Summary page: Electrostatic boundary conditions for conductors

1
Summary page Electrostatic boundary conditions
for conductors

• in the absence of current flow we have the
  following conditions for a conductor
• the electric field (and D as well) inside is
  identically zero
• at the surface of a conductor, the field is
  everywhere normal to that surface
• the conductor is an equipotential
• at the surface of a conductor, any normal
  component of the field induces a surface charge
  that is proportional to the field strength

2
Summary page Electrostatic boundary conditions
for dielectrics

• at the interface between two dielectrics
• the component of the electric field tangent to
  the interface between two dielectrics is
  continuous
• Dtan is discontinuous
• the component of the D field normal to the
  interface between two dielectrics is continuous
• if there is no free charge at the interface
• if there is a surface charge
• Enorm is discontinuous
• material properties

3
Summary page Electrostatic boundary conditions
as you cross the surface between two media

• at the interface between two materials
• the component of the electric field tangent to
  the interface between two materials (conductor or
  dielectric!) is continuous
• for a conductor with no currents Einside Etan2
  0 ? Etan conductor 0
• for a dielectric Dtan is discontinuous ? Dtan
  1/e1 Dtan 2/e2
• the component of the D field normal to the
  interface (i.e., perpendicular to the interface)
  between two materials is changes by the free
  surface charge density
• for a conductor with no currents Dinside D- 2
  0 ? D- cond rS
• for a dielectric with no surface charge D- 1 D-
  2

4
Conductor/vacuum example parallel plates

• consider two conductive parallel plates that are
  much wider than the distance by which they are
  separated
• assume we have charged these two plates, top
  plate with Q and bottom plate with Q
• pretty much all the charge will be on the inside
  faces of the plates
• rsurf Q/plate area

conductor
eo
D eoE
d
conductor
gaussian surface ? DDS rsurfDS ? D rsurf
? E rsurf/eo
gaussian surface, no charge inside ? DDS 0 ?
D 0 ? E 0
5
Conductor/dielectric example parallel plates

• now what happens if we carefully add a slab of
  dielectric that completely fills the gap between
  the plates?
• the charge on the two plates does not change
• so D does not change since rsurf Q/(plate area)
  did not change
• but E DOES change, in fact it is REDUCED by er
  compared to the original value
• with eo E E rsurf/eo
• with dielectric E (rsurf/eo)/er REDUCED as
  expected

conductor
D ereoE
ereo
d
conductor
gaussian surface ? DDS rsurfDS ? D rsurf
? E rsurf/ereo
6
Potential difference between the two plates

• integrate between bottom and top
• notice that here the ratio of charge to voltage
  is a function only of geometry (plate area A and
  plate separation d) and the dielectric constant e
  

7
Capacitance

• for two oppositely charged conductors the ratio
  of charge to potential difference is the
  capacitance of the system
• the capacitance is a function only of the
  geometry and the dielectric constant(s)

8
Parallel plates, partially filled

• what happens if we only partially fill the gap
  between the plates with a dielectric?
• the total charge on the two plates does not
  change
• symmetry insures that the field points straight
  from one plate to the other
• BCs at a dielectric interface
• Dnorm is continuous (if there is no free surface
  charge at the interface)
• so D does not change since rsurf Q/(plate area)
  did not change

? DDS rsurfDS ? D rsurf Q/A
9
non-uniformly filled parallel plate capacitor

• D is continuous
• but E is NOT
• lower region with dielectric 1 E1 rsurf/eoer1
  Q/(Aeoer1)
• upper region with dielectric 2 E2 rsurf/eoer2
  Q/(Aeoer2)
• so the voltage difference between the plates is

10
non-uniformly filled parallel plate capacitor

• the voltage difference between the plates is
• and the capacitance is then

11
non-uniformly filled parallel plate capacitor

• the capacitance is
• where C1 and C2 are the capacitances you would
  get from two separate capacitors with respective
  plate areas, separations and dielectric constants
  

12
non-uniformly filled parallel plate capacitor

• C1 and C2 are the capacitances you get from two
  separate capacitors, then connect them in series
  to get the total!
• does this make sense?
• looks like two capacitors in series

13
non-uniformly filled parallel plate capacitor
conductor
this is an equipotential surface
er2eo
d
er1eo
d1
conductor

• a conductor is also an equipotential surface
• you can always insert a perfectly conducting
  sheet (i.e., its infinitely thin) on an
  equipotential surface

14
Parallel plates, partially filled

• what happens if we fill the different areas
  between the plates with different dielectrics?
• symmetry insures that the field points straight
  from one plate to the other
• the two metals are equipotentials
• potential difference between the two plates is
  the same in both region 1 and 2
• BCs at a dielectric interface
• Etan is continuous
• so E is the same on both sides of the dielectric
  interface
• ? E V/d
• but how much charge is there?

