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10 Physics Formulary by ir. J.C.A. Wevers Here, the freedom remains to apply a gauge transformation . The fields can be derived from the potentials as follows: E = -∇ V - A t , B = ∇ × A Further holds the relation: c 2 B = v × E . 2.3 Gauge transformations The potentials of the electromagnetic fields transform as follows when a gauge transformation is applied: A = A - ∇ f V = V + f t so the fields E and B do not change. This results in a canonical transformation of the Hamiltonian. Further, the freedom remains to apply a limiting condition. Two common choices are: 1. Lorentz-gauge: · A + 1 c 2 V t = 0 . This separates the differential equations for A and V : V = - ρ ε 0 , A = - μ 0 J . 2. Coulomb gauge: · A = 0 . If ρ = 0 and J = 0 holds V = 0 and follows A from A = 0 . 2.4 Energy of the electromagnetic field The energy density of the electromagnetic field is: dW d Vol = w = HdB + EdD The energy density can be expressed in the potentials and currents as follows: w mag = 1 2 J · A d 3 x , w el = 1 2 ρ V d 3 x 2.5 Electromagnetic waves 2.5.1 Electromagnetic waves in vacuum The wave equation Ψ ( r, t ) = - f ( r, t ) has the general solution, with c = ( ε 0 μ 0 ) - 1 / 2 : Ψ ( r, t ) = f ( r, t
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