Transcript Document

EEE 498/598
Overview of Electrical
Engineering
Lecture 6:
Electromotive Force; Kirchoff ’s Laws;
Redistribution of Charge; Boundary
Conditions for Steady Current Flow
1
Lecture 6 Objectives

To study electromotive force; Kirchoff ’s
laws; charge redistribution in a conductor;
and boundary conditions for steady current
flow.
2
Lecture 6
Electromotive Force
Steady current flow requires a closed
circuit.
 Electrostatic fields produced by stationary
charges are conservative. Thus, they
cannot by themselves maintain a steady
current flow.

3
Lecture 6
Electromotive Force (Cont’d)
 
increasing
potential

I

4
The current I must
be zero since the
electrons cannot
gain back the energy
they lose in traveling
through the resistor.
Lecture 6
Electromotive Force (Cont’d)
 

I
+
-

5
To maintain a steady
current, there must be
an element in the
circuit wherein the
potential rises along
the direction of the
current.
Lecture 6
Electromotive Force (Cont’d)


For the potential to rise along the direction of the
current, there must be a source which converts some
form of energy to electrical energy.
Examples of such sources are:




batteries
generators
thermocouples
photo-voltaic cells
6
Lecture 6
Inside the Voltage Source
• Eemf is the electric field
established by the energy
conversion.
• This field moves positive
charge to the upper plate,
and negative charge to the
lower plate.
• These charges establish an
electrostatic field E.
+++
E
E emf
--In equilibrium:
E emf  E  0
Source is not connected
to external world.
7
Lecture 6
Electromotive Force (Cont’d)
 
I

+
Eemf
-
E
E
At all points in the circuit, we must
have
J
 E total  E emf  E

exists only in battery
8
Lecture 6
Electromotive Force (Cont’d)

Integrate around the circuit in the direction of
current flow
E
total
 dl  
C
C
1


 E  dl   E

C
emf
J  dl
 d l 
C
1

J  dl
0
9
Lecture 6
Electromotive Force (Cont’d)

Define the electromotive force (emf) or
“voltage” of the battery as

Vemf   E emf  d l

10
Lecture 6
Electromotive Force (Cont’d)
We also note that
1
l
C  J  d l  A I  RI
 Thus, we have the circuit relation

Vemf  RI
11
Lecture 6
Overview of Electromagnetics
Fundamental laws of
classical electromagnetics
Special
cases
Electrostatics
Statics:
Input from
other
disciplines
Maxwell’s
equations
Magnetostatics
Electromagnetic
waves

0
t
Geometric
Optics
Transmission
Line
Theory
Circuit
Theory
Kirchoff’s
Laws
12
d  
Lecture 6
Kirchhoff ’s Voltage Law

For a closed circuit containing voltages sources
and resistors, we have
V
emf
 IR
• “the algebraic sum of the emfs around a closed circuit
equals the algebraic sum of the voltage drops over the
resistances around the circuit.”
13
Lecture 6
Kirchhoff ’s Voltage Law
Strictly speaking KVL only applies to
circuits with steady currents (DC).
 However, for AC circuits having
dimensions much smaller than a
wavelength, KVL is also approximately
applicable.

14
Lecture 6
Conservation of Charge
Electric charges can neither be created nor
destroyed.
 Since current is the flow of charge and
charge is conserved, there must be a
relationship between the current flow out
of a region and the rate of change of the
charge contained within the region.

15
Lecture 6
Conservation of Charge (Cont’d)

Consider a volume V
bounded by a closed
surface S in a
homogeneous
medium of
permittivity e and
conductivity 
containing charge
density qev.
S
qev
V
ds
16
Lecture 6
Conservation of Charge (Cont’d)

The net current
leaving V through S
must be equal to the
time rate of decrease
of the total charge
within V, i.e.,
S
qev
V
dQenc
I 
dt
ds
17
Lecture 6
Conservation of Charge (Cont’d)

The net current leaving the region is given
by
I   J ds
S

The total charge enclosed within the region
is given by
Q   qev dv
V
18
Lecture 6
Conservation of Charge (Cont’d)

Hence, we have
d
S J  d s   dt V qev dv
net outflow
of current
net rate of
decrease of
total charge
19
Lecture 6
Continuity Equation

Using the divergence theorem, we have
 J  d s     J dv
S

V
We also have
qev
d
qev dv  
dv

dt V
t
V
20
Becomes a
partial
derivative when
moved inside of
the integral
because qev is a
function of
position as well
as time.
Lecture 6
Continuity Equation (Cont’d)

Thus,

Since the above relation must be true for
any and all regions, we have

V   J dv  V t dv  0

 J 
0
t
21
Continuity
Equation
Lecture 6
Continuity Equation (Cont’d)

For steady currents,

Thus,

0
t
 J  0
J is a solenoidal vector field.
22
Lecture 6

Continuity Equation in Terms
of Electric Field
Ohm’s law in a conducting medium states
J  E

For a homogeneous medium
  J    E  0    E  0


But from Gauss’s law,
E 
qev
e
Therefore, the volume charge density, , must be
zero in a homogeneous conducting medium
23
Lecture 6
Kirchhoff ’s Current Law

Since J is solenoidal,
we must have
S
J

d
s

0

S


In a circuit, steady
current flows in wires.
Consider a “node” in
a circuit.
24
Lecture 6
Kirchoff ’s Current Law (Cont’d)

We have for a node in a circuit
I  0
• “the algebraic sum of all currents leaving a
junction must be zero.”
25
Lecture 6
Kirchoff ’s Current Law (Cont’d)
Strictly speaking KCL only applies to
circuits with steady currents (DC).
 However, for AC circuits having
dimensions much smaller than a
wavelength, KCL is also approximately
applicable.

