Special Relativity of Electric and Magnetic Fields

stewsystemElectronics - Devices

Oct 18, 2013 (3 years and 10 months ago)

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Special Relativity of Electric and
Magnetic Fields
Branislav K. Nikolić
Department of Physics and Astronomy, University of Delaware, U.S.A.
PHYS 208 Honors: Fundamentals of Physics II
http://www.physics.udel.edu/~bnikolic/teaching/phys208/phys208.html
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Electric and Magnetic Fields Depend
on the Reference Frame
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Galilean Relativity
ddd
rrRvvV
dtdtdt

=+=+


rrR

=+


dd
vvaaFmamaF
dtdt

=====


PHYS 208 Honors: Special Relativity of Electromagnetic Fields
“Galilean Transformations” of E and B Fields:
Weak Relativistic Approximation ( )
vc

1
vcFF


⇒=



EEvB

=+


￿To get transformation formulas for magnetic
field one has to use full special relativity
derivation and then take its limit for
vc

2
1
BBvE
c

=


Fields are measured at the same point in space by
experimenters at rest in each reference frame
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Bio-SavartLaw as Coulomb Law Transformed
Into Moving Reference Frame
2
0

,0
4
qr
EB
r

==

2
0
00
222

4
1
Biot-Savar:
44
qr
EE
r
qq
BvrVrVE
rrc




==

===



PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Faraday’s Law of EM Induction Revisited
Motional EMF detected by an inertial
observer may appearas a curly EMF to
another observer
LAB Frame:
LOOP Frame:
12
,10,200
0
,
(0,0,),(0,0,)
()
LL
motional
xvtxbvt
BBvtBBbBvt
vBdsvBA

==+
==+
=×=





0
0
0
no motion, ()
(0,0,)
z
FF
F
curly
A
BBxvt
B
Bv
t
B
dAvBA
t

=+

=


=×=

∫∫



()
curlyFL
ABCDABCD
EdsEvBds

=×=+×
∫∫



for
vc

PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Almost Special Relativity
0
1
11
22
0
1


,
44
qv
q
EjBk
rr


==

2
0
1
111
2222
00
00
11
111
222222
000
11


1
444
1111


10
444
qvqqv
EEVBjvikj
rrrc
qvqvq
BBVEkvijk
crcrrc






=+=+=




===




?
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Lorentz-Einstein Transformations of
Electric and Magnetic Fields
22
1
EE
EVB
E
vc



=
+

=





2
22
1
BB
BVEc
B
vc



=


=





￿Simple Corollaries:
2
222
1.0,0
2.0,0
3.EM Field Invariants: .;.
EBBBBVEc
BEEVB
EBconstEcBconst


=⇒=+=

=⇒=
×==




Electric and Magnetic Fields are different facets of a single electromagnetic field
whose particular manifestation (and division into its E and B components) depends
largely on the chosen reference frame!
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Electromagnetic Field of Freely Moving
Relativistic Charge
MOVING (with charge) S’-Frame:
22
2223/23
0
1
4(1sin)
qvcR
E
vcR


=



2
1
BvE
c
=


3
0
4
qR
E
R


=



0
B
=

22
cos,sin
xvtRyzR

=+=
LAB S-Frame:
Electric Field
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Electromagnetic Force Between
Two Moving Charges
2222
1
2223/22
222
00
11

4(1sin90)4
1
electric
qvcq
FqEjj
vcr
rvc


===




22
0
11
2
222
1

()
4
1
magnetic
qv
FqvBqvvEjj
c
rvc


===



2
2
00
2
2222
2
0
1

11
4
magnetic
electric
LorentzelectricmagneticLorentz
F
v
v
c
F
q
FFFvcjFvc
r


==

=+==



￿In the classical Galileo-Newton world signals propagate at
infinite velocity ( ) and magnetism is absent!
￿For ultrarelativisticparticles
magneticelectric
vcFF
®⇒®
c
®
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Transformation Laws for Charge and
Current Densities
￿Interaction of current a current carrying wire and a particle with charge q in two intertialframes:
2
0
2
2
0
12
4
2
magnetic
FqvB
Iqv
F
cr
A
qv
IvAF
rc






=
=
=⇒=


v

q
r
0
v

+
+
=
vv



=
I
S
q
r
vv

+
+


=
0
v





=
'
S
I

22
22
22
22
22
0
0
22
2222
0
0,0
',,1
,'1,
1
(')
2
21
2
11
magneticelectric
FF
QQLALALLvc
vc
vc
AAvc
E
r
rvc
AqvcF
FqE
r
vcvc









+
++
++
+

=

===

===


+

==


===



2222
,
11
properproper
v
Jv
vcvc


===



2222
Force Transformation Law:
'
,'
11
y y
p
FttF
tF
pFt
vcvc




==⇒=

