The Physics
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Opus in profectus

# Maxwell's Equations

## 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