Download Electromagnetism

Electromagnetism
We want to apply the reaction theory developed in the first few lectures to
electronuclear interactions. It is worthwhile reviewing some facts from
electricity and magnetism.
CONSERVATION of CHARGE
In time dt an amount of charge dQ is lost from the
volume V because of a flux of charge, current density


J which leaves the surface element dS. Integrate over the surface.
 
 3
J
 3
V t d r  S J  dS  V   Jd r
dS
V
S
We used the divergence theorem to convert the integral over the surface S
into an integral over the volume V. We assume that the volume doesn’t
change, so only the charge density  in the volume changes. This leads to
the continuity equation.


   J  0,
t
(1)
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ELECTROSTATICS
Coulomb’s Law: This is a fundamental law of electrostatics. We will use the
Gaussian system of units here. This is also measured in cgs. The unit of
charge is the stat-coulomb. This amount of charge exerts a force of one
dyne on an equal charge located 1 centimeter away.
The electric field is a vector field which for a point charge is defined by
 q
E  2 rˆ,
r
E
(2)
r
q
From this expression for the electric field we derive Gauss’ law.
 
 3
4  d r (r )   E  dS     Ed r
3
V
S
V
We can convert the integral expression into a differential law,

  E  4,
(3)
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MAGNETOSTATICS
This case means that currents are constant. In particular the charge density
(r) is independent of time. The consequence of this is that the divergence
of the current density J is zero. This is a result we can deduce from the
equation of continuity ( eqn 1). The fundamental experimental law in
magnetostatics is the Biot-Savart Law. The contribution dB, to the magnetic
field, B, at point P, due to the current I in a short length dl is given by.
 
 I dl  r
P
dB 
,
(4)
c | r |3
  3

ˆ
 JdS dlJ  r J r d r
dB 
 
3
c |r|
c | r |3
r
I
 
I  J  dS


dl  dlJˆ , where J  JJˆ.
JdS
 
We consider a current density J


1
r

r
'
into an area dS.
B(r )   dr '3 J (r ' )    3 , (5)
c
| r  r '|
 
r  r'
1
We can rewrite (5) using
  3    
3
| r  r '|
| r  r '|
 
 
1
3 J (r ' )
B(r )     dr '   ,
(6)
c
| r  r '|

  1 3 J (r' )

A(r )   d r '     (r ), (7)
c
| r  r '|
Eqn. 6 can be written in terms
of an auxiliary potential A,
called the vector potential.
Notice that the addition of the gradient of an arbitrary function  in eqn 7
does not change the result we get for B in eqn 6. This arbitrariness in the
definition of the vector potential A can be used to advantage to simplify
equations. This flexibility is called “gauge invariance”. Consequently we
write for B,
 
 
B(r )    A(r ), (8)
We can find another relation between B and J from Ampere’s Law.
  4
4
B

d
l

I

C
c
c

 

 J  dS  (  B)  dS
S
S
C is the path encircling a surface S. I is the total current through
surface S. We used Stokes theorem to connect the first and third
integral.
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So for magnetostatics we get an alternate expression for Ampere’s law,
 4 
 B 
J,
c
(9)
ELECTRODYNAMICS
Now consider the case where currents change in time. The charge density is
now a function of time too.
FARADAY’S LAW
B
The EMF generated around the curve C depends on
the changing magnetic flux through area S.
B
C
S
E
dl
 
1 d B
EMF   E  dl  
c dt
C
Take the case where the geometry
remains constant so that only B changes
in time. Then using eqn 8,


 
 
1 B 
1
A 
C E  dl  S   E  dS   c S t  dS   c S   t  dS , (10)
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From eqn 10 we note that an induced electric field arises from changing
vector potentials. However, we also know from electrostatics that the
electric field is also generated by a gradient of an electric potential. We
take into account both of these sources of the electric field by writing,


1 A
E   
, (11) and we also note that
c t



1   A
1 B
 E  

, (12)
c t
c t
In eqn (12) we used that fact that the curl of a gradient is zero.
In the case of the magnetic field we can see that Ampere’s law can not
be the whole story. It can not only be a current which gives rise to the
magnetic field.
B=0 ?
B
B
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I
C
I
Consider the case of charging up a capacitor C which is connected to very
long wires. The charging current is I. From the symmetry it is easy to see that
an application of Ampere’s law will produce B fields which go in circles
around the wire and whose magnitude is B(r) = 2I/(cR). But there is no charge
flow in the gap across the capacitor plates and according to Ampere’s law
the B field in the plane parallel to the capacitor plates and going through the
capacitor gap should be zero! This is clearly unphysical. We also note from
the differential form for Ampere’s law, eqn (9) that


4
  B  0 
 J
c
But the divergence of J is zero only for magnetostatics. Maxwell saw that
Ampere’s Law, in the form of eqn 9, needed to be to modified to included
an additional current-like term, JD, called the displacement current.
 4  



4
 B 
(J  J D )     B  0 
(  J    J D ), but from eqn 1
c
c
 


 J 
 0, so,
   JD.
t
t
We now use Gauss’ Law in the form of eqn 3.
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



  1
1 E
1 E
 (   E)    (
)  JD 
, so the complete
t t 4
4 t
4 t
form of Ampere' s Law which includes time dependent fields is

 4  1 E
B 
J
, (13)
c
c t
MAXWELL’S EQUATIONS for the VACUUM

  E  4,
(3)

  B  0, (14)
,


1 B
 E  
, (12)
c t

 4  1 E
B 
J
,
c
c t
(13)
Continuity equation


   J  0,
t
(1)
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ALTERNATE FORM OF MAXWELL’S EQUATIONS FROM THE POTENTIALS
From eqn’s 11 and 8 we see that we can write the electric and magnetic
fields E and B in terms of the scalar potential  and the vector potential A.
Substituting into Maxwell’s equations we obtain,

1 
 
(  A)  4, and
c t

2

 1 
1  A
4 
2
 A  2 2  (  A 
)
J
c t
c t
c
2
Because of gauge invariance referred to in eqn 7, we have some flexibility
in our choice of A. In order that both B and E remain unchanged we must
make the following changes to the scalar potential if the vector potential is
modified.


1 
A  A   , then change,    
, (15)
c t
This flexibility in choice of A allows us to choose  in such a way that
 1 
 A
 0, (16)
c t
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Using eqn 15 the equations for the scalar and vector potentials become
1
2
  2
c
 1
2
 A 2
c
 2
 4, (17) and
t

2
 A
4 

J , (18)
2
t
c
Eqns 16, 17, and 18 form an equivalent set of Maxwell’s equations. The choice
of gauge which yields eqn 16 is called the Lorenz gauge. This gauge is
particularly useful when the explicit Lorentz invariance of the theory is
needed. Another popular choice of gauge called the Coulomb gauge, is used
mainly in low energy situations, such as atomic physics. The Coulomb gauge
yields a Poisson equation for the electric potential.

Coulomb gauge,   A  0, (19)
Poisson' s equation,  2  4,
(20)
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REFERENCES
1) “Classical Electrodynamics”, 2nd Edition, John David Jackson, John
Wiley and Sons, 1975
2) “Electrodynamics”, Fulvio Melia, University of Chicago Press, 2001
3) “Relativistic Quantum Mechanics and Field Theory”, Franz Gross,
John Wiley and Sons, 1993
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