FI 2201 Electromagnetism
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1 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
2 Cuvilinea Coodinate Sytem Cateian coodinate: ued to decibe ytem without any appaent ymmety. Cuvilinea coodinate: ued to decibe ytem with ymmety. We will often find pheical ymmety o aial ymmety in the poblem we will do thi emete, and will thu ue Spheical coodinate Cylindical coodinate Aleande A. Ikanda Electomagnetim 3 Spheical Coodinate Sytem he location of a point P can be defined by pecifying the following thee paamete: Radiu : ditance of P fom the oigin. Pola angle : angle between the poition vecto of P and the ai. (Like 9 latitude Aimuthal angle : angle between the pojection of the poition vecto P and the ai. (Like longitude Aleande A. Ikanda Electomagnetim 4
3 Spheical Coodinate Sytem he Cateian coodinate of P ae elated to the pheical coodinate a follow: y in co y y in in actan co y actan he unit vecto of pheical coodinate ytem ae not contant: thei diection change when the poition of point P change. Aleande A. Ikanda Electomagnetim 5 y Spheical Coodinate Sytem In Cateian coodinate, an infiniteimal diplacement fom point P i equal to d ydy d In pheical coodinate, an infiniteimal diplacement fom point P i equal to d d in d whee i paallel to, i pependicula to and lie in the - plane, i pependicula to thi plane, and point in the diection of inceaing and. Aleande A. Ikanda Electomagnetim 6 3
4 Spheical Coodinate Sytem he pheical-coodinate unit vecto in tem of the Cateian one. (, in co y in in co π (, (, co co y co in in co y co ( Aleande A. Ikanda Electomagnetim 7 Spheical Coodinate Sytem In a Cateian coodinate, an infiniteimal volume element aound point P i equal to d τ d dy d In a pheical coodinate, an infiniteimal volume element aound point P i equal to dτ in d d d Aleande A. Ikanda Electomagnetim 8 4
5 Spheical Coodinate Sytem In Cateian coodinate, an infiniteimal aea element on a plane containing point P i da d dy o y dy d o y d d In pheical coodinate, the infiniteimal aea element on a phee though point P i da in d d Aleande A. Ikanda Electomagnetim 9 ecto Deivative in Spheical Cood. What if you want to epe a vecto deivative in pheical coodinate? Stat fom the Cateian-coodinate veion, and fit ue the chain ule to tanfom the deivative, e.g. ( Ue the coodinate definition to educe the emaining deivative and eliminate all Cateian coodinate, e.g. y anfom the unit vecto, a we did ealie today. hen multiply the whole me out and implify. y in co Aleande A. Ikanda Electomagnetim 5
6 ecto Deivative in Spheical Cood. he following opeation will be encounteed fequently enough in ( ( in in in in in in in in Aleande A. Ikanda Electomagnetim in Cylindical Coodinate Sytem Spheical coodinate ae ueful motly fo pheically ymmetic ituation. In poblem involving ymmety about jut one ai, cylindical coodinate ae ued: he adiu : ditance of P fom the ai. he aimuthal angle : angle between the pojection of the poition vecto P and the ai. (Same a the pheical coodinate of the ame name. he coodinate: component of the poition vecto P along the ai. (Same a the Cateian. Aleande A. Ikanda Electomagnetim 6
7 Cylindical Coodinate Sytem he Cateian coodinate of P ae elated to the cylindical coodinate by co y in y y actan Again, the unit vecto of cylindical coodinate ytem ae not contant; thei diection change when the poition of point P change. y Aleande A. Ikanda Electomagnetim 3 Cylindical Coodinate Sytem In cylindical coodinate: unit vecto ( co y in π in y co ( ( infiniteimal diplacement d d d infiniteimal volume element dτ d d d infiniteimal aea element da d d ( top of d d cylinde ( cylinde wall Aleande A. Ikanda Electomagnetim 4 7