15
Parallel plates, partially filled

• E V/d
• recall that the change in the normal component of
  D equals the interface surface charge
• Dnorm dielectric Dnorm conductor rsurface
• for an interface with a conductor Din conductor
  0
• ? Dnorm dielectric rsurface

16
Parallel plates, partially filled

• the charge is just arearsurface
• looks like two capacitors in parallel

17
Examples coax
z

• inner conducting cylindrical wire, radius a
• since its a conductor, all charge is on the
  outside
• outer conducting cylinder
• all the charge is on the inside
• by symmetry, the field points radially outward
  from the outer surface of the inner conductor to
  the inner surface of the outer conductor

18
Field between coaxial cylinders

• the field produced by the charge on the inner
  conductor can be found using Gausss law, exactly
  the same way we did it for a line charge
• this looks just like a line charge
• for line charge on the z-axis, field is
  cylindrically symmetric

a

• gaussian surface, radius r

19
Field between coaxial cylinders

• the field produced by the charge on the outer
  conductor can also be found using Gausss law
• the charge on the outer conductor produces NO
  field inside (since no charge would be enclosed)
• now use superposition and just add the two

20
Coaxial cylinders

• weve already found the potential difference
  between two points in the field of a line charge
• so the capacitance is just
• or the capacitance per length is just

21
Coaxial cylinders

• summary
• check
• if b ? a then this should look like a parallel
  plate capacitor

l
22
Non-uniformly filled coax

• dielectric ring, inner radius d1, outer radius
  d2
• since cylinders are equipotentials we can insert
  thin metal sheets on the surfaces of the
  dielectric ring
• now we have three capacitors in
• series?
• parallel?
• this is exactly analogous to the parallel plate
  problem with the sheet of dielectric added
• that was a series connected problem, so this is
  too!

23
Non-uniformly filled coax

• dielectric wedge, filling a region of angle q
• the two metal surfaces must be equipotentials
• use continuity of Etan at dielectric interfaces
• find Ds, then surface charge densities on the
  conductors
• this is exactly analogous to the parallel plate
  problem with the part of the width filled with
  dielectric
• that was a parallel connected problem, so this is
  too!

• q measured in radians!!!!

24
Our first piece of the telegraph puzzle

• we now have something were going to need to
  understand the submarine telegraph cable the
  capacitance of coax!
• Im not quite ready yet to tackle the whole
  thing, so lets look at another pair of wires
  twin lead

25
Examples twin lead

• this problem is NOT the same as superposition of
  two uniformly charged cylindrical wires
• as the two wires get closer together
• the attractive forces between the opposite
  charges causes a redistribution of charge on the
  conductor surfaces
• results in a higher surface charge density on the
  inside faces

26
Twin line-charges

• lets look at two line charges
• in this case theres no surface to redistribute
  charge on
• we can use simple superposition to do this one
• this should be approximately correct for two
  finite radius wires that are far apart
• weve already found the fields and the potential
  difference between two points in the field of a
  single line charge located on the z-axis
• where a is the distance from the line charge to
  the end point, and b is the distance from the
  line charge to the starting (reference) point

27
Twin line-charge electric field

• superimpose two line charges, equal but opposite
  charges

y
R2
P(x,y)
R1
z
X
a
a
28
Twin line-charge potential

• superimpose two line charges, equal but opposite
  charges

y
reference point B V 0
P(x,y)
R2
b
b
R1
z
X
a
a
29
Twin-line charge capacitance

• can we get the capacitance?
• looks like all we need is the voltage difference
  between the two line charges
• i.e. our observation point needs to be on one of
  the line charges
• but if either R1 or R2 0 the potential is
  infinite!!!
• you CANNOT find the capacitance for point
  problems!!

30
Twin line charges equipotential surfaces

• what do the equipotential surfaces look like?
• set V constant V1 , then find the curve (xep,
  yep) that keeps V constant

31
Twin line charges equipotential surfaces

• what do the equipotential surfaces look like?
• set V constant V1

• equipotential surfaces are circles!!!
• with center at
• and radius

32
Twin line-charge equipotentials

• equipotential surfaces are circles
• with center
• and radius
• so what can this tell us about two finite radius
  wires??
• recall that we can insert a metal sheet on any
  equipotential surface
• so since the equipotentials are circles, why not
  make them our wires!
• we need to relate the centers of our wires to the
  centers of the equivalent line charges

calculation
33
Twin lead

• from the twin line-charge problem fill an
  equipotential circle with a conducting wire
• the fields outside the wires will be unchanged!
• we need to relate the centers of our wires to the
  centers of the equivalent line charges
• wire centers equipotential center
• wire radius equipotential radius
• the equivalent line charges have
• centers at /- a

wires
34
Twin lead
wires
line charges

• if we know the potential, radius, and separation
  of the wires we really want
• call the wire center-to-center separation 2h
• call the wire radius b
• set the wire potential to Vo
• - wire is at Vo
• can we find the position and charge on the
  equivalent line charges?
• find a in terms of h, b, and Vo
• find rl Q/l in terms of h, b, and Vo
• we have two equations and two unknowns!!!

a
h
b
wire center
wire radius
35
Twin lead

• we have two equations and two unknowns!!!

