Lecture 14. Time Harmonic Fields
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- Philip Webb
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1 Lcu 4 Tim amic Filds I his lcu u will la: Cmpl mahmaics f im-hamic filds Mawll s quais f im-hamic filds Cmpl Pig vc C 303 Fall 007 Faha aa Cll Uivsi Tim-amic Filds ad -filds f a pla wav a (fm las lcu: + + (, cs( (, ( cs( c Filds f which h im vaiai is siusidal a calld im-hamic filds Pla wavs a us ampl f im-hamic filds I h s f his cus, 95% f h maial will dal wih im-hamic filds C 303 Fall 007 Faha aa Cll Uivsi
2 Tim-amic igals i Cicuis iusidal ad a Csid a C cicui div b a siusidal vlag suc: ( V cs( V ~ I( + - V ( + C - V C ( C mmb phass fm C0 V ( [ V ] [ ] V ( V ( I ( I( [ ] V ( C V + C I ( C V + C Tim-avag pw dissipai i h sis: V V C (( I V ( I ( + ( C C 303 Fall 007 Faha aa Cll Uivsi Basic ida: m usful igmic Idiis call bf w sa: θ θ + θ cs( θ cs( θ + si( θ θ θ θ si( θ cs( θ si( θ pssi f h -fild f a pla wav i cmpl ai: (, cs( (, Tim-amic Filds ad Cmpl Nai If h im-vaiai f filds is w a-pii b siusidal (i h filds a w b im-hamic h, i d simplif h mah, ma ca aud h im dpdc plicil i calculais Ls l a pla wavs as a ampl s hw h cmpl ai ca b usd fac u h siusidal im dpdc ( ( + ( C 303 Fall 007 Faha aa Cll Uivsi
3 F h -fild f a pla wav w had ( cs(, (, D a lil m maipulai (, Tim-amic Filds ad Vc Phass (, ( wh: Th quai (, which is a im-idpd cmpl vc, is a vc phas f h pla wav I h b, h vc phas has a addiial ud-li ad wi as: ( C 303 Fall 007 Faha aa Cll Uivsi F h -fild f a pla wav w had ( cs(, (, ( C 303 Fall 007 Faha aa Cll Uivsi [ ] All im-hamic filds ( us pla wavs ca b wi i h fm:, Cmpl Nai Nw gali all im-hamic filds: ( wh is a cmpl im-idpd Giv a vc phas ( vc phas f a im-hamic fild, ca fid h acual im-dpd fild as fllws:, ampl: upps I giv u h fllwig vc phas f a pla wav: A Th u ca fid h acual im-dpd -fild as fllws: ( [ ] [ A, ] Acs 3
4 Mawll s quais f Phass - I L h im-hamic ad -filds b:, (, ( [ ] Assum ha h im-vaiais f chag dsi ad cu dsi a als siusidal:, (, Nw w subsiu hs pssis i Mawll s quais b Gauss Law: ε (, (, [ ε ] [ ] Th l wa h abv ca b u f all im is if: ε Gauss Law f h Magic Fild:, 0 0 ( ( C 303 Fall 007 Faha aa Cll Uivsi Mawll s quais f Phass - II Th im-hamic ad -filds a: ( [ ],,, ( (, Faada s Law:,, [ ] [ ] C 303 Fall 007 Faha aa Cll Uivsi [ ] [ ] Th l wa h abv ca b u f all im is if: (3 Amp s Law: ε,,, + v [ ] [ ( + ε ] Th l wa h abv ca b u f all im is if: v ( + ε (4 4
5 5 C 303 Fall 007 Faha aa Cll Uivsi Mawll s quais f Phass - III L h im-hamic ad -filds b giv as:,,,, Mawll s quais f h vc phass f im-hamic filds a h: ε 0 v ε + Gauss Law: Gauss Law f h Magic Fild: Faada s Law: Amp s Law: C 303 Fall 007 Faha aa Cll Uivsi Calculais i h Cmpl Nai upps f a pla wav w w h -fild b:, wh w ds fid h vc phas f h -fild? Us Faada s law f im-hamic filds: v,
6 6 C 303 Fall 007 Faha aa Cll Uivsi Cmpl Pig Vc upps f a pla wav w w h -fild ad -fild phass b: w ds fid h im-avag pw p ui aa caid b h wav? Dfi a cmpl Pig vc as: Claim: Th im-avag pw p ui aa is -half f h al pa f h cmpl Pig vc Chc:, which is idd h igh asw C 303 Fall 007 Faha aa Cll Uivsi M Calculais i h Cmpl Nai - I ampl: Csid a pla wav wih -fild f ampliud ad piig i a dici 45-dgs w h -ais (as shw ad avlig i h +-dici Wi pssi f h -fild phas: + Wi pssi f h -fild phas: + +
7 7 C 303 Fall 007 Faha aa Cll Uivsi M Calculais i h Cmpl Nai - II Th -fild phas is: + Th -fild phas is: Fid h cmpl Pig vc: + Fid h im-avag pw p ui aa:, C 303 Fall 007 Faha aa Cll Uivsi
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