notes6 2317

# notes6 2317 - Prof David R Jackson ECE Dept Spring 2011...

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Unformatted text preview: Prof. David R. Jackson ECE Dept. Spring 2011 Notes 6 ECE 2317 ECE 2317 Applied Electricity and Magnetism Applied Electricity and Magnetism Notes prepared by the EM Group University of Houston Review of Coordinate Systems Review of Coordinate Systems x y z P ( x,y,z ) An understanding of coordinate systems is important for doing EM calculations. Kinds of Integrals That Often Occur Kinds of Integrals That Often Occur 2 ˆ 4 C AB C C v V Q d V E dr R E d R Q dV ρ ρ πε ρ = = ⋅ = = ∫ ∫ ∫ ∫ l l l l (scalar integral, scalar result) (vector integral, scalar result) (vector integral, vector result) (scalar integral, scalar re Line integrals: Volume integrals: 2 ˆ 4 v V R E dV R ρ πε = ∫ sult) (vector integral, vector result) 2 ˆ ˆ 4 s S S s S Q dS I J n dS R E dS R ρ ρ πε = = ⋅ = ∫ ∫ ∫ (scalar integral, scalar result) (vector integral, scalar result) (vector integral, vector result) Surface integrals: We wish to be able to perform all of these in various coordinates. Rectangular Coordinates Rectangular Coordinates ˆ ˆ ˆ r xx yy zz = + + Short hand: ( 29 , , r x y z = Note: Different notations are used for vectors in the books. dx dy dz dS = dxdy dS = dxdz dS = dydz dV dx dy dz = x y z r P ( x,y,z ) Rectangular (cont.) Rectangular (cont.) ˆ ˆ ˆ dr x dx y dy z dz = + + Path Integral (need dr ) Note on notation: The symbol dl is often used instead of dr x y z A B C dr r r +dr ˆ ˆ ˆ r xx yy zz = + + Cylindrical Coordinates Cylindrical Coordinates . x y ρ φ ( 29 2 2 1 cos sin tan / x y z z x y y x z z ρ φ ρ φ ρ φ- = = = = + = = x y z ρ . z φ P ( ρ , φ , z ) Cylindrical (cont.) Cylindrical (cont.) Unit Vectors Note: and depend on ( x, y ) ˆ ρ ˆ φ x φ ˆ ρ y \$ φ x y z . ˆ ρ \$ φ ˆ z ρ This is why we often prefer to express them in terms of ˆ ˆ x y and Note: A unit vector direction is defined by increasing one coordinate variable while keeping the other two fixed. 1 2 1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ cos cos x A x x A y x A x A ρ ρ φ φ ⋅ = ⋅ + ⋅ = ⋅ = = Expressions for unit vectors (illustrated for ) 2 ˆ ˆ cos 2 sin A y ρ π φ φ = ⋅ =- = Hence ˆ ˆ ˆ cos sin x y ρ φ φ = + x y φ ˆ...
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notes6 2317 - Prof David R Jackson ECE Dept Spring 2011...

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