36
Twin lead

• we have two equations and two unknowns!!!

37
Twin lead

• if h b are given
• then we can find the line charge separation
• we would also like a relation between charge and
  voltage between the wires Vo

• but recall we also found

38
Twin lead capacitance

• so for twin lead
• wire separation 2h
• wire radius b
• potential difference between the two wires 2Vo
• the relation between wire voltage and wire charge
  is
• and so the capacitance is just

39
Twin lead capacitance

• what happens when the wires are close together
• h b d
• h/b 1 d/b , d/b 0

• even as the wires get close together it NEVER
  looks like a parallel plate capacitor!

40
Twin lead capacitance

• what happens when the wires are far apart
• h/b gtgt 1
• the approximation works pretty well for h/b gt 2

41
Twin lead equipotentials

• E-field is everywhere perpendicular to the
  equipotential surfaces
• recall that E spatial rate of change of
  potential
• densely packed equipotential lines indicate
  large electric field

steps between Vs are constant size
42
Circular wire over a ground plane

• can we do this one?

43
Calculators

• coax, twinlead
• http//www.mogami-wire.co.jp/e/cad/electrical.html
  

44
non-uniformly filled parallel plate capacitor

• Dnormal is continuous at dielectric charge free
  interface ? D is the same everywhere (D
  rsurf Q/A)
• but E is NOT continuous
• lower region with dielectric 1 E1 rsurf/eoer1
  Q/(Aeoer1)
• upper region with dielectric 2 E2 rsurf/eoer2
  Q/(Aeoer2)

45
Parallel plates, partially filled

• Etan is continuous ? E is the same everywhere (E
  V/d)
• but D is NOT continuous

46
What about this???
conductor
-
-
-
-
-
-
-
-
-
-
-
-
-
er3eo
er1eo
er2eo













conductor

• Etan tries to be continuous across the dielectric
  interface ? E tries to be the same everywhere
  ? D tries not to be the same everywhere

• but Dnormal also tries to be continuous at
  dielectric interface ? D tries to be the same
  everywhere ? E tries not to be the same
  everywhere

• THESE ARE INCONSITENT WITH ONE ANOTHER!!!!
• the field simply cant point straight up
  everywhere, in particular theres a problem near
  the corner where the three dielectrics meet!

47
Capacitors with multiple (non-uniform)
dielectrics

• if you have a simple uniform dielectric
  solution
• if the interfaces between any added dielectrics
  coincide with an equipotential surface in the
  original problem
• the field lines are then perpendicular to the
  interfaces
• make use of continuity of Dnormal to get new
  solution
• imagine inserting a thin sheet of metal at the
  interface between the dielectrics
• nothing changes
• you now have two caps in series
• if the interface between the dielectrics
  coincides with a field line (a stream line) that
  goes ALL the way from one conductor to the other
  in the original problem
• the field line is then parallel to the interface
• make use of continuity of Etan to get new
  solution
• you have two caps in parallel

48
Planar wire examples (cross sections)

• in general this is hard
• again, charge distribution on surfaces is not
  uniform, with higher surface charge density where
  the conductors are closest together
• coplanar strips
• coplanar waveguide (CPW)
• stripline
• microstrip

49
Alternative definition of capacitance

• so far we have used the idea that
• what we really always used was not the potential
  V but the potential DIFFERENCE between the two
  plates
• hence we have really been using
• for all the capacitors we have seen so far we
  always had a linear relationship, in other
  words, if you double voltage you got double the
  charge Q since C was a constant
• that suggests we could have defined capacitance
  slightly differently
• use a derivative that shows much the charge
  changes on a plate in response to the change in
  voltage difference between the plates
• this gives the same answer for all the cases we
  have considered before

50
What do we do for the (really) hard problems??

• what we have so far
• for the vector field D, the flux density, we
  have
• interpretation if the volume dv encloses charge
  dQ rvdv then the net flux emerging from the
  point is the charge density rv
• we also have the electric field as the gradient
  of potential

• can we combine these? ? Laplace/Poisson equations