Chapter 1: Vector Analysis

Size: px
Start display at page:

Download "Chapter 1: Vector Analysis"

Transcription

1 Chapte 1: Vecto Analysis Campos Electomagnéticos 2º Cuso Ingenieía Industial Dpto.Física Aplicada III Cuso 2007/2008 Dpto. Física Aplicada III - Univ. de Sevilla Joaquín Benal Méndez 1 Chapte 1: Index (I) Intoduction Scala and Vecto Fields Integal Calculus Ciculation Flux Diffeential Calculus The Gadient The Divegence The Cul Dpto. Física Aplicada III - Univ. de Sevilla 2

2 Chapte 1: Index (II) Mathematic tools Cylindical and Spheical coodinates The del opeato The Diac delta function Special types of fields: Cul-less o Iotational fields Divegence-less o Solenoidal fields Hamonic fields The Helmholtz Theoem Dpto. Física Aplicada III - Univ. de Sevilla 3 Intoduction Scala: quantities chaacteized by its magnitude. Examples: weight, chage, tempeatue Vecto: quantities that have diection as well as magnitude. Examples: velocity, acceleation, foce Dpto. Física Aplicada III - Univ. de Sevilla 4

3 Scala Fields (I) Definition: It is a function that associates a scala value to evey point in space. Tempeatue: T(x,y,z) Geogaphical altitude: h(x,y) Density field of a body A scala field should give a single value of some vaiable fo evey point in space (monovalued function). Dpto. Física Aplicada III - Univ. de Sevilla 5 Scala Fields (II) Gaphical epesentation: Equipotential sufaces ϕ ( x, yz, ) = C Dpto. Física Aplicada III - Univ. de Sevilla 6

4 Scala Fields (III) Example: Atmospheic pessue ϕ ( x, yz, ) = C Dpto. Física Aplicada III - Univ. de Sevilla 7 Vecto Fields (I) Definition: It is a function that associates a vecto to evey point in space. Gavitational field Velocity of a flowing liquid Electic and magnetic fields Monovalued functions Dpto. Física Aplicada III - Univ. de Sevilla 8

5 Vecto Fields (II) Example: Wind field Dpto. Física Aplicada III - Univ. de Sevilla 9 Vecto Fields (III) Gaphical epesentation: A Field line is dawn such that the tangent to the field line at any point is paallel to the vecto field at that point dx dy dz = = F F F x y z Dpto. Física Aplicada III - Univ. de Sevilla 10

6 Vecto Fields (IV) Example: Electic field of a point chage Positive chage Dpto. Física Aplicada III - Univ. de Sevilla 11 Vecto Fields (V) Example: Field lines of two point chages q q q -q Positive Chages Positive and negative chages Dpto. Física Aplicada III - Univ. de Sevilla 12

7 Chapte 1: Index (I) Intoduction Scala and Vecto Fields Integal Calculus Ciculation Flux Diffeential Calculus The Gadient The Divegence The Cul Dpto. Física Aplicada III - Univ. de Sevilla 13 Integal calculus We have both scala and vecto fields We can pefom line, suface and volume integals of these fields Fom these, two types of integals ae impotant fo us: Ciculation Flux Dpto. Física Aplicada III - Univ. de Sevilla 14

8 Ciculation (I): definition Line integal of a vecto field: Γ = B A, γ F d It measues how much the vecto field is aligned with the cuve The esult of the integal is a scala The line integal depends on the path Example: wok pefomed by a foce Dpto. Física Aplicada III - Univ. de Sevilla 15 Ciculation (II): Physical meaning Closed path: indicates how much the vecto field tends to ciculate aound the cuve C = F dl L L C = 0 C 0 Fo a foce vecto field this would be the wok done in a closed path Dpto. Física Aplicada III - Univ. de Sevilla 16 L

9 Ciculation: Futhe Examples What is the sign of the ciculation of these vecto fields? Dpto. Física Aplicada III - Univ. de Sevilla 17 Ciculation (III): calculation Γ = A, γ Paametize the cuve: B F d γ : = ( ), t (, ) { t t t } A B Calculate the integal: t B Γ= F( t ( )) t A d dt dt Dpto. Física Aplicada III - Univ. de Sevilla 18

10 Flux (I): definition Suface integal of a vecto field Φ= F ds S The esult is a scala magnitude It depends on the suface S The diection of s Fo closed sufaces: must be specified s is pointing outwads Dpto. Física Aplicada III - Univ. de Sevilla 19 Flux (II): Physical meaning The flux of a vecto field though a suface measues how much field cossed that suface Example: velocity of a fluid Φ = VS Φ=VS dφ = V ds Φ= VdS S Dpto. Física Aplicada III - Univ. de Sevilla 20

11 Flux (III): calculation Paametize the suface: S: ( α, β), α ( α1, α2), β ( β1, β2) ds = dα dβ α β { = } We have to calculate this double integal: Φ= F( ( α, β)) dα dβ α β Dpto. Física Aplicada III - Univ. de Sevilla 21 Summay We will wok with scala and vecto fields Equipotential sufaces ae useful fo epesenting scala fields Field lines ae employed fo epesenting vecto fields Ciculation of a vecto field measues the twist of the vectos The flux of a vecto field though a suface is popotional to the numbe of field lines passing though that suface Dpto. Física Aplicada III - Univ. de Sevilla 22

12 Chapte 1: Index (I) Intoduction Scala and Vecto Fields Integal Calculus Ciculation Flux Diffeential Calculus The Gadient The Divegence The Cul Dpto. Física Aplicada III - Univ. de Sevilla 23 Diffeential calculus The deivative of scala and vecto fields can be pefomed in diffeent ways. Scala fields: gadient Vecto fields: divegence and cul Dpto. Física Aplicada III - Univ. de Sevilla 24

13 Gadient (I) Fo a function of one vaiable: f ( x) df ( x) f ( x + ε ) f ( x ) 0 0 = lim dx ε 0 ε x 0 f ( x) The deivative tell us how apidly the function vaies when we change x f ( x + ε ) 0 f ( x ) 0 Dpto. Física Aplicada III - Univ. de Sevilla 25 ε x Gadient (II): diectional deivative How can we give an idea of the vaiation of a scala function of seveal vaiables? A diection must be specified: Unit vecto: v = ( vx, vy, vz) Diectional deivative: dϕ ϕ( x + εvx, y+ εvy, z+ εvz) ϕ( x, y, z) = lim ds ε 0 ε Dpto. Física Aplicada III - Univ. de Sevilla 26

14 Gadient (III) By using the concept of patial deivative: d ϕ ϕ v ϕ = + v + ϕ v ds x y z Definition: Theefoe: x y z ϕ ϕ ϕ gadϕ u x u = + y + uz x y z d ϕ =gad ϕ v ds ϕ ϕ ϕ =,, v x y z Dpto. Física Aplicada III - Univ. de Sevilla 27 Gadient (IV): geometical intepetation gadϕ gadϕ v α α = 0 cosα v= 1 d ϕ =gad ϕ v ds d ϕ =gad ϕ cos α ds The gadient vecto points in the diection of maximun diectional deivative (maximun incease of the function at that point) The magnitude of the gadient at one point is the maximum value of the deivative at that point Dpto. Física Aplicada III - Univ. de Sevilla 28

