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Unformatted text preview: : Curl and Divergence •  Curl: A measure of the magnitude of rota[on in a vector field at a given point •  Divergence: A measure of the flux into or out of a given point http://apollo.lsc.vsc.edu/classes/met130/notes/chapter12/graphics/divergence.free.gif http://en.wikipedia.org/wiki/Curl_(mathematics) 15 Review: Gradient •  Gradient: A vector field that provides a measure of the slope and direc[on of greatest change of a scalar field (func[on) at a given point Scalar function f(x,y) Cartesian Coordinates: ∂ f ( x, y, z ) ∂ f ( x, y, z ) ∂ f ( x, y, z ) ˆ ˆ ˆ ∇f ( x, y, z ) = x+ y+ z ∂x ∂y ∂z Vector field http://upload.wikimedia.org/wikipedia/commons/3/31/Gradient99.png Cylindrical Coordinates: ∂ f ( ρ ,φ , z ) 1 ∂ f ( ρ ,φ , z ) ˆ ∂ f ( ρ ,φ , z ) ˆ ˆ ∇f ( ρ,φ , z ) = ρ+ φ+ z ∂ρ ρ ∂φ ∂z Spherical Coordinates: ∇f (r,θ ,φ ) = ∂ f (r ,θ ,φ ) 1 ∂ f (r ,θ ,φ ) ˆ 1 ∂ f (r ,θ ,φ ) ˆ ˆ r+ θ+ φ ∂r r ∂θ r sin θ ∂φ 16 Key Points: Fields & Waves Charge creates divergence in the electric displacement field: ∇i D = ρ The electric field is related to the electric displacement field through the permittivity: D = εE The electric field is the negative of the gradient of the scalar potential field: E = −∇V dV ( x ) In one dimesion: E ( x ) = − dx V ( x ) = − ∫ E ( x ) dx http://www.pstcc.edu/departments/natural_behavioral_sciences/Web%20Physics/E2020D0103.gif 17 Gauss’ Law ∇•D = ρ ρ is the charge density with units of Coulombs m3 •  The divergence of the electric displacement flux density is equal to the charge density •  Can be derived from Ampere’s Law using the Con[nuity Equa[on http://www.ibiblio.org/links/devmodules/Gauss/graphics/blob1.gif 18 Key Points: Circuit Con...
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