Maxwell's Equations
Discussion
qualitative
- 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
- No one's law
- There are no magnetic monopoles.
- Magnetic field lines neither converge nor diverge (have no beginning or end).
- Faraday's law
- Electric field lines don't curl…
- except when the magnetic field changes.
- Ampère's law
- Magnetic field lines curl around electric current…
- and also curl when the electric field changes.
integral notation
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Gauss's law | ||||||
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No one's law | ||||||
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Faraday's law | ||||||
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Ampère's law |
differential notation
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Gauss's law | ||||||
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No one's law | ||||||
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Faraday's law | ||||||
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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 |