8 8 ecto Deivative in Cylindical Cood. he following opeation will be encounteed fequently enough ( Electomagnetim Aleande A. Ikanda 5 Diac Delta Function Conide the vecto field v he divegence (in pheical coodinate On the othe hand, conideing a pheical volume and calculating the flu of the vecto field we obtain v π π v Electomagnetim Howeve, with divegence theoem, thi i a contadiction Aleande A. Ikanda 6 ( π π π 4 in d d da v S S dτ v da v
9 Diac Delta Function Why the contadiction? Recall that the divegence of v i evaluated at any othe point ecept at the oigin, and note that at thi oigin the field i ingula. While the uface flu integation captue the net outgoing flu, i.e. infomation of ouce and ink inide the cloed uface of integation, in the cae at hand it i a ouce at the oigin. Hence, we need a function that ha the popety that it vanihe eveywhee ecept at one paticula point, in thi cae the oigin. Phyicit know (in fact a phyicit invented a function of thi behaviou, it i the Diac delta function. Aleande A. Ikanda Electomagnetim 7 Diac Delta Function Conide evaluating the aea unde the cuve: 4 δ( A 4 R A R A ( R d We can continue thi poce until the ange become infiniteimal and we have the definition of the Diac delta function δ ( d A ( R d Aleande A. Ikanda Electomagnetim 8 9
10 Diac Delta Function In fact the Diac delta function can alo be defined a δ( and integal i not neceay be taken fom until, but a long a the point [a, b] b a Anothe popety of the Diac delta function i when it act on anothe function a follow δ( b a δ ( d f ( δ ( d f ( f( In othe wod, the Diac delta function pick out the pecial value of the f( function at. Aleande A. Ikanda Electomagnetim 9 Diac Delta Function he one dimenional (D Diac delta function δ( can be genealied into a 3D Diac delta function a follow δ ( dτ,, f ( δ ( dτ f ( whee 3 δ ( δ ( δ ( δ ( y δ ( hu the divegence of the vecto field v i v 4 ( πδ Aleande A. Ikanda Electomagnetim
11 Diac Delta Function Why i the delta function ueful? It a nice way mathematically to epe compact entitie uch a point chage, when we have to decibe them with diffeential equation. Fo intance, the chage denity electic chage pe unit volume of a point chage q can be witten a ρ ( qδ ( Aleande A. Ikanda Electomagnetim heoy of ecto Field A unique olution of a diffeential equation can be obtained povided we ae given the bounday condition. In electomagnetic, ou bounday condition i that the field goe to eo at infinity. Helmholt heoem : Solution of a vecto field can be obtained uniquely povided that F D with the cala function D i known F C C : with the vecto function C i known the vecto field F goe to eo at infinity Aleande A. Ikanda Electomagnetim
12 heoy of ecto Field he olution i given a F A whee D( ( dτ 4π C( A( dτ 4π hi can be poved by taking divegence and cul of F. Aleande A. Ikanda Electomagnetim 3 Potential Scala Potential If cul of vanihe evey whee (i.e. C evey whee, then F he function i called the Scala Potential. Fo cul-le o iotational o conevative field F the following condition ae equivalent F eveywhee b F i independent of path F d l a, path C F, F i a gadient of a cala potential that i not unique (a contant can be added Aleande A. Ikanda Electomagnetim 4
13 Potential ecto Potential If divegence of vanihe evey whee (i.e. D evey whee, then F A he function A( i called the ecto Potential Fo divegence-le o olenoidal F the following condition ae equivalent F eveywhee F da i independent of uface fo any given bounday line S F da, F S F A i a cul of a vecto potential that i not unique ( a gadient of a cala function can be added Aleande A. Ikanda Electomagnetim 5 Homewok of Chapte Poblem.3,.6,.3,.3,.33,.45 Due 3 Feb Aleande A. Ikanda Electomagnetim 6 3
Static Electric Fields. Coulomb s Law Ε = 4πε. Gauss s Law. Electric Potential. Electrical Properties of Materials. Dielectrics. Capacitance E.
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