Field Derivation

Assuming we already know the Lagrangian for a charge in an electromagnetic field:

∘ ----2-L = - mc2 1- v- + qA ⋅V - qφ c2

then according to Lagrange's equation

( )∂- ∂L- ∂L-∂t ∂V = ∂r

we have

dP-= - q∂A-- q∇ φ +qV ×(∇ × A )dt ∂t.

Comparing this with the Lorentz force equation

f = q(E + V × B )

we have

E = ∇φ - ∂A- ∂t

and

B = ∇ × A.

If we calculate the curl of E and the divergence of B we will find:

∇ × E = - ∂B- ∂t

and

∇ ⋅B = 0.

The second pair of Maxwell's Equations can be derived in almost the same way, but require more detail. So for convenience we will just list the results:

∇ ⋅D = ρ

and

∇ × H = J + ∂D- ∂t.

This derivation can be found via the reference [1].

References

[1]   L.D. Landau and E.M.Lifshitz, The Classical Theory of Fields , Revised Second Edition

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