15 Gadient (V) Incease of a given scala function: dϕ = gad ϕ v ds = gadϕ d The gadient in a point is pependicula to the equipotential suface at that point: gadϕ d ϕ = Cte dϕ = 0= gadϕ d gad ϕ d Dpto. Física Aplicada III - Univ. de Sevilla 29 Gadient (VI): Summay The Gadient is a vecto field. Its magnitude gives the maximun value of the deivative of the function in that point. The gadient points in the diection of maximun ate of change of the scala function. The gadient in a point is pependicula to the equipotential suface at that point. Dpto. Física Aplicada III - Univ. de Sevilla 30

16 Chapte 1: Index (I) Intoduction Scala and Vecto Fields Integal Calculus Ciculation Flux Diffeential Calculus The Gadient The Divegence The Cul Dpto. Física Aplicada III - Univ. de Sevilla 31 Divegence (I) The divegence of a vecto field is a scala field: 1 div F( ) = lim F ds Δτ 0 Δτ Divegence in catesian coodinates: S Δτ F F x y F div F ( ) = + + x y z z Dpto. Física Aplicada III - Univ. de Sevilla 32

17 Execise Calculate the divegence of the vecto of position F F x y Fz div F( ) = + + with F( ) = x y z a) div = 0 Solution: b) div = x+ y+ z c) div = 3 Dpto. Física Aplicada III - Univ. de Sevilla 33 Divegence (II): Geometical Intepetation The divegence is a measue of how much the vecto field speads out (diveges) fom a given point P Lage positive divegence at P Zeo divegence points Dpto. Física Aplicada III - Univ. de Sevilla 34

18 Divegence (III) The Divegence Theoem: z τ τ div F( d ) τ = S τ FdS x S τ y ds vecto pointing outwad Useful fo evaluating integals Vey impotant fo deivation of theoetical esults Dpto. Física Aplicada III - Univ. de Sevilla 35 Cul (I) Definition: 1 ot F( ) = lim ds F Δτ 0 Δτ The Cul in catesian coodinates: ux uy uz F Fy z Fx Fz Fy F x ot F( ) = ot F u ( ) + = u + u y z xz y x z x y S Δτ x y z F F F x y z Dpto. Física Aplicada III - Univ. de Sevilla 36

19 Cul (II): Geometical Intepetation The culs measues how much the vecto field culs aound a given point: Nonzeo cul Zeo cul Zeo cul Dpto. Física Aplicada III - Univ. de Sevilla 37 Cul (III) Stokes Theoem: otf ds = F d Right hand ule: S If you finge point in the diection of the line integal then you thumb fixes the diection of ds It is useful fo: Calculation of integals Deivation of impotant theoetical esults γ s Dpto. Física Aplicada III - Univ. de Sevilla 38

20 Divegence and Cul: Summay Deivatives of vecto fields Divegence: scala field. Related with the existence of souces and sinks of the vecto field Rotacional: vecto field. Related with the existence of whilpools in the field lines Fundamental Theoems: Divegence theoem Stokes theoem Dpto. Física Aplicada III - Univ. de Sevilla 39 Chapte 1: Index (I) Intoduction Scala and Vecto Fields Integal Calculus Ciculation Flux Diffeential Calculus The Gadient The Divegence The Cul Dpto. Física Aplicada III - Univ. de Sevilla 40

21 Chapte 1: Index (II) Mathematic tools Cylindical and Spheical coodinates The del opeato The Diac delta function Special types of fields: Cul-less o Iotational fields Divegence-less o Solenoidal fields Hamonic fields The Helmholtz Theoem Dpto. Física Aplicada III - Univ. de Sevilla 41 Cuvilinea coodinates We have seen seveal examples using catesian coodinates Many poblems can be moe easily solved by using othe coodinates: Cylindical coodinates Spheical coodinates Thee ae moe coodinate systems but we will estict ouselves to these. Dpto. Física Aplicada III - Univ. de Sevilla 42

22 Catesian Coodinates (I) Any point is epesented by thee signed numbes, (x,y,z), whee the coodinate is the pependicula distance fom the plane fomed by the othe two axes Coodinate lines: staight lines paallel to the axis Coodinate sufaces: planes paallel to the coodinate planes Z z Z y z x Y x X P z = cte y X y = cte x = cte Dpto. Física Aplicada III - Univ. de Sevilla 43 Y Catesian Coodinates (II) Othogonal basis set: X Z k 0 u x u z P Y i j ds xy u y Vecto of position: = xu + yu + zu x y z d = dxu + dyu + dzu x y z Diffeential elements of suface: = dxdyu Infinitesimal displacement: z ds yz = dzdyu x ds zx = dxdzu y Diffeential volume elements: dτ = dxdydz Dpto. Física Aplicada III - Univ. de Sevilla 44

23 Cylindical Coodinates (I) Thee coodinates fo a point P : X Z φ ρ z 0 ρ < 0 ϕ < 2π < z < Y ρ: pependicula distance fom the z axis φ (azimuthal angle): angle aound fom the x axis z (vetical c.): distance fom the XY plane x = ρ cosϕ Y x y = ρ senϕ ρ y z = z φ X Dpto. Física Aplicada III - Univ. de Sevilla 45 Z Cylindical Coodinates (II) Coodinate lines: ρ: hoizontal staight half-lines φ: Hoizontal cicumfeences z: Vetical staight lines Coodinate sufaces: ρ=cte.: vetical cylindes φ=cte: vetical half-planes z=cte: hoizontal planes X ρ ϕ=cte z P z=cte Z ρ=cte ϕ Y Dpto. Física Aplicada III - Univ. de Sevilla 46

24 Cylindical coodinates (III) Othogonal basis set X u x Z u z φ ρ 0 u y u z z P u φ u ρ Y Vecto of position: = ρ cosϕu + ρsenϕu + zu = ρu + zu z ρ x y z Infinitesimal displacement: d = dρ u + ρdϕu + dzu z ρ The unit vectos change diection as P moves aound Dpto. Física Aplicada III - Univ. de Sevilla 47 ϕ Cylindical coodinates (IV) X ρ ϕ=cte z P z=cte Z ρ=cte ϕ Y Elements of suface: ρ = cte : ds = ρdϕdzu ρ z = cte : ds = ρdϕdρuz ϕ = cte : ds = dzdρ u ϕ Element of volume: dτ = ρdρdzdϕ Sometimes lette is used in instead of ρ Dpto. Física Aplicada III - Univ. de Sevilla 48

25 Gadient, divegence and Cul in Cylindical Coodinates f 1 f f gad f = uρ + uϕ + u ρ ρ ϕ z div F ρ ϕ 1 ( ρf ) 1 F F = + + ρ ρ ρ ϕ z 1 F F F z ϕ ρ Fz 1 Fρ ot F = u + u + ( F) u z ρ z ϕ ρ ϕ ρ ϕ ρ ρ ρ ϕ z z z Dpto. Física Aplicada III - Univ. de Sevilla 49 Spheical Coodinates (I) X Z φ θ 0 < 0 θ π 0 ϕ < 2π Z Y θ ρ (adial): distance fom the oigin θ (pola): angle down fom the positive z axis φ (azimuthal): angle fom the positive x-axis to the othogonal pojection of the position vecto in the XY plane z x= senθ cosϕ y = senθ senϕ z = cosθ Y Z x ρ φ y X Dpto. Física Aplicada III - Univ. de Sevilla 50

