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Classical Electrodynamics
Jingbo Zhang
Harbin Institute of Technology
Chapter 5
Special Theory of Relativity
Section 6 Covariant Electrodynamics
Section 6 Covariant Electrodyamics
Chapter 5
Review

Lorentz’s
Transformation
 x0   
 x  
 1     
 x2   0
  
 x3   0
 

0
0
0
0
1
0
0

0
0

1 
 x0 
x 
 1
 x2 
 
 x3 
v
1
1
The Principle of
 Relativity
,


c equation should
Any physical
under the
1   2 be 1invariance
 v2 / c2
Lorentz’s transformation.
May, 2008
Classical Electrodynamics
Jingbo Zhang
Section 6 Covariant Electrodyamics
Chapter 5
Question?



How about the electrodynamical equations?
How to transform for electromagnetic fields
and potentials?
Is Electrodynamics covariant in SR?
May, 2008
Classical Electrodynamics
Jingbo Zhang
Section 6 Covariant Electrodyamics
Chapter 5
1. Electromagnetic Field
Einstein’s belief that Maxwell’s equations describe electromagnetism
in any inertial frame was the key that led Einstein to the Lorentz
transformations.

Maxwell’s result that all electromagnetic waves travel at the speed
of light led Einstein to his postulate that the speed of light is invariant
in all inertial frames.

Einstein was convinced that magnetic fields appeared as electric
fields when observed in another inertial frame. That conclusion is the
key to electromagnetism and relativity.

May, 2008
Classical Electrodynamics
Jingbo Zhang
Section 6 Covariant Electrodyamics
Chapter 5
But how can a magnetic field appear as an electric field simply due
to motion?
Electric
field lines (and hence
the force field for a positive
test charge) due to positive
charge.
Magnetic
field lines circle a
current but don’t affect a test
charge unless it’s moving.
Wire
with
current
How can one become the other and still give the right answer?
May, 2008
Classical Electrodynamics
Jingbo Zhang
Section 6 Covariant Electrodyamics
Chapter 5
A Conducting Wire
F  qE  qv  B
Suppose that a positive test
charge and negative charges in a
wire have the same velocity. And
positive charges in the wire are
stationary.
The electric field due to charges
in the wire will be zero, so the
force on the test charge will be
magnetic.
May, 2008
F  qv  B
The magnetic field at the test
charge will point into the page, so
the force on the test charge will
be up.
Classical Electrodynamics
Jingbo Zhang
Section 6 Covariant Electrodyamics
Chapter 5
A Conducting Wire 2
F  qE  qv  B
Now transform to the frame of the
previously moving charges.
Now it’s the positive charges in the
wire that are moving. And they will be
Lorentz-contracted, so their density
will be higher.
There will still be a magnetic field, but
the test charge now has zero velocity,
so its force will be zero. The excess of
positive charges will yield an electric
field.
May, 2008
F  qE
The electric field will point radially
outward, and at the test charge it
will point upward, so the force on
the test charge will be up. The two
cases can be shown to be identical.
Classical Electrodynamics
Jingbo Zhang
Section 6 Covariant Electrodyamics
Chapter 5
2 The Four Current


j   v , where   0



 0U  0 (c, v )  ( c, j )
Current 4vector

Continuity
equation
 

1 

 J  
,   c , j  
 j  0
t
 c t

Charge-current
transformations
May, 2008
J

v jx 

j x    j x   v ,        2 
c 

Classical Electrodynamics
Jingbo Zhang
Section 6 Covariant Electrodyamics
Chapter 5
3 The Four Potential
Potential
4-vector
1 
A   , A
c

Lorentz
Gauge

1 
1 
 1 




A

0

,



,
A


A





c 2 t
 c t
 c

de’Lambert
equation
May, 2008



   A  0 J
Classical Electrodynamics

Jingbo Zhang
Section 6 Covariant Electrodyamics
Chapter 5
Relativistic Transformations of E and B





 
E   E  v  B , E//  E//





vE 
B    B  2  , B//  B//
c 

May, 2008
Classical Electrodynamics
Jingbo Zhang
Section 6 Covariant Electrodyamics
Chapter 5
Homework 5.6

To derive the transformation
of four-potential by using
Lorentz matrix.
 '   
 c 
 A' x    

  0
A
'
 y 
 A' z   0
May, 2008
 

0
0
0 0  
 c

0 0  Ax 
1 0  Ay 
 
0 1  Az 
Classical Electrodynamics
Jingbo Zhang