The Physics
Hypertextbook
Opus in profectus

Maxwell's Equations

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Discussion

qualitative

  1. Gauss's law
    • There are two types of charge, positive and negative, just as there are two types of real numbers, positive and negative.
    • Electric field lines diverge from positive charge and converge on negative charge
  2. No one's law
    • There is no magnetic monopole
    • Magnetic field lines neither converge nor diverge (have no beginning or end)
  3. Faraday's law
    • Electric field lines don't curl
    • … except when the magnetic field changes.
  4. Ampère's law
    • Magnetic field lines curl around electric current
    • … and also curl when the electric field changes.

integral notation

∯ E · dA =  Q
ε0
  Gauss's law
∯ B · dA =  0  
 
  No one's law
E · ds =  −  ∂ΦB
t
  Faraday's law
B · ds =  μ0ε0  ∂ΦB  + μ0I
t
  Ampère's law

differential notation

∇ · E =  ρ
ε0
  Gauss's law
∇ · B =  0  
 
  No one's law
∇ × E =  −  B
t
  Faraday's law
∇ × B =  μ0ε0  E  + μ0J
t
  Ampère's law

tensor notation

Gauss's law and Ampere's law together…

αFαβ = µ0Jβ

No one's law and Faraday's law together…

αFβγ + ∂βFγα + ∂γFαβ = 0

The new symbols are matrices.

4 gradient…

α = 
1      −     −     −   
c t , x , y , z  

4 current density (usually just called 4 current)…

Jβ = 




cρ




Jx
Jy
Jz

Electromagnetic field tensor, contravariant form…

Fαβ = 




0 Ex/c Ey/c Ez/c




+Ex/c 0 Bz +By
+Ey/c +Bz 0 Bx
+Ez/c By +Bx 0

Electromagnetic field tensor, covariant form…

Fαβ = 




0 +Ex/c +Ey/c +Ez/c




Ex/c 0 Bz +By
Ey/c +Bz 0 Bx
Ez/c By +Bx 0