334 lecoagneic Theoy I Todd Kaise Maxwell s quaions: Maxwell s equaions wee developed on expeienal evidence and have been found o goven all classical elecoagneic phenoena. They can be wien in diffeenial o inegal fo. Gauss' s Law No agneiconopoles Faaday's Law Modified Apee's Law 0 + d dl d 0 dl whee: lecic Field Inensiy (/) lecic Flux ensiy (/ ) Magneic Field Inensiy (A/) Magneic Flux ensiy (T) lecic uen ensiy (A/ ) lecic hage ensiy (/ 3 ) dv Q d d + d The oninuiy quaion fo cuen is consisen wih Maxwell s quaions and he consevaion of chage. I can be used o deive Kichhoff s uen Law: + 0 if 0 0 iplies KL onsiuive elaionships: The field inensiies and flux densiies ae elaed by using he consiuive equaions. In geneal, he peiiviy () and he peeabiliy (µ) ae ensos (diffeen values in diffeen diecions) and ae funcions of he aeial. In siple aeials hey ae scalas. µ 0 µ µ whee: elaive peiiviy 0 acuu peiiviy µ elaive peeabiliy µ 0 acuu peeabiliy 0 ounday ondiions: A abup inefaces beween diffeen aeials he following condiions hold: ( ) ˆ 0 n ( ) ( ) ( ) 0 whee: n is he noal veco fo egion- o egion- s is he suface cuen densiy (A/) s is he suface chage densiy (/ )
lecosaic Fields: When hee ae no ie dependen fields, elecic and agneic fields can exis as independen fields. The elecic fields ae poduced by chage disibuions govened by: 0 d dl 0 Q dv Gauss' s Law Gauss s Law can be used o find he elecic field of highly syeic pobles whee only a single veco coponen is equied. Using he pinciple of supeposiion, he field and he poenial due o an abiay chage disibuion ae: ( ) ( ') ˆ ( ') dv dv whee is he posiion veco of elecicfield 'is he posiion veco of he chagedisibuion ' is he disance beween he elecic field and chage disibuion ince he cul of he elecic field is zeo he field can be wien as he gadien of a scala poenial: a 0 The las equaion iplies Kichhoff s olage Law. a b dl dl b Fo Gauss s Law he poenial us saisfy Poisson s quaion o Laplace s equaion fo chage fee egions: 0 ufaces of consan poenials ae called equipoenial sufaces, and ae found o inesec elecic fields a igh angles. Pefec conducing sufaces ae equipoenial sufaces. The foce exising beween poin chages was found o obey oulob s Law: qq F ˆ Whee is he disance beween he chages and is he peiiviy of he aeial beween he chages, and foce diecion is on he line connecing he poin chages. Like chages epel and opposies aac. hage soage devices can be ceaed using uliple conducos sepaaed by a dielecic. The capaciance and enegy soage in elecosaics ae given by: d Q W dv dl e
lecosaic foces can be calculaed using he gadien of he enegy funcion: F W e Moving hages: A chage (q) oving wih a velociy (u) in an elecic and agneic field will expeience a foce: Loenz Foce Law F q( + u ) This was how he elecic field and agneic fields wee discoveed and defined. Moving chages give ise o a cuen densiy: u onvecioncuen densiy This also applies o chages oving in conducos, howeve he elecons scae in he aeial coplicaing he odel. xpeienally, i was found he ne dif velociy of elecons is popoional o he applied elecic field in he conduco. σ Poin fo of Oh's Law The conduciviy (σ (/)) is a popey of he aeial. Any excess of chage in a conduco will edisibue iself o an equilibiu sae wih zeo ne chage wih a ie consan τ d /σ which is he dielecic elaxaion ie. The oal cuen is elaed o he cuen densiy by: I d The esisance of a unifo wie of conducing aeial is given by: dl dl L d σ d σ A whee L is he lengh of he wie and A is he coss-secional aea. Magneosaic Fields: aic agneic fields ae govened by: No agneic onopoles Apee's Law 0 d 0 dl d I aic agneic fields ae poduced by seady cuens (). Apee s Law can be used o find agneic fields of highly syeic configuaions when only one coponen of he field is equied. io-ava Law is used o copue he agneic field fo an abiay cuen disibuion: Idl ˆ ( ) dv ' ˆ ( ) dl' ( ) L 3
Fo a long wie: I ˆ φ π A wie caying a cuen expeiences a foce in a agneic field given by: F I dl when he wie is saigh, his educes o a foce pe lengh, he foce will be pependicula o boh he cuen and he field. The agneic flux is given by: Ψ d The agneic flux acoss a closed suface is always zeo: Ψ d 0 The aio of he oal agneic flux and he cuen is called he inducance. Fo a cicui wih N loops he inducance and he soed agneic enegy ae given by: N L d d NΨ I W dv ince he induced agneic fields ae also popoional o he nube of uns he inducance usually vaies as N. Magneosaic foces ae copued as he gadien of he enegy: LI F W Tie-vaying Fields Maxwell s equaions ae geneal and apply fo all elecoagneic phenoena. oweve, sinusoidal vaying fields ae os coonly used wih he ie dependence of (cos ω). The fields ae wien in a phaso fo: jω { e } (. ) e ( ) Maxwell's quaions and coninuiy equaions in phaso - fo ae : -jω + jω 0 jω 4
5 negy, Powe, and Poyning Theoe: The powe densiy caied by he elecic and agneic fields is given by he Poyning veco: ( ) W / The diecion of he Poyning veco is he diecion of powe flow. Using Maxwell s equaions, he flux of he Poyning veco ou of a closed suface () is Poyning s Theoe. ( ) dv d + + This epesens hee fos of enegy: Ohic Losses (ea) oed Magneic negy ensiy oed lecic negy ensiy p w w e σ µ σ In phaso-fo he coplex Poyning veco is: This gives he insananeous powe densiy, usually he aveage powe densiy is oe ipoan and ha is given by: { } ( ) / e - aveaged Powe ensiy Tie W P A Teinology: oogeneous: A ediu in which he aeial consans ae unifo in he aea of inees. Inhoogeneous: The aeial vaies in coposiion such ha he aeial popeies ae funcions of posiion. xaples: (x) µ(x) σ(x) Isoopic: The aeial popeies do no depend on he polaizaion of he elecoagneic fields. Anisoopic: Maeial popeies depend on he polaizaion of he elecoagneic fields and us be wien as ensos. ouce-fee: egion whee no elecoagneic souces ae found heefoe 0 Non-agneic: The aeial has µµ 0 ispesive: The aeial popeies ae a funcion of fequency. xaples: (ω), µ(ω)