Maxwell's equations in different mathematical forms ✍️
Differential form (SI, vacuum with sources)
∇ · E = ρ / ε₀
∇ · B = 0
∇ × E = − ∂B/∂t
∇ × B = μ₀ J + μ₀ε₀ ∂E/∂t
Integral form
∮ E · dA = Q_enc / ε₀
∮ B · dA = 0
∮ E · dℓ = − dΦ_B/dt
∮ B · dℓ = μ₀ I_enc + μ₀ε₀ dΦ_E/dt
Covariant form
∂_μ F^μν = μ₀ J^ν
∂_[λ F_μν] = 0
The last term in the fourth equation, μ₀ε₀ ∂E/∂t, is Maxwell's own addition. Set ρ and J to zero and it gives ∇²E = μ₀ε₀ ∂²E/∂t², a wave travelling at 1/√(μ₀ε₀), which turned out to be the speed of light.
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