Download ELEC 3105 Lecture 7

ELEC 3105 Basic EM and
Power Engineering
Conductivity / Resistivity
Current Flow
Resistance
Capacitance
Boundary conditions
Conductivity and resistivity
The relaxation time model for conductivity works for most metals and
semiconductors.
In a conductor at room temperature, electrons are in random thermal motion,
with mean time  between collisions.
collision
Random motion of the electron
in the metal. Electron
undergoes collisions then moves
off in different direction.
electron

E0
Conductivity and resistivity
The relaxation time model for conductivity works for most metals and semiconductors.
In a conductor at room temperature, electrons are in random thermal motion, with mean time 
between collisions.
collision
Electrons acquire a small
systematic velocity v*
component in response to
applied electric field
electron

E
Conductivity and resistivity
For a weak electric field v* can be obtained.



F
qE
q 

v*   

E  E
m
m
m
collision
m = mass of electron
 = carrier mobility (ELEC 2507)
{} units of
electron

E
 m2 
 
 Vs 
Conductivity and resistivity
For strong electric fields,
electrons acquire so much
energy between collision that
mean time between collisions
is reduced.
FOR A STRONG
ELECTRIC FIELD
v*

E
for low fields v* proportional to E
E
Conductivity and resistivity
v*
As long as we stay in the weak electric
field regime, i.e. the linear region of the
curve in the previous slide, then the
current density can be defined as:


J  qNE
This region
E


J  E
Conductivity
  qN
Resistivity

1


E
Conductivity of elements
Current flow
The total amount of charge moving through a given
cross section per unit time is the current, usually
denoted by I.
Conductor ???
dq
v
vdt
CURRENT
dq
I
dt
Current flow
If we consider the current per unit cross-sectional area,
we get a value which can be defined any point in space
as a vector, typically denoted

J
cross-sectional area A
dq
v
vdt
N charged particles per unit volume moving at v meters per second
dq = N q vdt A
Charge moving through cross-sectional area A in time dt
Current flow
The charge density is simply this
quantity divided by the unit time and
area. The current density is:


J  Nqv
dq = N q vdt A
cross-sectional area A
dq
v
vdt
N charged particles per unit volume moving at v meters per second
dq = N q vdt A
Charge moving through cross-sectional area A in time dt
Current flow
The total current through the end face can
be obtained from the current density as an
integration over the cross-sectional area of
the conducting medium.
cross-sectional area A
dq
v
vdt
TOTAL CURRENT


I   J  da   Nqv  da
A
A
Current flow
The total charge passing through the
cross-sectional area A over a time
interval from t1 to t2 can be obtained
from:
cross-sectional area A
Q
v
vdt
 

Q   Idt     J  da  dt
t1
t1  A

t2
TOTAL CHARGE
t2
MOSFET
Resistance of conductors:
any shape

Vab   E  d
b
a


I ab   J  dA   E  dA
A
RESISTANCE
A
Vab
Rab 
I ab
Resistance of conductors:
any shape
A uniform rectangular bar
Vab
Rab 
I ab

Vab   E  d
b
a
Electric field is uniform and in the
direction of a bar length L.
Vab  EL
Resistance of conductors:
any shape
A uniform rectangular bar
Vab
Rab 
I ab


I ab   J  dA   E  dA
A
A
Electric field is normal to the crosssectional area A.
I ab  EA
Resistance of conductors:
any shape
A uniform rectangular bar
Vab
Rab 
I ab
Vab  EL
L
Rab 
A
I ab  EA
Rab 
L
A
SUPERCONDUCTORS
Capacitance
•Capacitance is a property of a geometric configuration, usually
two conducting objects separated by an insulating medium.
•Capacitance is a measure of how much charge a particular
configuration is able to retain when a battery of V volts is
connected and then removed.
•The amount of charge Q deposited on each conductor will be
proportional to the voltage V of the battery and some constant C,
called the capacitance.
Q
C
V
Capacitance {C/V}
Parallel plate capacitor

Vz    E  d
z
z
V = V volts
+Q
0
 s
E
o
s
Vz 
z
o
-Q
V = 0 volts
Plate area A
Plate separation D
Free space between plates
V
Q
D
o A
Rearrange
s 
o
Between plates
Q
A
Q
D
At z = D V 
o A
C
o A
D
Capacitance of parallel
plate capacitor
C
Q
V
CAPACITORS IN SERIES/ PARALLEL/
DECOMPOSITION
SERIES
PARALLEL
C1
C2
C1
C2
Ceq
Ceq  C1  C2
Ceq
1
1
1
 
Ceq C1 C2
CAPACITORS IN SERIES/ PARALLEL/
DECOMPOSITION
DECOMPOSITION

Ceq 


Ceq
1
1
1

C1 C2
 C3
C2
C1

C3
CAPACITANCE OF A COAXIAL
TRANSMISSION LINE
L  b 
Va  Vb 
ln  
2  a 
If we consider  L as the charge per unit length
then:
C
2


b
ln  
a
Q
C
V
C  r 55.6  pF 




b  m 
ln  
a

C
Prove this result as part of
next assignment.
on each of the two coaxial surface,
 L
Va  Vb
(ELEC 3909)
C
2
b
ln  
a
CHARGE CONSERVATION AND THE
CONTINUITY EQUATION
Qin    v dv
Charge in volume v
v
Current through surface A

I   J  dA
A
Also recall
I
dQ
dt
The main ingredients to the pie
CHARGE CONSERVATION AND THE
CONTINUITY EQUATION
Current out of volume is
Using previous expressions
From divergence theorem
Then:
Qin
t


