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
Electric and Magnetic Fields Depend
on the Reference Frame
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Galilean Relativity
d d v = v′ ⇔ a = a′ ⇔ F = ma = ma′ = F ′
dt
dt
r = r′ + R
d d d r = r′ + R ⇔ v = v′ + V
dt
dt
dt
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
“Galilean Transformations” of E and B Fields:
Weak Relativistic Approximation ( v ≪c )
v c ≪ 1 ⇒ F = F′
⇓ E′ = E + v × B
To get transformation formulas for magnetic
field one has to use full special relativity
derivation and then take its limit for v ≪ c
1 B′ = B − 2 v × E
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-Savart Law as Coulomb Law Transformed
Into Moving Reference Frame
E =
q
4π ε 0
q rˆ
rˆ ′
, B = 0 E =E=
2
2
4
πε
r
0
r
µ0 q µ0 q 1 Biot-Savar: B′ =
v ×r = −
V ×r = − 2 V × E
2
2
4π r
4π r
c
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Faraday’s Law of EM Induction Revisited
Motional EMF detected by an inertial
observer may appear as a curly EMF to
another observer
LOOP Frame:
LAB Frame:
x1 = vt , x2 = b + vt
BL ,1 = (0, 0, B0 vt ), BL ,2 = (0, 0, B0b + B0 vt )
ε motional = ∫ (v × B ) ⋅ ds = −vB0 A
ε curly = ∫ E F ⋅ ds = ∫ ( E L + v × B ) ⋅ ds
ABCD
ABCD
for v ≪ c
no motion, BFz = B0 ( x F + vt )
∂BF
= (0, 0, B0 v )
∂t
∂B ⋅ dA = − vB0 A
ε curly = − ∫∫
∂t
A
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Almost Special Relativity
E1 =
1
4π ε 0
µ 0 q1 v ˆ
q ˆ j
,
B
=
k
1
2
2
r
4π r
1 q ˆ ˆ µ0 q1v ˆ
1 q  v2  ˆ
E1′ = E1 + V × B1 =
j + vi ×
k=
1 −  j
4πε 0 r 2
4π r 2
4πε 0 r 2  c 2 
1 µ qv
1
1 q ˆ  µ0 q1v 
1 ˆ
B1′ = B1 − 2 V × E1 = 0 12 kˆ − 2  viˆ ×
j
=
1
−


k ≡ 0
c
4π r
c 
4πε 0 r 2  4π r 2  ε 0 µ0c2 
?
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Lorentz-Einstein Transformations of
Electric and Magnetic Fields
E ⊥′ =
E ′ = E E⊥ + V × B
1− v
2
c
2
B⊥′ =
B′ = B
2
B⊥ − V × E c
1− v c
2
2
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!
Simple Corollaries:
2
1. E ≠ 0, B = 0 ⇒ B′ = B′ + B⊥′ = −V × E′ c
2. B ≠ 0, E = 0 ⇒ E′ = V × B′
3. EM Field Invariants: E ⋅ B = const.; E 2 − c 2 B2 = const.
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Electromagnetic Field of Freely Moving
Relativistic Charge
Electric Field
MOVING (with charge) S’-Frame:
q R′
E=
4πε 0 R′3
B=0
LAB S-Frame:
2
2
q
1− v c
R
E=
4πε 0 (1 − v2 sin 2 θ c2 )3/ 2 R3
x − vt = R cosθ , y 2 + z 2 = R sin θ
1 B = 2 v×E
c
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Electromagnetic Force Between
Two Moving Charges
Fmagnetic
v2
2
= ε0µ0v = 2
c
Felectric
1 q
2 2ˆ
2 2
′
FLorentz = Felectric + Fmagnetic =
v
c
j
F
v
c
1
−
=
1
−
Lorentz
2
4πε0 r
In the classical Galileo-Newton world signals propagate at
infinite velocity ( c →∞) and magnetism is absent!
For ultrarelativistic particles
v → c ⇒ Fmagnetic → Felectric
2
q2
q
1 − v2 c2
1
ˆj =
ˆj
Felectric = qE1 =
2
2
2 3/ 2 2
4πε 0 (1 − v sin 90 c ) r
4πε 0 r 2 1 − v2 c2
2 2
µ
1
q
v
ˆj
Fmagnetic = qv × B1 = qv 2 vE1 (− ˆj) = − 0
c
4π r 2 1 − v2 c2
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
Transformation Laws for Charge and
ρ
ρ
v
ρ
=
,
J
=
ρ
v
=
Current Densities
1− v c
1− v c
proper
2
proper
2
2
2
Interaction of current a current carrying wire and a particle with charge q in two intertial frames:
v
q
S
r
ρ+
v+ = 0
q
S'
ρ +′
v− = v
v +′ = − v
ρ −′
v −′ = 0
I′
I
F m a g n etic = q v × B
′
′
Fmagnetic = 0, Felectric
≠0
1
Q = Q ', ρ LA = ρ ′L′A, L′ = L 1 − v 2 c 2
2 Iq v
F =
4 πε 0 c 2 r
I = ρ − vA ⇒ F =
q
2 πε 0
⇓
ρ− A v2
r
c
2
Force Transformation Law:
∆p′y
r
ρ−
∆t ′
F ' ∆t ′
F
=
, ∆t =
⇒ F'=
∆p y
F ∆t
1 − v2 c2
1 − v2 c2
ρ +′ =
ρ+
1 − v2 c2
, ρ '− = ρ − 1 − v 2 c 2 , ρ − = − ρ +
( ρ +′ + ρ '− ) A
ρ+ A v2 c2
E′ =
=
2πε 0 r
2πε 0 r 1 − v 2 c 2
F ′ = qE ′ =
PHYS 208 Honors: Special Relativity of Electromagnetic Fields
q
2πε 0
ρ+ A
v2 c2
r
1 − v2 c2
=
F
1 − v2 c2