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Formula sheet - Formula Sheet J = E(Ohm's law d E = dt(flux...

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Formula Sheet ~ J = σ ~ E (Ohm’s law) E = - d dt Φ (flux rule) Φ = MI (self-inductance M) M 12 = μ 0 4 π H S 1 H S 2 d ~ l 1 · d ~ l 2 | ~ r 1 - ~ r 2 (mutual inductance) ∂t ρ = - ~ ∇ · ~ J (conservation laws) ∂t ( u mech + u em ) = - ~ ∇ · ~ S ∂t ( p mech + p em ) = ~ ∇· T ~ S = 1 μ 0 ~ E × ~ B (Poynting vector) T ij = 0 ( E i E j - 1 2 δ ij E 2 ) + 1 μ 0 ( B i B j - 1 2 δ ij B 2 ) (Maxwell’s stress tensor) ~ A 0 = ~ A + λ (Gauge transformation) V 0 = V - ∂t λ ~ ∇ · ~ A = 0 (Coulomb gauge) ~ ∇ · ~ A = - μ 0 0 ∂t V (Lorentz gauge) Λ μ ν = γ - γβ 0 0 - γβ γ 0 0 0 0 1 0 0 0 0 1 , β = v c , γ = 1 1 - β 2 (Lorentz transformation with boost in x-direction) g μν = g μν = - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 (metric tensor) 1
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F μν =
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