ECE 3318 Applied Electricity and Magnetism. Spring Prof. David R. Jackson ECE Dept. Notes 13

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1 ECE 338 Applied Electicity and Magnetism ping 07 Pof. David R. Jackson ECE Dept. Notes 3

2 Divegence The Physical Concept Find the flux going outwad though a sphee of adius. x ρ v0 z a y ψ = D nˆ d = D ˆ d = = D D d 4π pheical egion of unifom volume chage density

3 Divegence The Physical Concept z Fom Gauss s law: x ρ v0 a y D D ρ v0 = C/m ρ 3 3 v0 = C/m 3 a ( < a) ( > a) Hence ψ = π ρv ψ πa ρv0 ψ = D ( < a) = ( > a) 4π (inceasing with distance inside sphee) (constant outside sphee) 3

4 Divegence -- Physical Concept (cont.) Obsevation: Moe flux lines ae added as the adius inceases (as long as we stay inside the chage). ψ = D nˆ d > 0 The net flux out of a small volume inside the chage is not zeo. Divegence is a mathematical way of descibing this. 4

5 Divegence Definition Definition of divegence: div D lim 0 D nˆ d ˆn = outwad nomal Note: The limit exists independent of the shape of the volume (poven late). A egion with a positive divegence acts as a souce of flux lines. A egion with a negative divegence acts as a sink of flux lines. 5

6 Gauss s Law -- Diffeential Fom Apply divegence definition to a small volume inside a egion of chage: div D lim 0 D nˆ d Gauss's law: ˆ encl v D n d = Q ρ ( ) ρ v ( ) Hence div D( ) = lim ( ) v 0 = ρ v ( ) ( ρ ) 6

7 Gauss s Law -- Diffeential Fom (cont.) The electic Gauss law in point (diffeential) fom: div D = ρ ( ) ( ) v This is one of Maxwell s equations. 7

8 Calculation of Divegence ( x /,0,0) z div D lim V 0 x y z D nˆ d x z y ( 0,0,0) x Assume that the point of inteest is at the oigin fo simplicity (the cente of the cube). The integals ove the 6 faces ae appoximated by sampling at the centes of the faces. y x D nˆ d Dx,0,0 y z x Dx,0,0 y z y + Dy 0,,0 x z y Dy 0,,0 x z z + Dz 0,0, x y z Dz 0,0, x y 8

9 Calculation of Divegence (cont.) div D lim 0 x y z D nˆ d x D nˆ d Dx,0,0 y z x Dx,0,0 y z y + Dy 0,,0 x z y Dy 0,,0 x z z + Dz 0,0, x y z Dz 0,0, x y x y z D nˆ d x x Dx,0,0 Dx,0,0 x y y Dy 0,,0 Dy 0,,0 + y + D z z z 0,0, Dz 0,0, z 9

10 Calculation of Divegence (cont.) 0 D D div D lim + + x D y z x x,0,0 Dx,0,0 x y y 0,,0 Dy 0,,0 y z z 0,0, Dz 0,0, z Hence div D x y z x y = + + z 0

11 Calculation of Divegence (cont.) Final esult in ectangula coodinates: div D x y = + + x y z z

12 The del opeato xˆ + yˆ + zˆ x y z This is a vecto opeato. Examples of deivative opeatos: scala vecto d dx d : ( sin x) = cos x dx Note: The del opeato is only defined in ectangula coodinates. ˆ ˆ ρ + φ + zˆ φ z ˆ ˆ ˆ + θ + φ d d θ φ xˆ : xˆ ( sin x) = xˆcos x dx dx d d xˆ ( xˆsin x) = xˆ xˆ ( sin x) = cos x dx dx d d xˆ ( yˆsin x) = xˆ yˆ ( sin x) = zˆ cos x dx dx

13 Divegence Expessed with del Opeato Now conside: D= xˆ + yˆ + zˆ xd ˆ + yd ˆ + zd x y z x y z = xˆ xˆ + yˆ yˆ + zˆ zˆ x y z ( ˆ ) x y z Hence D x y = + + x y z z This is the same as the divegence. 3