26 Spheical Coodinates (II) Coodinate lines: : adial half-lines fom the oigin φ: hoizontal cicumfeences (paallels) θ: vetical cicumfeences (meidians) Coodinate sufaces: =constant: Concentic sphees φ=constant: Vetical half-planes θ=constant: Cones ϕ=cte X θ P Z =cte θ=cte ϕ Y Dpto. Física Aplicada III - Univ. de Sevilla 51 Spheical Coodinates (III) Othogonal basis set u x Z u z φ θ ρ u y 0 z P u u θ u φ Y Vecto of position: = senθ cosϕu + senθsenϕu + cosθu = u d Line = d u element: + dθ u + senθdϕu x y z θ ϕ X These unit vectos change diection as P moves aound Dpto. Física Aplicada III - Univ. de Sevilla 52

27 Spheical Coodinates (IV) X ϕ=cte P Z θ=cte ϕ Suface elements: = ds = d d u θ = cte : ds = senθdϕdu θ ϕ = cte : ds = dθdu ϕ 2 cte : senθ ϕ θ θ =cte Y Volume elements: dτ = 2 senθddθdϕ Dpto. Física Aplicada III - Univ. de Sevilla 53 Gadient, Divegence and Cul in Spheical Coodinates f 1 f 1 f gad f = u + uθ + uϕ θ senθ ϕ Fϕ div F = ( F ) (sen ) 2 + θ Fθ + senθ θ senθ ϕ 1 (sen θfϕ) F 1 1 ( F ) θ F ϕ otf = u + uθ + sen θ sen θ ϕ θ ϕ 1 ( Fθ ) F u ϕ θ Dpto. Física Aplicada III - Univ. de Sevilla 54

28 Chapte 1: Index (II) Mathematic tools Cylindical and Spheical coodinates The del opeato The Diac delta function Special types of fields: Cul-less o Iotational fields Divegence-less o Solenoidal fields Hamonic fields The Helmholtz Theoem Dpto. Física Aplicada III - Univ. de Sevilla 55 The Opeato (I) It allows a shothand notation Definition: = ux + uy + uz x y z Vecto opeato : It acts upon (diffeentiate) the function to the ight It behaves like an odinay vecto Dpto. Física Aplicada III - Univ. de Sevilla 56

29 The opeato (II) ϕ ϕ ϕ gad ϕ = ux + uy + uz = ϕ x y z F F x y F z div F = + + = F x y z u u x uy z otf = = F x y z F F F x y z Dpto. Física Aplicada III - Univ. de Sevilla 57 The opeato (III) can be expessed in any system of coodinates. Calculus caied out with the help of ae independent of the coodinate system. Any identity that can be poved by using the catesian coodinates vesion of emains valid fo any othe system of coodinates. Dpto. Física Aplicada III - Univ. de Sevilla 58

30 Poduct Rules (I) A scala field can be obtained as the poduct of two othe fields: ψϕ Poduct of two scala fields: Dot poduct of two vecto fields: F G What is the gadient of the poduct? ( ϕψ) = ϕ ψ + ψ ϕ ( F G) = F ( G) + ( F ) G+ G ( F) + ( G ) F F = Fx + Fy + Fz x y z ( F ) G F( G) Dpto. Física Aplicada III - Univ. de Sevilla 59 Poduct Rules (II) Also, a vecto field can be obtained fom a poduct of fields: Scala and vecto fields: ϕf G Coss poduct of two vecto fields: F Divegence: ( ϕf) = ϕ F + ( ϕ) F ( F G) = ( F) G F ( G) ( ϕf) = ϕ F + ( ϕ) F ( F G) = F( G) ( F ) G G( F) + ( G ) F Cul: Dpto. Física Aplicada III - Univ. de Sevilla 60

31 Poduct Rules : Summay ( ϕψ) = ϕ ψ + ψ ϕ ( F G) = F ( G) + ( F ) G+ G ( F) + ( G ) F Gadient: Divegence: ( ϕf) = ϕ F + ( ϕ) F ( F G) = ( F) G F ( G) ( ϕf) = ϕ F + ( ϕ) F ( F G) = F( G) ( F ) G G( F) + ( G ) F Cul: Dpto. Física Aplicada III - Univ. de Sevilla 61 Second deivatives (I) By applying twice we can constuct five species of second deivatives: The gadient is a vecto field: Divegence of gadient ϕ ( ) Cul of gadient ϕ ( ) The divegence is a scala field: Gadient of divegence ( F) The cul is a vecto field: Divegence of cul F ( ) Cul of cul F ( ) Dpto. Física Aplicada III - Univ. de Sevilla 62

32 Second deivatives (II) ϕ ϕ ϕ 2 ( ϕ) = + + = ϕ x y z ( ) ( ϕ) = 0 Vey impotant ( F) Seldom occus 2 ( F) ( ) F = F! ( F) = 0 Vey impotant 2 ( F) = ( F) F Laplacian Aleady defined Dpto. Física Aplicada III - Univ. de Sevilla 63 Chapte 1: Index (II) Mathematic tools Cylindical and Spheical coodinates The del opeato The Diac delta function Special types of fields: Cul-less o Iotational fields Divegence-less o Solenoidal fields Hamonic fields The Helmholtz Theoem Dpto. Física Aplicada III - Univ. de Sevilla 64

33 The Diac Delta Function (I) Conside this vecto field: u v = = 2 3 Radial and pointing outwads, but: v = 0 2 = 2 Howeve, by integating ove a sphee (R): τ = = senθ θ ϕ = 4π π 2π u 2 vd v ds u R d d R τ S Divegence theoem τ Dpto. Física Aplicada III - Univ. de Sevilla 65 The Diac Delta Function (II) The souce of the poblem is the point = v =!! 2 2 = = 0 Summing up, the function u fulfills: 2 u 0 0 u = con d τ = π = 0 τ We have found a weid function: the Diac delta function Dpto. Física Aplicada III - Univ. de Sevilla 66

34 The Diac Delta Function (III) The one-dimensional Diac delta function: 0 x 0 δ ( x) = con δ ( xdx ) = 1 x = 0 - Distibution: the limit of a sequence of functions 1 δ ( x) = lim δ ε( x) = lim e ε 0 ε 0 ε π x ε 2 2 Dpto. Física Aplicada III - Univ. de Sevilla 67 The Diac Delta Function (IV) δε(x) δε( x) = e ε π x ε 2 2 ε=1 ε=0.5 ε=0.25 ε= x Dpto. Física Aplicada III - Univ. de Sevilla 68

35 The Diac Delta Function (IV) Poduct with an odinay function: δ ( x) f( x) = δ( x) f(0) δ( x) f( x) dx= f(0) It is sufficient that the domain extend acoss the delta function: ε ε The spike can be shifted: δ ( x a) f( x) dx= f( a) δ ( x) f( x) dx= f(0) Dpto. Física Aplicada III - Univ. de Sevilla 69 The Diac Delta Function (V) The thee-dimensional delta function: δ 3 ( ) = δ( x) δ( y) δ( z) 3 ( a) = ( x a ) ( y a ) ( z a ) δ δ δ δ In geneal: τ ϕ ( a) ϕ( ) δ( a) dτ = 0 x y z a τ a τ z a τ a y x Dpto. Física Aplicada III - Univ. de Sevilla 70

