Download The Displacement Current and Maxwell`s Equations

Motivation: Displacement Current and Maxwell’s
Equations
It was known in Maxwell’s time that
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 0 0
 c  speed of light in vacuum=2.99  108 m/s
Maxwell first realized that formulation of Ampere’s Law was
incomplete and required the addition of a term proportional to the time
derivative of the electric field called the displacement current. With the
displacement current included in Ampere’s Law, it is possible for a time
varying electric field to induce a magnetic field in a manner analogous
to Faraday Induction, where a changing magnetic field induces an
electric field. This process is now called Maxwell Induction. Once the
displacement current is included, the two induction equations become
symmetric and imply that electric and magnetic fields can propagate in
the absence of charges and currents.
Displacement Current: Generalizing Ampere’s
Law
In Maxwell’s time Ampere’s law as originally stated related the line
integral of the magnetic field around a closed loop to the current
passing through the loop
 B  dl
 0 I
where I is the net current due to moving charges.
Faraday’s Law was also well established and related the changing
magnetic flux through a loop to the line integral of the electric field
around the loop
d
B  dA
dt 
dB

dt
 E  dl  

Maxwell reasoned that if changing magnetic flux could produce a
circulating electric field, then a changing electric flux might likewise
produce a circulating magnetic field. To accomplish this he added a
second term to Ampere’s Law
 d

B

dl


I



E

dA
0
0
0

 dt 



 dE 
B

dl


I


0
0  0

dt 

The quantity in square brackets is called the displacement current, and
enters Ampere’s Law as a current equivalent to the conventional
current I produced by moving charges. With this addition Maxwell’s
Equations provide a complete description of all electromagnetic
phenomena:
 E  dA 
Qenc
0
 B  dA  0
Gauss' Law for E
Gauss' Law for B
d
B  dA Faraday's Law
dt 
 d

B

dl


I



E

dA
0
0
0

 dt 
 General Ampere's Law
 E  dl  
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Maxwell’s reasoning was correct. Changing electric fields do indeed
generate magnetic fields in a manor analogous to Faraday Induction.
The generation of magnetic fields by changing electric fields is called
Maxwell Induction. The line integral of the magnetic field around a
closed loop is sometimes referred to as a “magneto motive force” or
MMF.
Once the displacement current had been included in the equations
governing electromagnetism, Maxwell was able to derive a traveling
wave equation from Faraday’s Law and Ampere’s Law in the absence
of charges or currents due to moving charges which are shown below
 E  dl  
d
d
B

dA


B
dt 
dt
d
1 d
 d

E

dA

E

dA




0 I Disp
0
0
E
2




dt
c dt
 dt

 B dl   0 0
 E  dl  
dB
(Faraday)
dt
 B dl 
1 dE
(Maxwell)
c 2 dt
With the displacement current I Disp included in Maxwell’s equations,
the combination
 0 0 
1
c2
naturally appears in the equation for
Maxwell induction.
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We have studied Maxwell’s equation in integral form. To derive a
traveling wave equation, Maxwell actually worked with the equations in
differential form. While we will not study this formulation in this
course, it is still informative to display Maxwell’s equations in
differential form
 E 
E  

B
t

0
 B  0

E 
  B  0  J   0


t


J the current density. We have
already mentioned the divergence operation (   ), for example (   E ),
where
is the charge density and
which measures how the flux of a vector field flows into or out of a
region of space. A region with a nonzero divergence contains a source
or sink for the field (charge in the case of electrostatic fields). The curl
operation (   ), for example (   B ), similarly measures the
circulation of a vector field--its tendency to form closed loops. If you
think of the velocity field for water flowing in a river, a region with a
nonzero divergence contains either a spring (source) of water or a drain
(sink). A region of a flowing river with a nonzero curl will cause a
paddle wheel to turn if placed in the flow. Whirlpools have a nonzero
curl. It is very easy to obtain the integral form of Maxwell’s equations
from the differential form by integrating both sides of each equation
over a volume (Gauss’ Law for E and B ) or over a surface (Faraday
induction and Ampere’s Law with displacement current) and then
applying two theorems from vector calculus, the Divergence Theorem
and Stokes’ Theorem respectively. You will soon learn this approach in
your study of vector calculus. The point to emphasize is that this is very
straightforward to learn and offers powerful new techniques and
insights for the further study of Maxwell’s equation and
electromagnetism.
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