ECE350lectures - 2 Static elds and potentials Static elds E...

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2Stat icfe ldsandpotent ia ls Static felds E = E ( r ) , D = D ( r ) , B = B ( r ) , H = H ( r ) independent oF the time variable t are produced by static source distributions ρ = ρ ( r ) and J = J ( r ) which only depend on position vector r =( x, y, z ) .I nc a s eo Fs t a t i cf e l d s Maxwell’s equations simpliFy and decouple as ± ² ³ ´ Time-dependent: ∇· D = ρ B =0 ∇× E = - B ∂t H = J + D ± ² ³ ´ Electrostatics: (curl-Free) D = ρ E D = ± o E Magnetostatics: (divergence-Free, solenoidal) B H = J B = μ o H 1
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Important vector identities: •∇× ( V )=0 •∇· ( ∇× A )=0 •∇×∇× A = ( ∇· A ) -∇ 2 A . ± ² ³ ´ Electrostatics: (curl-free) D = ρ E =0 D = ± o E Since all curl-free Felds can be expressed in terms of a scalar gradient, we choose E = -∇ V, where V = V ( x, y, z ) is called electrostatic potential . ± ² ³ ´ Magnetostatics: (divergence-free) B H = J B = μ o H Since all divergence-free Felds can be ex- pressed in terms of a curl, we choose B = A where A = A ( x, y, z ) is called vector potential . 2
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± ² ³ ´ Electrostatics: (curl-free) ∇· D = ρ ∇× E =0 D = ± o E such that E = -∇ V. Electrostatic potential V = V ( x, y, z ) signiFes the kinetic energy available (i.e., stored potential energy) — total energy be- ing 1 2 m v · v + qV —p e run i tcha rg eina static Feld measured from a convenient refer- ence point (ground). ± ² ³ ´ Magnetostatics: (divergence-free) B H = J B = μ o H such that B = A . If we apply the constraint A —known as Coulomb gauge and discussed in more detail next lecture — then the vector po- tential A = A ( x, y, z ) can be interpreted as kinetic momentum m v available — total (canonical) momentum be- ing m v + q A —pe i rgeinas ta t
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ECE350lectures - 2 Static elds and potentials Static elds E...

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