36 The Diac Delta Function (VI) u 2 u 0 0 u = with d τ = π = 0 τ u = 4 πδ ( ) 2 0 = 4 πδ ( 3 0) = 2 1 = 4 πδ ( 0 0) 0 0 Coming back to the function It can be witten as: Dpto. Física Aplicada III - Univ. de Sevilla 71 3 Chapte 1: Index (II) Mathematic tools Cylindical and Spheical coodinates The del opeato The Diac delta function Special types of fields: Cul-less o Iotational fields Divegence-less o Solenoidal fields Hamonic fields The Helmholtz Theoem Dpto. Física Aplicada III - Univ. de Sevilla 72

37 Iotational Fields Cul-less vecto fields: Equivalent conditions: F d = 0 γ B B F d = F d A, γ A, γ 1 2 F = 0 F d = ( F) ds = 0 A Thee exits a scala field such that: γ γ 2 S γ γ 1 B F = ϕ Dpto. Física Aplicada III - Univ. de Sevilla 73 Solenoidal Fields Divegence-less vecto fields: Popiedades: S S F ds = 0 τ Sτ F ds = F ds si γ s =γ S Flux is constant though a field tube: s F = 0 F ds = Fdτ= 0 τ γ =γ s1 s2 ds 2 ds 1 S 1 S 2 Thee exists a vecto field such that: S L F = A Dpto. Física Aplicada III - Univ. de Sevilla 74

38 Types of Vecto Fields F 0 F 0 F = 0 F 0 Solenoidal F 0 F = 0 Iotational F = 0 F = 0 Solenoidal and iotational Dpto. Física Aplicada III - Univ. de Sevilla 75 Hamonic Fields Scala fields satisfying: 2 ϕ= Example: conside a vecto field which is iotational and solenoidal: F = 0 F = 0 0 Laplace equation F = ϕ 2 ϕ=0 ( ϕ ) = 0 Pactical case: electostatic field in a egion without chages Dpto. Física Aplicada III - Univ. de Sevilla 76

39 Chapte 1: Index (II) Mathematic tools Cylindical and Spheical coodinates The del opeato The Diac delta function Special types of fields: Cul-less o Iotational fields Divegence-less o Solenoidal fields Hamonic fields The Helmholtz Theoem Dpto. Física Aplicada III - Univ. de Sevilla 77 The Helmholtz Theoem Given we can calculate: F and Given and Is it possible to get? Let: F F F =ρ F = c F Scala souces Vecto souces F F ( c = 0) If this is insufficient infomation: many solutions If this is too much infomation: no solution Dpto. Física Aplicada III - Univ. de Sevilla 78

40 Helmholtz theoem: statement The system F =ρ ; F = c with c = 0 defined in all the space with: 2 2 lim ρ ( ) = 0 ; lim c( ) = 0 ; lim F( ) = 0 has a single solution given by: F = ϕ+ A with: Souce point Field point 1 ρ( 1) dτ 1 1 cd ( ) ϕ ( ) = y 4 π A ( ) = τ esp 4 π esp Scala potential 1 Vecto potential Dpto. Física Aplicada III - Univ. de Sevilla 79 Chapte 1: Index (I) Intoduction Scala and Vecto Fields Integal Calculus Ciculation Flux Diffeential Calculus The Gadient The Divegence The Cul Dpto. Física Aplicada III - Univ. de Sevilla 80

41 Chapte 1: Index (II) Mathematic tools Cylindical and Spheical coodinates The del opeato The Diac delta function Special types of fields: Cul-less o Iotational fields Divegence-less o Solenoidal fields Hamonic fields The Helmholtz Theoem Dpto. Física Aplicada III - Univ. de Sevilla 81

(read nabla or del) is defined by, k. (9.7.1*)

(read nabla or del) is defined by, k. (9.7.1*) 9.7 Gadient of a scala field. Diectional deivative Some of the vecto fields in applications can be obtained fom scala fields. This is vey advantageous because scala fields can be handled moe easily. The

More information

2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum

2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum 2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known

More information

working pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50

working pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50 woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,

More information

Final Review of AerE 243 Class

Final Review of AerE 243 Class Final Review of AeE 4 Class Content of Aeodynamics I I Chapte : Review of Multivaiable Calculus Chapte : Review of Vectos Chapte : Review of Fluid Mechanics Chapte 4: Consevation Equations Chapte 5: Simplifications

More information

ENGI 4430 Non-Cartesian Coordinates Page xi Fy j Fzk from Cartesian coordinates z to another orthonormal coordinate system u, v, ˆ i ˆ ˆi

ENGI 4430 Non-Cartesian Coordinates Page xi Fy j Fzk from Cartesian coordinates z to another orthonormal coordinate system u, v, ˆ i ˆ ˆi ENGI 44 Non-Catesian Coodinates Page 7-7. Conesions between Coodinate Systems In geneal, the conesion of a ecto F F xi Fy j Fzk fom Catesian coodinates x, y, z to anothe othonomal coodinate system u,,

More information

THE LAPLACE EQUATION. The Laplace (or potential) equation is the equation. u = 0. = 2 x 2. x y 2 in R 2

THE LAPLACE EQUATION. The Laplace (or potential) equation is the equation. u = 0. = 2 x 2. x y 2 in R 2 THE LAPLACE EQUATION The Laplace (o potential) equation is the equation whee is the Laplace opeato = 2 x 2 u = 0. in R = 2 x 2 + 2 y 2 in R 2 = 2 x 2 + 2 y 2 + 2 z 2 in R 3 The solutions u of the Laplace

More information

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

ECE 3318 Applied Electricity and Magnetism. Spring Prof. David R. Jackson ECE Dept. Notes 13 ECE 338 Applied Electicity and Magnetism ping 07 Pof. David R. Jackson ECE Dept. Notes 3 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

More information

Stress, Cauchy s equation and the Navier-Stokes equations

Stress, Cauchy s equation and the Navier-Stokes equations Chapte 3 Stess, Cauchy s equation and the Navie-Stokes equations 3. The concept of taction/stess Conside the volume of fluid shown in the left half of Fig. 3.. The volume of fluid is subjected to distibuted

More information

As is natural, our Aerospace Structures will be described in a Euclidean three-dimensional space R 3.

As is natural, our Aerospace Structures will be described in a Euclidean three-dimensional space R 3. Appendix A Vecto Algeba As is natual, ou Aeospace Stuctues will be descibed in a Euclidean thee-dimensional space R 3. A.1 Vectos A vecto is used to epesent quantities that have both magnitude and diection.

