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Unformatted text preview: Lecture 40: Electrodynamics II (7 Dec 09) A. Review – Electrodynamics 1. Recall: 4-vector transforms as A * μ = X ν a μν A ν and that x μ = ( ct, r ) is a 4-vector. 2. Lorentz invariance of scalar products of two 4-vectors: ( A μ ,B μ ) ≡ g μ A μ B μ ; g =- 1 ,g 1 = g 2 = g 3 = 1 3. gradient as a 4-vector: 2 μ = g μ ∂ ∂x μ and ( 2 μ , 2 μ ) is the wave equation operator. 4. 4-velocity using proper time dt = γdτ : u μ = ( u , u ); u = d r dτ = γ ( v ) v ; u = cdt dτ = γ ( v ) c and then to the energy-momentum 4-vector p μ = ( E/c,m u ) 5. 4-vectors in electrodynamics: current density j μ = ( cρ, ~ j ) and vector potential A μ = ( φ/c, ~ A ), with ( 2 μ ,j μ ) = 0 (equation of continuity) and ( 2 μ ,A μ ) = 0 (Lorentz gauge) and the Maxwell source equations 2 2 A μ =- μ j μ . 6. Dynamics using the Lorentz force: d~ p dt = q ( ~ E + ~v × ~ B ); dE dt = q~v · ~ E ↔ dp μ dτ = F μ with a 4-vector force F μ = q X ν g ν u ν ( 2 μ A ν- 2 ν...
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- Fall '09
- Fundamental physics concepts, dt dt dτ, Fµ, gν uν