A J  dA   t v v dv
 

   J dv   J  dA
v
 



J
dv


v dv
v
t v
I 
A
 
 v
 J  
31t
CHARGE CONSERVATION AND THE
CONTINUITY EQUATION
 
 v
 J  
t
Interpretation of equation: The
amount of current diverging from
am infinitesimal volume element
is equal to the time rate of change
decrease of charge contained in
the volume. I.e. conservation of
charge.
In circuits:
I
in
0
If no accumulation of charge at node.
CHARGE CONSERVATION AND THE
CONTINUITY EQUATION
A charge is deposited in a medium.
 
 v
 J  
t


J  E


 v
  E  
t
Also


 

t
 
  D   free
 
  0
t 
   free
E 

 t   oe
t 
33
T
CHARGE CONSERVATION AND THE
CONTINUITY EQUATION
A charge is deposited in a medium.
 t   oe
 t T 

T

If you place a charge in a volume v, the charge will redistribute itself in
the medium (repulsion???). The rearrangement of charge is governed
by the constant
T = REARRANGEMENT TIME CONSTANT
TCu, Ag=10-19 s
Tmica=10 h
ELECTROSTATICS
Boundary conditions
Tangential Component of
Boundary

E
1
2
Potential around
closed path
a

b

E2
Et 2
Around closed path (a, b, c, d, a)
d
c

E1
tˆ
Et 1

 E  d  0
c
V  0
ELECTROSTATICS
Boundary conditions

Tangential Component of E
1
Boundary
2
a

b
d

E2
c

E1
Et 1
tˆ
Et 2
lim
ab,cd 0 , 0

c 
a 
 E  d   E2  d   E1  d  0
b
c
d
ELECTROSTATICS
Boundary conditions

Tangential Component of E
lim
ab,cd 0 , 0
a

c 
a 
 E  d   E2  d 2   E1  d1  0
b
c


b
d
d
c
tˆ
The tangential components of the electric
field across a boundary separating two
media are continuous.


E2  E1  d  0
 E
t2
 Et1 d  0
Et 2  Et1  0
Et 2  Et1
ELECTROSTATICS
Boundary conditions

Tangential Component of E
1

E1  E1nˆ
a

E2  0

b
metal
Et 2  Et1  0
Et 2  Et1
d
c
tˆ
Et1  0
At the surface of a metal the electric
field can have only a normal
component since the tangential
component is zero through the
boundary condition.
ELECTROSTATICS
Boundary conditions 
Normal Component of E
Boundary
2
1
En1
A

E2

E1
n̂
n
n̂
En 2
Gauss’s law over pill box surface

 D  dA  qenclosed
c
ELECTROSTATICS
Boundary conditions 
Normal Component of E
1
Boundary
En1
2
n̂
A

E2

E1
n
n̂
En 2

 
lim  D  dA   D  D  dA    dA
n 0
1
c
2
s
ELECTROSTATICS
Boundary conditions 
Normal Component of E
1
2
A
En1

 
lim  D  dA   D  D  dA    dA
n 0
1
2
 D
En 2
n1
 D
 Dn 2 dA    s dA
n1
The normal components
of the electric flux density
are discontinuous by the
surface charge density.
s
c
 Dn 2   s dA  0
Dn1  Dn 2   s
 E  E  
41
1
n1
2
n2
s
ELECTROSTATICS
Boundary conditions 
Normal Component of E
1

E1  E1nˆ
Dn1  Dn 2   s
En 2  Dn 2  0
metal

E2  0
At the surface of a metal the electric field
magnitude is given by En1 and is directly
related to the surface charge density.
Dn1   s
s
En1  42
1
ELECTROSTATICS
Boundary conditions
Normal Component of

D
Dn1  Dn 2   s
Gaussian surface on metal interface
encloses a real net charge s.
Dn1   s
Gaussian surface on dielectric
interface encloses a bound surface
charge sp , but also encloses the
other half of the dipole as well. As a
result Gaussian surface encloses no
net surface charge.
Air Dielectric
Dn1  Dn 2  0
Dn1  Dn 2
Gaussian Surface
ELEC 3105 Basic EM and
Power Engineering
Extra extra read all about it!
44
Electric fields in metals
45
Electric fields in metals
(a) no current Einside = 0
(b) with current Einside  0
Inhomogeneous dielectrics
We can consider an inhomogeneous dielectric as being made up of small homogeneous
pieces, at the interfaces of which bound charge will accumulate.

D
x
dielectric
Suppose that we have a dielectric
whose permittivity is a function of x,
and a constant D field is directed
along x as well.
Inhomogeneous dielectrics
We can consider an inhomogeneous dielectric as being made up of a stack of thin sheets
of thickness dx and permittivity (x).

D
In each sheet, positive charges will
accumulate on the right and negative
ones on the left, according to the
permittivity of the sheet.
x
Inhomogeneous dielectrics
We can consider an inhomogeneous dielectric as being made up of a stack of thin sheets
of thickness dx and permittivity (x).

D
The charges will mostly cancel by
adjacent sheets, but any difference in
permittivity between adjacent sheets
d will leave some net charge density.
x
Inhomogeneous dielectrics
We can consider an inhomogeneous dielectric as being made up of a stack of thin sheets
of thickness dx and permittivity (x).

D
We can express this net bound charge
easily as the difference in
polarizations, so that we have:
x
 bound  
Px  dx   Px 
dP

dx
dx
Inhomogeneous dielectrics
In the more general case when the permittivity is varies in all directions, i. e. (x,y,x).
z

D
 bound  
Px  dx   Px 
dP

dx
dx
We can express this net bound charge
easily as the difference in
polarizations, so that we have:
y
 bound
 
   P
x
  
  
  D    oE    P
Inhomogeneous dielectrics
In the more general case when the permittivity is varies in all directions, i. e. (x,y,x).
z

D

 
D  oE  P
Take divergence on each side:
y
x
  
  
  D    oE    P
total  bound   free