More information

Review: Electrostatics and Magnetostatics

Review: Electrostatics and Magnetostatics Review: Electostatics and Magnetostatics In the static egime, electomagnetic quantities do not vay as a function of time. We have two main cases: ELECTROSTATICS The electic chages do not change postion

More information

Math 2263 Solutions for Spring 2003 Final Exam

Math 2263 Solutions for Spring 2003 Final Exam Math 6 Solutions fo Sping Final Exam ) A staightfowad appoach to finding the tangent plane to a suface at a point ( x, y, z ) would be to expess the cuve as an explicit function z = f ( x, y ), calculate

More information

Vectors, Vector Calculus, and Coordinate Systems

Vectors, Vector Calculus, and Coordinate Systems Apil 5, 997 A Quick Intoduction to Vectos, Vecto Calculus, and Coodinate Systems David A. Randall Depatment of Atmospheic Science Coloado State Univesity Fot Collins, Coloado 80523. Scalas and vectos Any

More information

Math 259 Winter Handout 6: In-class Review for the Cumulative Final Exam

Math 259 Winter Handout 6: In-class Review for the Cumulative Final Exam Math 259 Winte 2009 Handout 6: In-class Review fo the Cumulative Final Exam The topics coveed by the cumulative final exam include the following: Paametic cuves. Finding fomulas fo paametic cuves. Dawing

More information

AE301 Aerodynamics I UNIT B: Theory of Aerodynamics

AE301 Aerodynamics I UNIT B: Theory of Aerodynamics AE301 Aeodynamics I UNIT B: Theoy of Aeodynamics ROAD MAP... B-1: Mathematics fo Aeodynamics B-2: Flow Field Repesentations B-3: Potential Flow Analysis B-4: Applications of Potential Flow Analysis AE301

More information

EELE 3331 Electromagnetic I Chapter 4. Electrostatic fields. Islamic University of Gaza Electrical Engineering Department Dr.

EELE 3331 Electromagnetic I Chapter 4. Electrostatic fields. Islamic University of Gaza Electrical Engineering Department Dr. EELE 3331 Electomagnetic I Chapte 4 Electostatic fields Islamic Univesity of Gaza Electical Engineeing Depatment D. Talal Skaik 212 1 Electic Potential The Gavitational Analogy Moving an object upwad against

More information

Electrostatics (Electric Charges and Field) #2 2010

Electrostatics (Electric Charges and Field) #2 2010 Electic Field: The concept of electic field explains the action at a distance foce between two chaged paticles. Evey chage poduces a field aound it so that any othe chaged paticle expeiences a foce when

More information

Physics 235 Chapter 5. Chapter 5 Gravitation

Physics 235 Chapter 5. Chapter 5 Gravitation Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

More information

Gauss Law. Physics 231 Lecture 2-1

Gauss Law. Physics 231 Lecture 2-1 Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

More information

Ch 30 - Sources of Magnetic Field! The Biot-Savart Law! = k m. r 2. Example 1! Example 2!

Ch 30 - Sources of Magnetic Field! The Biot-Savart Law! = k m. r 2. Example 1! Example 2! Ch 30 - Souces of Magnetic Field 1.) Example 1 Detemine the magnitude and diection of the magnetic field at the point O in the diagam. (Cuent flows fom top to bottom, adius of cuvatue.) Fo staight segments,

More information

University Physics (PHY 2326)

University Physics (PHY 2326) Chapte Univesity Physics (PHY 6) Lectue lectostatics lectic field (cont.) Conductos in electostatic euilibium The oscilloscope lectic flux and Gauss s law /6/5 Discuss a techniue intoduced by Kal F. Gauss

More information

! E da = 4πkQ enc, has E under the integral sign, so it is not ordinarily an

! E da = 4πkQ enc, has E under the integral sign, so it is not ordinarily an Physics 142 Electostatics 2 Page 1 Electostatics 2 Electicity is just oganized lightning. Geoge Calin A tick that sometimes woks: calculating E fom Gauss s law Gauss s law,! E da = 4πkQ enc, has E unde

More information

Physics 2B Chapter 22 Notes - Magnetic Field Spring 2018

Physics 2B Chapter 22 Notes - Magnetic Field Spring 2018 Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field

More information

6 Vector Operators. 6.1 The Gradient Operator

6 Vector Operators. 6.1 The Gradient Operator 6 Vecto Opeatos 6. The Gadient Opeato In the B2 couse ou wee intoduced to the gadient opeato in Catesian coodinates. Fo an diffeentiable scala function f(x,, z), we can define a vecto function though (

More information

DonnishJournals

DonnishJournals DonnishJounals 041-1189 Donnish Jounal of Educational Reseach and Reviews. Vol 1(1) pp. 01-017 Novembe, 014. http:///dje Copyight 014 Donnish Jounals Oiginal Reseach Pape Vecto Analysis Using MAXIMA Savaş

More information

Do not turn over until you are told to do so by the Invigilator.

Do not turn over until you are told to do so by the Invigilator. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Seies UG Examination 2015 16 FLUID DYNAMICS WITH ADVANCED TOPICS MTH-MD59 Time allowed: 3 Hous Attempt QUESTIONS 1 and 2, and THREE othe questions.

More information

ME 210 Applied Mathematics for Mechanical Engineers

ME 210 Applied Mathematics for Mechanical Engineers Tangent and Ac Length of a Cuve The tangent to a cuve C at a point A on it is defined as the limiting position of the staight line L though A and B, as B appoaches A along the cuve as illustated in the

More information

The Divergence Theorem

The Divergence Theorem 13.8 The ivegence Theoem Back in 13.5 we ewote Geen s Theoem in vecto fom as C F n ds= div F x, y da ( ) whee C is the positively-oiented bounday cuve of the plane egion (in the xy-plane). Notice this

More information

B da = 0. Q E da = ε. E da = E dv

B da = 0. Q E da = ε. E da = E dv lectomagnetic Theo Pof Ruiz, UNC Asheville, doctophs on YouTube Chapte Notes The Maxwell quations in Diffeential Fom 1 The Maxwell quations in Diffeential Fom We will now tansfom the integal fom of the

More information

PHZ 3113 Fall 2017 Homework #5, Due Friday, October 13

PHZ 3113 Fall 2017 Homework #5, Due Friday, October 13 PHZ 3113 Fall 2017 Homewok #5, Due Fiday, Octobe 13 1. Genealize the poduct ule (fg) = f g +f g to wite the divegence Ö (Ù Ú) of the coss poduct of the vecto fields Ù and Ú in tems of the cul of Ù and

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II September 15, 2012 Prof. Alan Guth PROBLEM SET 2

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II September 15, 2012 Prof. Alan Guth PROBLEM SET 2 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.07: Electomagnetism II Septembe 5, 202 Pof. Alan Guth PROBLEM SET 2 DUE DATE: Monday, Septembe 24, 202. Eithe hand it in at the lectue,

More information

3 VECTOR CALCULUS I. 3.1 Divergence and curl of vector fields. 3.2 Important identities

3 VECTOR CALCULUS I. 3.1 Divergence and curl of vector fields. 3.2 Important identities 3 VECTOR CALCULU I 3.1 Divegence and cul of vecto fields Let = ( x, y, z ). eveal popeties of φ fo scala φ wee intoduced in section 1.2. The gadient opeato may also be applied to vecto fields. Let F =

More information

7.2. Coulomb s Law. The Electric Force

7.2. Coulomb s Law. The Electric Force Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat

More information

1 Spherical multipole moments

1 Spherical multipole moments Jackson notes 9 Spheical multipole moments Suppose we have a chage distibution ρ (x) wheeallofthechageiscontained within a spheical egion of adius R, as shown in the diagam. Then thee is no chage in the

More information

F Q E v B MAGNETOSTATICS. Creation of magnetic field B. Effect of B on a moving charge. On moving charges only. Stationary and moving charges

F Q E v B MAGNETOSTATICS. Creation of magnetic field B. Effect of B on a moving charge. On moving charges only. Stationary and moving charges MAGNETOSTATICS Ceation of magnetic field. Effect of on a moving chage. Take the second case: F Q v mag On moving chages only F QE v Stationay and moving chages dw F dl Analysis on F mag : mag mag Qv. vdt

