Chapter 1: Vector Analysis
|
|
- Tabitha Bridges
- 5 years ago
- Views:
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*)
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 information2. 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 informationworking 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 informationFinal 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 informationENGI 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 informationTHE 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 informationECE 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 informationStress, 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 informationAs 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 informationReview: 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 informationMath 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 informationVectors, 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 informationMath 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 informationAE301 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 informationEELE 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 informationElectrostatics (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 informationPhysics 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 informationGauss 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 informationCh 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 informationUniversity 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
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 informationPhysics 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 information6 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 informationDonnishJournals
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 informationDo 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 informationME 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 informationThe 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 informationB 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 informationPHZ 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 informationMASSACHUSETTS 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 information3 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 information7.2. 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 information1 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 informationF 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 informationPhys-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 informationQuestion 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 informationLecture 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 informationJ. 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 informationAST 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 informationVector 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 informationLecture 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 informationVectors, 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 informationAppendix 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 informationANTENNAS. 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 informationB. 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 informationFlux. 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 informationLiquid 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 informationPhysics 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 informationHopefully 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 informationPage 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 informationLook 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 informationEM 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 informationπ(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 informationIntroduction: 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 informationPhysics 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 informationForce 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 information1 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 informationMath 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 informationMAGNETIC 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 informationAn 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 informationChapter 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
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 informationFI 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 informationIX 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 informationCHAPTER 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 information4. 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 informationWelcome 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 informationOn 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 informationdq 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 information2 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 informationMAGNETIC 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 informationWhen 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 informationChapter 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 informationPhysics 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 informationVectors 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 informationEM-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 informationFaraday 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 informationPHYS 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.
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 informationtransformation 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 informationElectric 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 informationSources 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 informationTo 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 informationVoltage ( = 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 information17.1 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 informationUnit 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 information3. 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 informationPhysics 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 informationFields. 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 informationarxiv: 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 informationAn 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 informationApplied 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 informationChapter 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 informationSuperposition. 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 informationLecture 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 informatione.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 informationUNIT 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
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 information1 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 informationChapter 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