14 Divegence with del Opeato (cont.) D = div D Note that the dot afte the del opeato is impotant; any symbol following it tells us how it is to be used and how it is ead: Φ = "gadient" V = "divegence" V = "cul" 4

15 ummay of Divegence Fomulas Rectangula: D x y = + + x y z z ee Appendix A. in the Hayt & Buck book fo a geneal deivation that holds in any coodinate system. Cylindical: φ z D = ( ρdρ ) + + ρ ρ ρ φ z pheical: φ D= ( D) + ( Dθ sinθ ) + sinθ θ sinθ φ 5

16 Maxwell s Equations (Maxwell s equations in point o diffeential fom) B E = t H = J + t D = ρ B = 0 v Faaday s law Ampee s law Electic Gauss law Magnetic Gauss law Divegence appeas in two of Maxwell s equations. Note: Thee is no magnetic chage density! (Magnetic lines of flux must theefoe fom closed loops.) 6

17 Example Evaluate the divegence of the electic flux vecto inside and outside a sphee of unifom volume chage density, and veify that the answe is what is expected fom the electic Gauss law. x z ρ v0 a < a ρv0 D ˆ = 3 y D = ρ v0 D= ( D) ρ 3 ρv0 ( 3 ) 3 v0 = = = ρ v0 This agees with the electic Gauss law. 7

18 Example (cont.) z > a D 3 ρv0 ˆ a = 3 x ρ v0 a y ( D= D) 3 ρv0a = 3 3 ρv0a = 0 = 3 D = 0 This agees with the electic Gauss law. 8

19 Divegence Theoem ˆn = outwad nomal ˆn V V A dv = A nˆ d In wods: A = abitay vecto function The volume integal of "flux pe volume" equals the total flux! 9

20 Divegence Theoem (cont.) Poof The volume is divided up into many small cubes. n V 0 n = ( ) A dv = lim A N n Note: The point n is the cente of cube n. 0

21 Divegence Theoem (cont.) Fom the definition of divegence: n n = suface of cube = A nˆ d V ( A) lim n 0 n n n A nˆ d Hence: N N N AdV = lim ( A) V = lim ˆ lim ˆ V 0 A n d V = A n d n 0 V 0 V n n V = = n= n n

22 Divegence Theoem (cont.) n V AdV = lim A nˆ d N 0 n = n Conside two adjacent cubes: ˆn ˆn An ˆ is opposite on the two faces. Hence, the suface integal cancels on all INTERIOR faces.

23 Divegence Theoem (cont.) n V A dv = lim A nˆ d N 0 n = = lim A nˆ d 0 outside faces n n But lim 0 outside faces n A nˆ d = A nˆ d Hence V AdV = A nˆ d (poof complete) 3

24 Given: Example A= xˆ ( 3x) z Veify the divegence theoem using this egion. x so 3 Dimensions in metes V y ( 33 ( ) )( ) ( 30 ( ) ( ))( ) A nˆd = xˆ xˆ + xˆ xˆ A A = 8 A x y = + + = ( 3x) = 3 x A x y z z ( ) A dv = 3 dv = 3V = 3 3 = 8 V Note: A is constant on the font and back faces. 4

25 Note on Divegence Definition div D lim 0 D nˆ d Is this limit independent of the shape of the volume? ˆn Use the divegence theoem fo RH: D nˆ d = D dv mall abitay-shaped volume div D = lim D dv 0 = lim ( ( D) V) = D 0 Hence, the limit is the same egadless of the shape of the limiting volume. 5

26 Gauss s Law (Diffeential to integal fom) We can convet the diffeential fom into the integal fom by using the divegence theoem. Integate both sides ove a volume: D = ρ v V D dv = V ρ dv v Apply the divegence theoem to the LH: D nˆ d = V ρ dv v Use the definition of Q encl : D nˆ d = Q encl 6

27 Gauss s Law (ummay of two foms) D nˆ d = Q encl Integal (volume) fom of Gauss s law Definition of divegence + Gauss's law Divegence theoem D = ρ v Diffeential (point) fom of Gauss s law Note: All of Maxwell s equations have both a point (diffeential) and an integal fom. 7

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