More information

Phys-272 Lecture 17. Motional Electromotive Force (emf) Induced Electric Fields Displacement Currents Maxwell s Equations

Phys-272 Lecture 17. Motional Electromotive Force (emf) Induced Electric Fields Displacement Currents Maxwell s Equations Phys-7 Lectue 17 Motional Electomotive Foce (emf) Induced Electic Fields Displacement Cuents Maxwell s Equations Fom Faaday's Law to Displacement Cuent AC geneato Magnetic Levitation Tain Review of Souces

More information

Question Bank. Section A. is skew-hermitian matrix. is diagonalizable. (, ) , Evaluate (, ) 12 about = 1 and = Find, if

Question Bank. Section A. is skew-hermitian matrix. is diagonalizable. (, ) , Evaluate (, ) 12 about = 1 and = Find, if Subject: Mathematics-I Question Bank Section A T T. Find the value of fo which the matix A = T T has ank one. T T i. Is the matix A = i is skew-hemitian matix. i. alculate the invese of the matix = 5 7

More information

Lecture 8 - Gauss s Law

Lecture 8 - Gauss s Law Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.

More information

J. N. R E DDY ENERGY PRINCIPLES AND VARIATIONAL METHODS APPLIED MECHANICS

J. N. R E DDY ENERGY PRINCIPLES AND VARIATIONAL METHODS APPLIED MECHANICS J. N. E DDY ENEGY PINCIPLES AND VAIATIONAL METHODS IN APPLIED MECHANICS T H I D E DI T IO N JN eddy - 1 MEEN 618: ENEGY AND VAIATIONAL METHODS A EVIEW OF VECTOS AND TENSOS ead: Chapte 2 CONTENTS Physical

More information

AST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1

AST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1 Please ead this fist... AST S: The oigin and evolution of the Univese Intoduction to Mathematical Handout This is an unusually long hand-out and one which uses in places mathematics that you may not be

More information

Vector d is a linear vector function of vector d when the following relationships hold:

Vector d is a linear vector function of vector d when the following relationships hold: Appendix 4 Dyadic Analysis DEFINITION ecto d is a linea vecto function of vecto d when the following elationships hold: d x = a xxd x + a xy d y + a xz d z d y = a yxd x + a yy d y + a yz d z d z = a zxd

More information

Lecture 23. Representation of the Dirac delta function in other coordinate systems

Lecture 23. Representation of the Dirac delta function in other coordinate systems Lectue 23 Repesentation of the Diac delta function in othe coodinate systems In a geneal sense, one can wite, ( ) = (x x ) (y y ) (z z ) = (u u ) (v v ) (w w ) J Whee J epesents the Jacobian of the tansfomation.

More information

Vectors, Vector Calculus, and Coordinate Systems

Vectors, Vector Calculus, and Coordinate Systems ! Revised Apil 11, 2017 1:48 PM! 1 Vectos, Vecto Calculus, and Coodinate Systems David Randall Physical laws and coodinate systems Fo the pesent discussion, we define a coodinate system as a tool fo descibing

More information

Appendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk

Appendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk Appendix A Appendices A1 ɛ and coss poducts A11 Vecto Opeations: δ ij and ɛ These ae some notes on the use of the antisymmetic symbol ɛ fo expessing coss poducts This is an extemely poweful tool fo manipulating

More information

ANTENNAS. Vector and Scalar Potentials. Maxwell's Equations. D = εe. For a linear, homogeneous, isotropic medium µ and ε are contant.

ANTENNAS. Vector and Scalar Potentials. Maxwell's Equations. D = εe. For a linear, homogeneous, isotropic medium µ and ε are contant. ANTNNAS Vecto and Scala Potentials Maxwell's quations jωb J + jωd D ρ B (M) (M) (M3) (M4) D ε B Fo a linea, homogeneous, isotopic medium and ε ae contant. Since B, thee exists a vecto A such that B A and

More information

B. Spherical Wave Propagation

B. Spherical Wave Propagation 11/8/007 Spheical Wave Popagation notes 1/1 B. Spheical Wave Popagation Evey antenna launches a spheical wave, thus its powe density educes as a function of 1, whee is the distance fom the antenna. We

More information

Flux. Area Vector. Flux of Electric Field. Gauss s Law

Flux. Area Vector. Flux of Electric Field. Gauss s Law Gauss s Law Flux Flux in Physics is used to two distinct ways. The fist meaning is the ate of flow, such as the amount of wate flowing in a ive, i.e. volume pe unit aea pe unit time. O, fo light, it is

More information

Liquid gas interface under hydrostatic pressure

Liquid gas interface under hydrostatic pressure Advances in Fluid Mechanics IX 5 Liquid gas inteface unde hydostatic pessue A. Gajewski Bialystok Univesity of Technology, Faculty of Civil Engineeing and Envionmental Engineeing, Depatment of Heat Engineeing,

More information

Physics 2A Chapter 10 - Moment of Inertia Fall 2018

Physics 2A Chapter 10 - Moment of Inertia Fall 2018 Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.

More information

Hopefully Helpful Hints for Gauss s Law

Hopefully Helpful Hints for Gauss s Law Hopefully Helpful Hints fo Gauss s Law As befoe, thee ae things you need to know about Gauss s Law. In no paticula ode, they ae: a.) In the context of Gauss s Law, at a diffeential level, the electic flux

More information

Page 1 of 6 Physics II Exam 1 155 points Name Discussion day/time Pat I. Questions 110. 8 points each. Multiple choice: Fo full cedit, cicle only the coect answe. Fo half cedit, cicle the coect answe and

More information

Look over Chapter 22 sections 1-8 Examples 2, 4, 5, Look over Chapter 16 sections 7-9 examples 6, 7, 8, 9. Things To Know 1/22/2008 PHYS 2212

Look over Chapter 22 sections 1-8 Examples 2, 4, 5, Look over Chapter 16 sections 7-9 examples 6, 7, 8, 9. Things To Know 1/22/2008 PHYS 2212 PHYS 1 Look ove Chapte sections 1-8 xamples, 4, 5, PHYS 111 Look ove Chapte 16 sections 7-9 examples 6, 7, 8, 9 Things To Know 1) What is an lectic field. ) How to calculate the electic field fo a point

More information

EM Boundary Value Problems

EM Boundary Value Problems EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do

More information

π(x, y) = u x + v y = V (x cos + y sin ) κ(x, y) = u y v x = V (y cos x sin ) v u x y

π(x, y) = u x + v y = V (x cos + y sin ) κ(x, y) = u y v x = V (y cos x sin ) v u x y F17 Lectue Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V φ = uî + vθˆ is a constant. In 2-D, this velocit

More information

Introduction: Vectors and Integrals

Introduction: Vectors and Integrals Intoduction: Vectos and Integals Vectos a Vectos ae chaacteized by two paametes: length (magnitude) diection a These vectos ae the same Sum of the vectos: a b a a b b a b a b a Vectos Sum of the vectos:

More information

Physics 2212 GH Quiz #2 Solutions Spring 2016

Physics 2212 GH Quiz #2 Solutions Spring 2016 Physics 2212 GH Quiz #2 Solutions Sping 216 I. 17 points) Thee point chages, each caying a chage Q = +6. nc, ae placed on an equilateal tiangle of side length = 3. mm. An additional point chage, caying