26
Lecture 6
Redistribution of Free Charge

Charges introduced into the interior of an
isolated conductor migrate to the conductor
surface and redistribute themselves in such a
way that the following conditions are met:
E = 0 within the conductor
 Et = 0 just outside the conductor
 qev = 0 within the conductor
 qes  0 on the surface of the conductor

27
Lecture 6
Redistribution of Free Charge
(Cont’d)
We can derive the differential equation
governing the redistribution of charge from
Gauss’s law in differential form and the
continuity equation.
 From Gauss’s law for the electric field, we
have
qev

  D  qev    E 
   J  qev
e
e

28
Lecture 6
Redistribution of Free Charge
(Cont’d)


From the continuity equation, we have

 J  
t
Combining the two equations, we obtain
 r , t  
  r , t   0
t
e
29
Describes the time
evolution of the
charge density at a
given location.
Lecture 6
Redistribution of Free Charge
(Cont’d)

The solution to the DE is given by
 r , t   0 r  e
 t /  r 
Initial charge distribution at t = 0
where r = e/ is the time constant of the
process called the relaxation time.
30
Lecture 6
Redistribution of Free Charge
(Cont’d)




The initial charge distribution at any point in the
bulk of the conductor decays exponentially to zero
with a time constant r.
At the same time, surface charge is building up on
the surface of the conductor.
The relaxation time decreases with increasing
conductivity.
For a good conductor, the time required for the
charge to decay to zero at any point in the bulk of
the conductor (and to build up on the surface of the
conductor) is very small.
31
Lecture 6
Relaxation Times for Some
Materials
copper
19
r
 1.5 10 s
H O
5
 r  10 s
amber
3
r
 4 10 s
mica
 r  10 to 20 hrs
quartz
r
 50 days
2
32
Lecture 6
Electrical Nature of Materials
as a Function for Frequency


The concept of relaxation time is also
used to determine the electrical nature
(conductor or insulator) of materials at a
given frequency.
A material is considered to be a good
conductor if
1
 r  T 
f
  r f  1
33
Lecture 6
Electrical Nature of Materials as a
Function for Frequency (Cont’d)

A material is considered to be a good insulator if
1
 r  T 
f

  r f  1
A good conductor is a material with a relaxation
time such that any free charges deposited within its
bulk migrate to its surface long before a period of
the wave has passed.
34
Lecture 6
Boundary Conditions for Steady
Current Flow
The behavior of current flow across the
interface between two different materials
is governed by boundary conditions.
 The boundary conditions for current
flow are obtained from the integral
forms of the basic equations governing
current flow.

35
Lecture 6
Boundary Conditions for Steady
Current Flow (Cont’d)
aˆ n
e1 ,  1
e 2 , 2
36
Lecture 6
Boundary Conditions for Steady
Current Flow (Cont’d)

The governing equations for steady
electric current (in a conductor) are:
 J ds  0
S
E

d
l

0



C
C
37
J

 dl  0
Lecture 6
Boundary Conditions for Steady
Current Flow (Cont’d)

The normal component of a solenoidal
vector field is continuous across a material
interface:
J1n  J 2 n

The tangential component of a conservative
vector field is continuous across a material
interface:
J1t J 2t
1

2
38
Lecture 6
Conductor-Dielectric Interface
 0
J=0
 0
39
J
Lecture 6
Conductor-Dielectric Interface
(Cont’d)



The current in the conductor must flow
tangential to the boundary surface.
The tangential component of the electric
field must be continuous across the interface.
The normal component of the electric field
must be zero at the boundary inside the
conductor, but not in the dielectric. Thus,
there will be a buildup of surface charge at
the interface.
40
Lecture 6
Conductor-Dielectric Interface
(Cont’d)
+++++
no current flow  Et = 0
E
----+++++
current flow  En >> Et
I
-----
41
Lecture 6
Conductor-Conductor Interface

The current bends as it cross the interface
between two conductors
1
 2  1
1
2
42
Lecture 6
Conductor-Conductor Interface
(Cont’d)

The angles are related by

Suppose medium 1 is a good conductor and
medium 2 is a good insulator (i.e., 1 >> 2).
Then  2  0 . In other words, the current
enters medium 2 at nearly right angles to the
boundary. This result is consistent with the
fact that the electric field in medium 2 should
have a vanishingly small tangential
component at the interface.
43
Lecture 6
Conductor-Conductor Interface
(Cont’d)

In general, there is a buildup of surface
charge at the interface between two
conductors.
 Only when the
J1n  J 2 n  J n   1 E1n   2 E2 n
 s  D1n  D2 n  e1 E1n  e 2 E2 n

 e1 e 2 
1 
 E1n  J n   
  e1  e 2
2 

 1  2 
 J n  r1   r 2 
44
relaxation times of
the two conductors
are equal is there no
buildup of surface
charge at the
interface.
Lecture 6