Lecture III-2: Light propagation in nonmagnetic

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1 A. La Rosa Lecture Notes ALIED OTIC Lecture III2: Light propagatio i omagetic materials 2.1 urface ( ), volume ( ), ad curret ( j ) desities produced by arizatio charges The objective i this sectio is to derive expressios for the charge surface desity, charge volume desity, ad curret desity j, i terms of the physical properties of a material. ubsequetly, i the ext sectio, those expressios will be properly itegrated ito the Maxwell equatios., j, ad come from charges boud to atoms (i.e. charges that do ot travel freely i a medium, i cotrast to the free charges i metals). That is, at this stage, we distiguish the boud charges (iside material classified as dielectric or ocoductive medium) from the free mobile charges (ecoutered i metals). Our objective is to idetify, later o, the effects of boud charges to the propagatio of light i such a ocoductig medium. 2.1.A The olarizatio vector (material property) A electrically eutral atom acquires a die momet upo the applicatio of a exteral electric field. Neutral atom Electric field E q p + q p p q p E + p Electric die Fig. 1 A electric field iduces electric dies o a eutral atom. A collectio of small (molecular) dies p iside a uit volume geerates a arizatio vector (total die per uit volume). r p

2 The iduced separatio of charge (idicated by ) i a molecule gives rise to a electric die p defied as as, p q p I a give material with electric dies i its iside (whether the dies are iduced by the applicatio of a exteral field or ot), the vector OLARIZATION is defied as total die momet per uit volume (1) Let N be the umber of molecules per uit volume i the material. Assumig that i a give uit volume each atom acquires the same die momet, the, N p Notice, i geeral (r ) (2) that is, may vary from poit to poit iside the material. 2.1.B Chargearizatio surface desity If the arizatio of a material is ot uiform, the applicatio of a exteral electric field will, i geeral, create accumulatio of charges o uiformly distributed across the material s volume as well as o the its surface boudaries. A special case occurs whe the material is isotropic, i.e. is uiform throughout the material; i that case oly et accumulatio of arized charge occurs at the material s boudaries. Case: Isotropic materials A uiform arizatio throughout the volume of the slab show i the figure below, implies that there is ot accumulatio of charge at the iterior of the slab. To illustrate this poit, otice that i the dashed volume #1 IN Fig. 2 that, as the egative ad positive charges of idividual molecules separate out due to the exteral electric field, the amout of iduced charges gettig iside that volume is the same as the oe leavig out. Accumulatio of charge happes oly at the boudaries of the dielectric slab. I volume #2, for example, oly egative charges leave that volume i the dow directio, thus leavig ucompesated positive charges; the latter costitutes the surface charge desity.)

3 Q Q Q +Q A Q = Q /A +Q Fig. 2 Absece of et charge accumulatio iside the bulk of a isotropic material uder a uiform exteral electric field. till, otice there exists a arizatio that is costat ( 0 ) throughout the iterior of the material; it is just that such uiform arizatio does ot give rise to a et accumulatio of charge. However, there is et charge accumulatio at the boudaries. Let s fid out a relatioship betwee the surface desity of the arizatio charge ad q p the arizatio. To that effect, cosider, for example, the top layer displayed i Fig. 3: Q = Q /A A Fig. 3 Zoomi of the top sectio of Fig. 2 i order to visualize better the charge distributio ad relate it to the arizatio vector. Total umber of molecules i the top layer is N x AEach molecule, havig a die p=q e cotributes with a amout of charge q e to the surface charge. Hece, the total amout of charge i the layer is, Q = (charge umber of charges) = q e (N A ) ad the surface charge desity is the give by, = Q / A = q e N (i) O the other had, the total die momet of the layer is: (Total umber of molecules i the layer) p = (N A p ice the volume of the layer is A, the die per uit volume (i the layer) is the give by = (NA p) / (A) = N p = N q e (ii)

4 From (i) ad (ii), (2) The arizatio surface charge desity is umerically equal to the arizatio iside the material. Notice, although the aalysis above cosidered a exteded area A, the argumet is valid for the case of a more localized area o the surface. That is, if varied alog the sides of the slab (but still uiform alog the vertical directio), oe would obtai, ( r ) ( r ) that is, the local surface charge desity depeds o the local value of the arizatio vector. Chargearizatio surface desity alog a arbitrary surface I the previous case we foud the surface desity of the arizatiocharge alog a surface perpedicular to the arizatio vector. I may occasios we will ecouter boudaries which do ot alig perpedicular to ; still we have to fid the correspodet surface charge desities o that surface. First, otice that whe the surface is perpedicular to, the the total arizatiocharge crossig the surface is times the area, or. But if were tagetial to the surface, o et charge crosses the surface ad 0. p + p + p + Fig. 4 Upo the applicatio of a electric field, the et amout of charge crossig a hypothetical surface (dashed lies) depeds o the relative orietatio of the iduced dies a c b Q Fig. 5 Aother view of the same situatio depicted i Fig. 4, this time to quatify the surface desity of the arizatio charge established alog local surfaces of differet orietatios relative to the arizatio vector. I figure 5, Q crosses the surface area ab ad establishes a surface desity =Q /(ab), which, accordig to expressio (2), we kow it is equal to the magitude of the arizatio vector,

5 = Q /(ab) = a c Q b Q However, the same charge establishes a differet charge desity alog the surface of area cb (the ormal of that surface is ), ice a= c Cos() =Q /(bc) = [Q /(ba) ] Cos() = [ ] Cos() = [ ] Cos() = Geeralizig this result, Net surface charge desity alog a hypothetical (3) surface perpedicular to the uit vector. r urface Fig. 6 A local surface charge desity ( r) ( r) establishes alog the surface of the material. The local orietatio of the surface is defied by the local ormal uit vector. 2.1.C Chargearizatio volume desity I aisotropic materials, the ouiformity of ca give rise to a et accumulatio of charges iside the dielectric. For example, the diagram below shows two differet

6 dielectric slabs. Because the arizatios are differet, the volume of the dashed boudary comprises a et arizatio charge. Q Q Fig.7 Accumulatio of charges occurs whe the arizatio is ot uiform. The localizatio of charges at the iterior of a volume is described better by a volume charge desity, istead of a surface charge desity. till, the result obtaied i (3) for surface charge desities will help to obtai. The diagram below displays a more geeral case, where it is assumed that (r ). (r ) Q olume urface Fig.8 Diagram to evaluate the volumetric chargearizatio desity. Accordig to expressio (3), the amout of boudcharge crossig the boudary of the volume would be da. A equal excess of charge, of opposite sig, is left behid iside the volume. Thus, a et chage of charge iside the volume Q will be equal to, Q da The charge i the volume ca be cosidered as a charge distributed accordig to a charge desity. Hece, Q d

7 d da The latter surface itegral ca be coverted to a volume itegral usig the Gauss mathematical theorem da d, d d Hece, arizati ochargevolumedesity attheiteriorof amaterial (4) r (r ) Fig. I a material of ouiform arizatio (r ) desity of charge i the material., its divergece gives the et 2.1.D olarizatio curret desity j Expressio (4) is also valid whe the drivig electric field varies with time. As varies with time, it will geerate a correspodig arizatio curret desity. ice the arizatio charges are real charges (ad ot fictitious charges) the coservatio of charges should apply. Accordigly, j 0 (statemet of charge coservatio) t Usig (4), j j ( j ) 0. This expressio t t t is satisfied if, j olarizatio curret desity (5) t j

SECTION 2 Electrostatics

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