1. Special relativity in electromagnetism
M.C. Chang
Dept of Phys
Special relativity in electromagnetism
Supplement to Jackson Chap 11
Refs:
Special relativity, by A.P. French
Electromagnetic fields and waves, by Lorrain and Corson

2. Special relativity in classical mechanics (I)x y z x’ y’ z’ S S’ v

3. Special relativity in classical mechanics (II)x y z x’ y’ z’ S S’ v

4. Transformation of electric force (I)x y z x’ y’ z’ S S’ q q v
Vector notation x y z
a central force in one frame is no longer a central force in another
Magnetic-like force exists for all moving central force, such as gravity!

5. Transformation of electric force (II)x y z x’ y’ z’ S S’ q q v
Vector notation

6. The force between two moving charges
Force exerted by qa on qb θa x y z S qa qb r vb va θb
Force exerted by qb on qa
just switch the subscripts a and b and let r → -r

7. Electric field of a moving chargeθ x y z S q r u

8. Transformation of electric and magnetic field
From the transformation of force, we can obtain the transformation of E and B field
The result is:
From which we can also get
That is, A=(ψ, A) also forms a 4-vector!

9. Motion in uniform E and B field
E⊥B
Can remove either E or B by jumping on a different frame x’ y’ z’ B’ S’ x y z E B S
E<B
E×B drift (～ Hall effect) x’ y’ z’ S’ E’ x z E B
E>B S y
E not ⊥ B
Cannot remove either E or B by jumping on a different frame
Note: Show that E2–B2 and E．B are Lorentz invariants

10. Electric charges on a straight line
Static charges x y z
Charge density λ b … …
Moving charges x y z
Charge density λ b … …

11. A straight current wirex y z
Charge density λ b … …
electric fields from positive and negative charges are cancelled
magnetic field , consistent with Ampere’s law
typically, v～1 mm/sec, ∴ β～10-12
It’s tiny but crucial.
Special relativity can be important at low velocity too!
(see below for another example)

12. Shockley-James paradox (Shockley and James, PRLs 1967)
A simpler version (Vaidman, Am. J. Phys. 1990)
A charge and a solenoid: E B S q
Poynting vector

13. Resolution of the paradox: needs to consider relativistic effect
Penfield and Haus, Electrodynamics of Moving Media, 1967
S. Coleman and van Vleck, PR 1968
Jackson, Classical Electrodynamics, the 3rd ed.
A stationary current loop in an E field
Smaller mass m
Gain energy
Lose energy
momentum flow
“hidden momentum” E
Larger mass
Force on a magnetic dipole (Jackson, p.189)
magnetic charge model
current loop model

14. Gravitomagnetism (Heaviside 1893)
Valid for weak field, low velocity
Electromagnetic wave → gravity wave
Frame-dragging effect (Lense andThirring, 1918) …
A rotating stick can generate gravity wave
And many more …

15. Relativistic notations
4-vectors
Space-time (ct, x)
energy-momentum (E/c, p)
scalar potential-vector potential (φ, A)
4-dim space-time operator (below)
charge density-charge current (below) …
but not the usual velocity, acceleration, force…

16. Covariant 4-vector and contravariant 4-vector
Vectors that transform like x are called contravariant vectors; Vectors that transform like are called covariant vectors. Their notations are distinguished by the position of the index.
inverse
Raising and lowering of the index
g (called a metric tensor) converts a contravariant vector to a covariant vector, and vice versa

17. Inner product between two 4-vectors
The inner product is invariant under Lorentz transformation because and transform oppositely.
Relativistic invariants
Conversely, any linear transformation that leaves x2 (or other inner product) invariant must be a Lorentz transformation (including spatial rotation).
Analogy: any linear transformation that leaves |x|2 (or other inner product) invariant must be a rotation.

18. Covariant form of the electromagnetic field
No relation with the same word in “covariant vector”
Covariant form of the electromagnetic field
Transformation of the field strength tensor or

19. local time, or “propre” time
Covariant form of the Maxwell equations
Covariant form of the Lorentz force equation
The usual velocity v is not part of a 4-vector since t is not invariant under the Lorentz transformation
4-velocity
local time, or “propre” time

20. Relativistic electrodynamics
Exactly the same as Maxwell’s electrodynamics!