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16 Νοε 2013 (πριν από 4 χρόνια και 8 μήνες)

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Classical Electrodynamics

Jingbo Zhang

Harbin Institute of Technology

Chapter 5

Special Theory of Relativity

Section 6

Covariant Electrodynamics

Jingbo Zhang

Section 6

Covariant Electrodyamics

Chapter 5

May, 2008

Classical Electrodynamics

/
1
1
1
1

,

2
2
2
c
v
c
v

Review

Lorentz’s
Transformation

3
2
1
0
3
2
1
0

1
0
0
0
0
1
0
0
0
0
0
0
x
x
x
x
x
x
x
x




The Principle of Relativity

Any physical equation should be invariance under the

Lorentz’s transformation.

Jingbo Zhang

Section 6

Covariant Electrodyamics

Chapter 5

May, 2008

Classical Electrodynamics

Question?

electrodynamical

equations?

How to transform for
electromagnetic fields
and

potentials
?

Is Electrodynamics covariant in SR?

Jingbo Zhang

Section 6

Covariant Electrodyamics

Chapter 5

May, 2008

Classical Electrodynamics

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.

Jingbo Zhang

Section 6

Covariant Electrodyamics

Chapter 5

May, 2008

Classical Electrodynamics

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?

Jingbo Zhang

Section 6

Covariant Electrodyamics

Chapter 5

May, 2008

Classical Electrodynamics

A Conducting Wire

v
F qE q 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.

v
F q B
 
The magnetic field at the test
charge will point into the page, so
the force on the test charge will
be

up
.

Jingbo Zhang

Section 6

Covariant Electrodyamics

Chapter 5

May, 2008

Classical Electrodynamics

A Conducting Wire 2

v
F qE q B
  
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.

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.

F qE

Jingbo Zhang

Section 6

Covariant Electrodyamics

Chapter 5

May, 2008

Classical Electrodynamics

2 The Four Current

Current 4
-
vector

)
,
(
)
,
(
where
,
0
0
0
j
c
v
c
U
J
v
j

Continuity
equation

0
,
,
1



j
t
j
c
t
c
J

Charge
-
current
transformations

2
,
c
j
v
v
j
j
x
x
x

Jingbo Zhang

Section 6

Covariant Electrodyamics

Chapter 5

May, 2008

Classical Electrodynamics

3 The Four Potential

A
c
A

,
1

A
A
c
t
c
A
t
c



,
1
,
1
0
1
2
Potential

4
-
vector

Lorentz
Gauge

J
A
0

de’Lambert
equation

Jingbo Zhang

Section 6

Covariant Electrodyamics

Chapter 5

May, 2008

Classical Electrodynamics

Relativistic Transformations of E and B

//
//
2
//
//
,
,
B
B
c
E
v
B
B
E
E
B
v
E
E

Jingbo Zhang

Section 6

Covariant Electrodyamics

Chapter 5

May, 2008

Classical Electrodynamics

Homework 5.6

To derive the transformation
of four
-
potential by using
Lorentz matrix.

z
y
x
z
y
x
A
A
A
c
A
A
A
c




1
0
0
0
0
1
0
0
0
0
0
0
'
'
'
'