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Chapter 10
Time-Varying Fields and Maxwell’s Equations
Two New Concepts:
The electric field produced by a changing magnetic field (Faraday)
The magnetic field produced by a changing electric field (Maxwell)
Faraday's Law
Any change in the magnetic environment of a coil of wire will cause a
voltage (emf) to be "induced" in the coil. No matter how the change is
produced, the voltage will be generated. The change could be produced by
changing the magnetic field strength, moving a magnet toward or away from
the coil, moving the coil into or out of the magnetic field, rotating the coil
relative to the magnet, etc.
Faraday's law is a fundamental relationship which comes from Maxwell's
equations. It serves as a succinct summary of the ways a voltage may be
generated by a changing magnetic environment. The induced emf in a coil is
equal to the negative of the rate of change of magnetic flux times the number
of turns in the coil. It involves the interaction of charge with magnetic field.
Faraday's Law
From GSU Webpage
Faraday's Law
Lenz’s Law
When an emf is generated by a change in magnetic flux according to Faraday's Law, the
polarity of the induced emf is such that it produces a current whose magnetic field opposes
the change which produces it. The induced magnetic field inside any loop of wire always acts
to keep the magnetic flux in the loop constant. In the examples below, if the B field is
increasing, the induced field acts in opposition to it. If it is decreasing, the induced field acts
in the direction of the applied field to try to keep it constant.
From GSU Webpage
Faraday's Law

d 
emf   
 dt 
A nonzero value of emf may result from of the situations:
1 – A time-changing flux linking a stationary closed path
2 – Relative motion between a steady flux and a closed path
3 – A combination of the two
Faraday's Law
emf
 N
emf



d

dt
E_dot_ d L
S
d 
 
dt 
B_dot_ d S

d
emf
E_dot_ d L 
B_dot_ d S
 dt

S
Applying Stokes' Theorem






( Del  E)_dot_ d S
S
S




d
B_dot_ d S
dt
Removing the intregals - assuming same surface
( Del  E)_dot_dS
Del  E
 d B

 dt 

 d B  _dot_dS

 dt 

Faraday's Law

emf
B y  d
d
 
dt
 d y d

 dt 
B 
B v  d
Consider example using concept of motional emf
F
Qv  B
F
v B
Q
Em
v B
The force per unit charge is called
the motional electric field intensity
Subject every portion of the moving conductor
Faraday's Law
Displacement Current
Faraday's experimental law led to Maxwell's equation
d
 B
dt
Ampere's circuital law for staedy magnetic fields
Del  E
Del  H
J
It shows its inadequacy for time-varying
Del_dot_Del  H
0
Del_dot_J
The divergence of the curl is zero, and so the Del of J.
However, the equation of continuity
Del_dot_J
d
 v
dt
Then it shows that
d
v
dt
Del  H
J
can be true only if
0
This is an unrealistic limitation and Del  H
must be amended before we can
accept it for time-varying fields
J
Displacement Current
Suppose we add a term G
Del  H
JG
Taking the divergence
0
Del_dot_J  Del_dot_G
d
v
dt
Replacing .v for Del_dot_D
Del_dot_G
G
d
D
dt
Del  H
J
d
D
dt
Displacement Current

 H_dot_ d L

emf
I

d 
I    D  _dot_ d S
  dt 
S
Vo  cos   t
 C Vo  sin   t 
Displacement current is associated with time-varying fields and exists in all imperfect
Conductors carrying a time-varying conduction current.
Maxwell’s Equations In Point Form
Maxwell’s Equations In Integral Form
The Retarded Potentials
V
A





v
4  R
dv
Knowledge of the distribution of charge density
and current density throughout space will enable
us to determine V and A
vol

 ( J)

dv
 4  R

vol
Every t appearing in v or (J) jas been replaced by
a retarded time
t1
v
t
R
v
1
 
D10.7

while

8
4 cos 10   t
A point charge of
8
4 cos 10   t


C
is located at P+(0, 0, 1.5)
is at P-(0, 0, -1.5), both in free space
Find V and P(r=450, ,  =0)
 0 
Pplus   0 


 1.5 
 0 
Pminus   0 


 1.5 
R1  P  Pplus
7
 12
0  8.854 10

 cos 
10    t 
6
Vplus  76.083
  0
 0 
P   0 


 450 
R1  448.5
 0  4  10
ro1  4 10
r  450
8


R1 

v 
v 
1
 0 0
Vplus 

9
t  15 10
ro1
 12
4  8.854187917 10
 R1
D10.7
R1  P  Pminus
7
 0  4  10
R1  451.5
 12
0  8.854 10
 cos 10    t 
6
ro1  4 10
8

Vminus  75.524
V  Vplus  Vminus
V  151.607

R1 
v


v 
1
9
t  15 10
 0 0
Vminus 

ro1
R1
 12
4  8.854187917 10