Maxwell's equations are the cornerstone of classical electromagnetism, describing how electric and magnetic fields arise from charges and currents, and how they evolve over time. They exist in different unit systems and can be elegantly expressed in covariant form, unifying electromagnetism with special relativity.
- Gauss's Law for Electricity:
∇⋅E=ε0ρ
- Gauss's Law for Magnetism:
∇⋅B=0
- Faraday's Law of Induction:
∇×E=−∂t∂B
- Ampère–Maxwell Law:
∇×B=μ0J+μ0ε0∂t∂E
Integral Form (SI Units)
- Gauss's Law for Electricity:
∮∂VE⋅dA=ε01∫VρdV
- Gauss's Law for Magnetism:
∮∂VB⋅dA=0
- Faraday's Law of Induction:
∮CE⋅dl=−dtd∫SB⋅dA
∮CB⋅dl=μ0∫SJ⋅dA+μ0ε0dtd∫SE⋅dA
- Gauss's Law for Electricity:
∇⋅E=4πρ
- Gauss's Law for Magnetism:
∇⋅B=0
- Faraday's Law of Induction:
∇×E=−c1∂t∂B
- Ampère–Maxwell Law:
∇×B=c4πJ+c1∂t∂E
Integral Form (Gaussian Units)
- Gauss's Law for Electricity:
∮∂VE⋅dA=4π∫VρdV
- Gauss's Law for Magnetism:
∮∂VB⋅dA=0
- Faraday's Law of Induction:
∮CE⋅dl=−c1dtd∫SB⋅dA
∮CB⋅dl=c4π∫SJ⋅dA+c1dtd∫SE⋅dA
- Gauss's Law for Electricity:
∇⋅E=ρ
- Gauss's Law for Magnetism:
∇⋅B=0
- Faraday's Law of Induction:
∇×E=−c1∂t∂B
- Ampère–Maxwell Law:
∇×B=c1J+c1∂t∂E
Integral Form (Heaviside–Lorentz Units)
- Gauss's Law for Electricity:
∮∂VE⋅dA=∫VρdV
- Gauss's Law for Magnetism:
∮∂VB⋅dA=0
- Faraday's Law of Induction:
∮CE⋅dl=−c1dtd∫SB⋅dA
∮CB⋅dl=c1∫SJ⋅dA+c1dtd∫SE⋅dA
Maxwell's equations can be compactly expressed in special relativity using the antisymmetric electromagnetic field tensor Fμν, it's hodge dual Gμν, and the four-current Jμ defined as
Fμν=⎝⎜⎜⎜⎛0Ex/cEy/cEz/c−Ex/c0Bz−By−Ey/c−Bz0Bx−Ez/cBy−Bx0⎠⎟⎟⎟⎞,
Gμν=21ϵμναβFαβ=Gμν=⎝⎜⎜⎜⎛0BxByBz−Bx0−cEzcEy−BycEz0−cEx−Bz−cEycEx0⎠⎟⎟⎟⎞,
and
Jμ=(cρ,J).
We can now express Maxwell's equations very compactly as
∂νFμν=μ0Jμ,∂νGμν=0
where the μ=0 case of the first equation gives Gauss's law, and the μ=0 case of the second equation gives the no magnetic monopole law. The μ=1,2,3 cases correspond respectively to the spatial components of the Maxwell–Ampère law (first equation) and Faraday's law (second equation).
This system can be further compactified by employing the potential formalism, whereby the electric and magnetic fields are expressed in terms of a scalar potential V and a vector potential A:
E=−∇V−∂t∂A,B=∇×A.
As you might have guessed, V and A together constitute a four-vector:
Aμ=(cV,Ax,Ay,Az),
which can be used to express the Faraday tensor as
Fμν=∂μAν−∂νAμ,
where ∂μ=∂xμ∂ is the four-gradient with respect to the covariant coordinate xμ.
By then using the Lorentz gauge condition, which in relativistic notation becomes
∂μAμ=0,
all four of Maxwell's equations can neatly be wrapped up into the wave equation for the four-potential:
□2Aμ=−μ0Jμ,
where □2 is the d'Alembertian operator,
□2=∂ν∂ν=∇2−c21∂t2∂2.
Throughout, all tensors and vectors are expressed using Einstein summation convention and index notation, where repeated indices imply summation over spacetime coordinates (μ,ν=0,1,2,3).
- Jackson, J. D., Classical Electrodynamics, 3rd Edition, Wiley (1998), Chapters 6 and 11.
- Griffiths, D. J., Introduction to Electrodynamics, 4th Edition, Pearson (2013), Chapters 5 and 12.
- Peskin, M. E., & Schroeder, D. V., An Introduction to Quantum Field Theory, Addison-Wesley (1995), Chapter 2.