the field in the surrounding dielectric
Snn 21 DD
Dr. Blanton - ENTC 3331 - Energy & Potential 31
• Conducting slab
• Bottom surface:• Normal and field are in opposite directions.
• Top surface:• Normal and field are in same directions.
+ + + + + + + + + + + + E
11 n
n
conductor
0bs
0ts
Dr. Blanton - ENTC 3331 - Energy & Potential 32
• Since the conductor is field-free
• And since , the magnitude of the surface charge densities is given by the product of permittivity and field strength.
bs
ts
Snn 21 DD
Dr. Blanton - ENTC 3331 - Energy & Potential 33
• Dielectric slabcapacitor
• Most general capacitor
Parallel plate
capacitor
++ + +
++
++ +
V
Conductor 1
Conductor 2
E
Dr. Blanton - ENTC 3331 - Energy & Potential 34
• Because the conductors must have inside,
• To achieve this, the charges distribute on the two surfaces.• There are equilibrium currents until
everything is stationary.
• Very fast—speed of light.
0E
0
t
E
Dr. Blanton - ENTC 3331 - Energy & Potential 35
• The surface charges on conductor 1 and conductor 2 give rise to the field with
• This implies that the total charge on either conductor is:
E
nS E
dsdsdsQSS
n
S
S EE
Definition of surface charge density
Boundary conditions (no tangential
component
Dr. Blanton - ENTC 3331 - Energy & Potential 36
• The potential difference V along any one of the field lines is given by:
C
dV lE ˆ
Dr. Blanton - ENTC 3331 - Energy & Potential 37
• Capacitance is the charge per potential difference.
FFaradsd
A
Ed
EA
d
d
V
QC
C
S
lE
sE
ˆ
ˆ
Dr. Blanton - ENTC 3331 - Energy & Potential 38
• The capacitance of a parallel-plate capacitor is• proportional to area A.• inversely proportional to separation, d.• proportional to the permittivity of the
dielectric filling.• independent of E
Dr. Blanton - ENTC 3331 - Energy & Potential 39
Summary of ElectrostaticsSummary of Electrostatics
• The sources of the electrostatic field are time-independent charge distributions.• That is, the charge distributions are static
(derivative is zero).• Electrostatics follows from the empirical facts
of• Coulomb’s law• The principle of linear, vectorial superposition of
forces and fields.• Energy conservation.
Dr. Blanton - ENTC 3331 - Energy & Potential 40
Summary of ElectrostaticsSummary of Electrostatics
• Electrostatics can be based on two fundamental Maxwell equations;• •
• The electric field is free from circulation ( ) and can always be expressed as the gradient of a potential ( ).
0 EE
rotD
div
0E
rot
VE
Dr. Blanton - ENTC 3331 - Energy & Potential 41
• Potential and Fields can be calculated for a given charge distribution, • from the field definition• using Gauss’s Law• using image charges
• Conducting and dielectric media can be distinguished.
• At boundaries between media, the following conditions hold:• •
tt21 EE
S
nn 21 DD