More information

Force and Work: Reminder

Force and Work: Reminder Electic Potential Foce and Wok: Reminde Displacement d a: initial point b: final point Reminde fom Mechanics: Foce F if thee is a foce acting on an object (e.g. electic foce), this foce may do some wok

More information

1 Fundamental Solutions to the Wave Equation

1 Fundamental Solutions to the Wave Equation 1 Fundamental Solutions to the Wave Equation Physical insight in the sound geneation mechanism can be gained by consideing simple analytical solutions to the wave equation. One example is to conside acoustic

More information

Math 209 Assignment 9 Solutions

Math 209 Assignment 9 Solutions Math 9 Assignment 9 olutions 1. Evaluate 4y + 1 d whee is the fist octant pat of y x cut out by x + y + z 1. olution We need a paametic epesentation of the suface. (x, z). Now detemine the nomal vecto:

More information

MAGNETIC FIELD INTRODUCTION

MAGNETIC FIELD INTRODUCTION MAGNETIC FIELD INTRODUCTION It was found when a magnet suspended fom its cente, it tends to line itself up in a noth-south diection (the compass needle). The noth end is called the Noth Pole (N-pole),

More information

An o5en- confusing point:

An o5en- confusing point: An o5en- confusing point: Recall this example fom last lectue: E due to a unifom spheical suface chage, density = σ. Let s calculate the pessue on the suface. Due to the epulsive foces, thee is an outwad

More information

Chapter 22: Electric Fields. 22-1: What is physics? General physics II (22102) Dr. Iyad SAADEDDIN. 22-2: The Electric Field (E)

Chapter 22: Electric Fields. 22-1: What is physics? General physics II (22102) Dr. Iyad SAADEDDIN. 22-2: The Electric Field (E) Geneal physics II (10) D. Iyad D. Iyad Chapte : lectic Fields In this chapte we will cove The lectic Field lectic Field Lines -: The lectic Field () lectic field exists in a egion of space suounding a

More information

( ) Make-up Tests. From Last Time. Electric Field Flux. o The Electric Field Flux through a bit of area is

( ) Make-up Tests. From Last Time. Electric Field Flux. o The Electric Field Flux through a bit of area is Mon., 3/23 Wed., 3/25 Thus., 3/26 Fi., 3/27 Mon., 3/30 Tues., 3/31 21.4-6 Using Gauss s & nto to Ampee s 21.7-9 Maxwell s, Gauss s, and Ampee s Quiz Ch 21, Lab 9 Ampee s Law (wite up) 22.1-2,10 nto to

More information

FI 2201 Electromagnetism

FI 2201 Electromagnetism FI Electomagnetim Aleande A. Ikanda, Ph.D. Phyic of Magnetim and Photonic Reeach Goup ecto Analyi CURILINEAR COORDINAES, DIRAC DELA FUNCION AND HEORY OF ECOR FIELDS Cuvilinea Coodinate Sytem Cateian coodinate:

More information

IX INDUCTANCE AND MAGNETIC FIELDS

IX INDUCTANCE AND MAGNETIC FIELDS IX INDUCTNCE ND MGNETIC FIELDS 9. Field in a solenoid vaying cuent in a conducto will poduce a vaying magnetic field. nd this vaying magnetic field then has the capability of inducing an EMF o voltage

More information

CHAPTER 25 ELECTRIC POTENTIAL

CHAPTER 25 ELECTRIC POTENTIAL CHPTE 5 ELECTIC POTENTIL Potential Diffeence and Electic Potential Conside a chaged paticle of chage in a egion of an electic field E. This filed exets an electic foce on the paticle given by F=E. When

More information

4. Electrodynamic fields

4. Electrodynamic fields 4. Electodynamic fields D. Rakhesh Singh Kshetimayum 1 4.1 Intoduction Electodynamics Faaday s law Maxwell s equations Wave equations Lenz s law Integal fom Diffeential fom Phaso fom Bounday conditions

More information

Welcome to Physics 272

Welcome to Physics 272 Welcome to Physics 7 Bob Mose mose@phys.hawaii.edu http://www.phys.hawaii.edu/~mose/physics7.html To do: Sign into Masteing Physics phys-7 webpage Registe i-clickes (you i-clicke ID to you name on class-list)

More information

On the Sun s Electric-Field

On the Sun s Electric-Field On the Sun s Electic-Field D. E. Scott, Ph.D. (EE) Intoduction Most investigatos who ae sympathetic to the Electic Sun Model have come to agee that the Sun is a body that acts much like a esisto with a

More information

dq 1 (5) q 1 where the previously mentioned limit has been taken.

dq 1 (5) q 1 where the previously mentioned limit has been taken. 1 Vecto Calculus And Continuum Consevation Equations In Cuvilinea Othogonal Coodinates Robet Maska: Novembe 25, 2008 In ode to ewite the consevation equations(continuit, momentum, eneg) to some cuvilinea

More information

2 Governing Equations

2 Governing Equations 2 Govening Equations This chapte develops the govening equations of motion fo a homogeneous isotopic elastic solid, using the linea thee-dimensional theoy of elasticity in cylindical coodinates. At fist,

More information

MAGNETIC FIELD AROUND TWO SEPARATED MAGNETIZING COILS

MAGNETIC FIELD AROUND TWO SEPARATED MAGNETIZING COILS The 8 th Intenational Confeence of the Slovenian Society fo Non-Destuctive Testing»pplication of Contempoay Non-Destuctive Testing in Engineeing«Septembe 1-3, 5, Potoož, Slovenia, pp. 17-1 MGNETIC FIELD

More information

When a mass moves because of a force, we can define several types of problem.

When a mass moves because of a force, we can define several types of problem. Mechanics Lectue 4 3D Foces, gadient opeato, momentum 3D Foces When a mass moves because of a foce, we can define seveal types of poblem. ) When we know the foce F as a function of time t, F=F(t). ) When

More information

Chapter 21: Gauss s Law

Chapter 21: Gauss s Law Chapte : Gauss s Law Gauss s law : intoduction The total electic flux though a closed suface is equal to the total (net) electic chage inside the suface divided by ε Gauss s law is equivalent to Coulomb

More information

Physics 122, Fall October 2012

Physics 122, Fall October 2012 hsics 1, Fall 1 3 Octobe 1 Toda in hsics 1: finding Foce between paallel cuents Eample calculations of fom the iot- Savat field law Ampèe s Law Eample calculations of fom Ampèe s law Unifom cuents in conductos?

More information

Vectors Serway and Jewett Chapter 3

Vectors Serway and Jewett Chapter 3 Vectos Sewa and Jewett Chapte 3 Scalas and Vectos Vecto Components and Aithmetic Vectos in 3 Dimensions Unit vectos i, j, k Pactice Poblems: Chapte 3, poblems 9, 19, 31, 45, 55, 61 Phsical quantities ae

More information

EM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1)

EM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1) EM- Coulomb s law, electic field, potential field, supeposition q ' Electic field of a point chage ( ') E( ) kq, whee k / 4 () ' Foce of q on a test chage e at position is ee( ) Electic potential O kq

More information

Faraday s Law (continued)

Faraday s Law (continued) Faaday s Law (continued) What causes cuent to flow in wie? Answe: an field in the wie. A changing magnetic flux not only causes an MF aound a loop but an induced electic field. Can wite Faaday s Law: ε

More information

PHYS 1444 Section 501 Lecture #7

PHYS 1444 Section 501 Lecture #7 PHYS 1444 Section 51 Lectue #7 Wednesday, Feb. 8, 26 Equi-potential Lines and Sufaces Electic Potential Due to Electic Dipole E detemined fom V Electostatic Potential Enegy of a System of Chages Capacitos

More information

( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx.

( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx. 9. LAGRANGIAN OF THE ELECTROMAGNETIC FIELD In the pevious section the Lagangian and Hamiltonian of an ensemble of point paticles was developed. This appoach is based on a qt. This discete fomulation can

More information

transformation Earth V-curve (meridian) λ Conical projection. u,v curves on the datum surface projected as U,V curves on the projection surface

transformation Earth V-curve (meridian) λ Conical projection. u,v curves on the datum surface projected as U,V curves on the projection surface . CONICAL PROJECTIONS In elementay texts on map pojections, the pojection sufaces ae often descibed as developable sufaces, such as the cylinde (cylindical pojections) and the cone (conical pojections),

More information

Electric field generated by an electric dipole

Electric field generated by an electric dipole Electic field geneated by an electic dipole ( x) 2 (22-7) We will detemine the electic field E geneated by the electic dipole shown in the figue using the pinciple of supeposition. The positive chage geneates

More information

Sources of Magnetic Fields (chap 28)

Sources of Magnetic Fields (chap 28) Souces of Magnetic Fields (chap 8) In chapte 7, we consideed the magnetic field effects on a moving chage, a line cuent and a cuent loop. Now in Chap 8, we conside the magnetic fields that ae ceated by

More information

To Feel a Force Chapter 7 Static equilibrium - torque and friction

To Feel a Force Chapter 7 Static equilibrium - torque and friction To eel a oce Chapte 7 Chapte 7: Static fiction, toque and static equilibium A. Review of foce vectos Between the eath and a small mass, gavitational foces of equal magnitude and opposite diection act on

More information

Voltage ( = Electric Potential )

Voltage ( = Electric Potential ) V-1 of 10 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage

More information

17.1 Electric Potential Energy. Equipotential Lines. PE = energy associated with an arrangement of objects that exert forces on each other

17.1 Electric Potential Energy. Equipotential Lines. PE = energy associated with an arrangement of objects that exert forces on each other Electic Potential Enegy, PE Units: Joules Electic Potential, Units: olts 17.1 Electic Potential Enegy Electic foce is a consevative foce and so we can assign an electic potential enegy (PE) to the system

More information

Unit 7: Sources of magnetic field

Unit 7: Sources of magnetic field Unit 7: Souces of magnetic field Oested s expeiment. iot and Savat s law. Magnetic field ceated by a cicula loop Ampèe s law (A.L.). Applications of A.L. Magnetic field ceated by a: Staight cuent-caying

More information

3. Magnetostatic fields

3. Magnetostatic fields 3. Magnetostatic fields D. Rakhesh Singh Kshetimayum 1 Electomagnetic Field Theoy by R. S. Kshetimayum 3.1 Intoduction to electic cuents Electic cuents Ohm s law Kichoff s law Joule s law Bounday conditions

More information

Physics 122, Fall October 2012

Physics 122, Fall October 2012 Today in Physics 1: electostatics eview David Blaine takes the pactical potion of his electostatics midtem (Gawke). 11 Octobe 01 Physics 1, Fall 01 1 Electostatics As you have pobably noticed, electostatics

More information

Fields. Coulomb s Law

Fields. Coulomb s Law Coulomb s Law q t -q q 2 Electic Field Vecto valued function ligned with foce F = q E -q q 2 Supeposition of Electic Field q t -q q 2 Potential Enegy U = U() U() = q du = F d = qe d U = F = qe E d E =

More information

arxiv: v1 [physics.pop-ph] 3 Jun 2013

arxiv: v1 [physics.pop-ph] 3 Jun 2013 A note on the electostatic enegy of two point chages axiv:1306.0401v1 [physics.pop-ph] 3 Jun 013 A C Tot Instituto de Física Univesidade Fedeal do io de Janeio Caixa Postal 68.58; CEP 1941-97 io de Janeio,

More information

An Exact Solution of Navier Stokes Equation

An Exact Solution of Navier Stokes Equation An Exact Solution of Navie Stokes Equation A. Salih Depatment of Aeospace Engineeing Indian Institute of Space Science and Technology, Thiuvananthapuam, Keala, India. July 20 The pincipal difficulty in

More information

Applied Aerodynamics

Applied Aerodynamics Applied Aeodynamics Def: Mach Numbe (M), M a atio of flow velocity to the speed of sound Compessibility Effects Def: eynolds Numbe (e), e ρ c µ atio of inetial foces to viscous foces iscous Effects If

More information

Chapter 13 Gravitation

Chapter 13 Gravitation Chapte 13 Gavitation In this chapte we will exploe the following topics: -Newton s law of gavitation, which descibes the attactive foce between two point masses and its application to extended objects

More information

Superposition. Section 8.5.3

Superposition. Section 8.5.3 Supeposition Section 8.5.3 Simple Potential Flows Most complex potential (invicid, iotational) flows can be modeled using a combination of simple potential flows The simple flows used ae: Unifom flows

More information

Lecture 3. Announce: Office Hours to be held Tuesdays 1-3pm in I've discussed plane wave solutions. When seeking more general solutions to

Lecture 3. Announce: Office Hours to be held Tuesdays 1-3pm in I've discussed plane wave solutions. When seeking more general solutions to Lectue 3. Announce: Office Hous to be held Tuesdays 1-3pm in 4115 I've discussed plane wave solutions. When seeking moe geneal solutions to ρ && u = (λ + 2µ) ( u) µ ( u) many people will decouple the PDE

More information

e.g: If A = i 2 j + k then find A. A = Ax 2 + Ay 2 + Az 2 = ( 2) = 6

e.g: If A = i 2 j + k then find A. A = Ax 2 + Ay 2 + Az 2 = ( 2) = 6 MOTION IN A PLANE 1. Scala Quantities Physical quantities that have only magnitude and no diection ae called scala quantities o scalas. e.g. Mass, time, speed etc. 2. Vecto Quantities Physical quantities

More information

UNIT 3:Electrostatics

UNIT 3:Electrostatics The study of electic chages at est, the foces between them and the electic fields associated with them. UNIT 3:lectostatics S7 3. lectic Chages and Consevation of chages The electic chage has the following

More information

. Using our polar coordinate conversions, we could write a

. Using our polar coordinate conversions, we could write a 504 Chapte 8 Section 8.4.5 Dot Poduct Now that we can add, sutact, and scale vectos, you might e wondeing whethe we can multiply vectos. It tuns out thee ae two diffeent ways to multiply vectos, one which

More information

1 Similarity Analysis

1 Similarity Analysis ME43A/538A/538B Axisymmetic Tubulent Jet 9 Novembe 28 Similaity Analysis. Intoduction Conside the sketch of an axisymmetic, tubulent jet in Figue. Assume that measuements of the downsteam aveage axial

More information

Chapter 25. Electric Potential

Chapter 25. Electric Potential Chapte 25 Electic Potential C H P T E R O U T L I N E 251 Potential Diffeence and Electic Potential 252 Potential Diffeences in a Unifom Electic Field 253 Electic Potential and Potential Enegy Due to Point

More information