Gauss Law in dielectrics

Transcription

1 Gauss Law in dielectics We fist deive the diffeential fom of Gauss s law in the pesence of a dielectic. Recall, the diffeential fom of Gauss Law is This law is always tue. E In the pesence of dielectics, we know that we can divide the chages up into two types, fee bound whee bound P E fee P Fom Gauss s law, E P) D ( PYL1_216_L5 fee 1

2 ( H.W.) Gauss Law in dielectics The new field D E P is called the electic displacement It is a vey useful concept, as in tems of the electic displacement, Gauss s law looks the same as in the case of the vacuum, namely, E D fee The souce fo the electic displacement is fee chage only Specifying these fee, specify the D completely Using divegence theoem, the Integal fom of Gauss law: D da Q fee compaed to case of vacuum, This esult is extemely useful in poblems having symmety. Howeve, knowledge of the D is not enough to detemine the E as D E P PYL1_216_L5 E da Q / 2

3 Ex. 4.4 A long staight wie, caying unifom line chage, is suounded by ubbe insulation out to a adius a. Find D. Fo cylindical Gaussian suface shown, the Gauss Law fo dielectic D da Q yields, D(2 sl) L fee D sˆ This holds both within and outside insulation 2 s Since in outside egion, P=, E D D sˆ 2 s E P gives: Howeve, inside ubbe as P is not known, we can not find E. fo s a PYL1_216_L5 3

4 As in the case of E, we wish to wite D in tems of a potential It can be easily shown that no such potential exists fo D, as cul of D is nonzeo D E P E P P e.g., conside the ectangula ba with unifom P inside along its length (& P= outside). Fo the loop shown, P= P dl, by Stokes theoem, P P P D within the loop [It is obvious that P has a non zeo deivative along diections pependicula to the length at the top and bottom sufaces. Hence, The Cul of Electic Displacement P D ] 4 PYL1_216_L5

5 Electostatic Bounday Conditions 2.3.5: How the field E change acoss any suface? By Gauss s Law, fo the wafe-thin Gaussian Pillbox (thickness=) Q A enc E da S No contibutions to flux fom the sides in the limit, so we obtain fo the Nomal component of the field E above E below (Since aea vecto A points in opposite diection on the top and bottom faces of the pillbox) Conclusion: the nomal component of amount of / at any bounday. E is discontinuous by an PYL1_216_L5 5

6 To find the longitudinal component of E : we use E dl fo path In the limit E above l E below Tangential component of the field, H.W. E Combining, Futhe as, ; no contibution fom ends l above E below E V above above E V below below nˆ b E dl When path length dl shinks to zeo a PYL1_216_L5 n = unit vecto nomal to suface V V above Potential is always continuous below 6

7 ( H.W.) The bounday conditions fo D As D obeys Gauss s law, the bounday conditions fo the electic displacement D ae mathematically identical to the bounday conditions fo the electic field E. At the bounday between two media 1 and 2, if thee is a fee suface chage density fee pesent, then the bounday conditions ae: D da Qf enc D P D D D D 2 1 P Dielectic mateials can be divided up into 2 categoies, based on the dependence of P on E. fee 2// 1// 2// P1// PYL1_216_L5 7

8 Electets In most mateials, emoving the electic field causes the polaization to disappea. Such mateials ae called dielectics o paa-electics If the polaization is caused by the pola natue of molecules (i.e., molecules having pemanent dipole moment), those substances ae called as dipola/feoelectic mateials. These dioples get aligned even in the absence of any applied field to give the mateial a pemanent polaization This second class of mateials is also called as Electets. They have vey many uses especially as micophones, piezo tansduces, etc. (Slides 25 onwads to be taken in next lectue on Monday 8 Aug 216) PYL1_216_L5 8

9 Linea dielectics Of the class of mateials fo which the P is induced by the field E, almost all have a P which is popotional to the applied E (povided E is not too stong). P ee Such mateials ae called linea dielectics and the constant of popotionality is called the electic susceptibility of the mateial. This constant is dimensionless. e Fo such linea dielectics, D E P E ee D is also popotion al to Defining, PYL1_216_L5 (1 e) E E ( 1 e) D E 9

10 Linea dielectics ( 1 e) D E Hee is called the pemittivity of the mateial, then by analogy is the pemittivity of the vacuum. Thei atio is called the dielectic constant K o and is dimensionless. K 1 e Dielectic Constants of some substances (at 2C, 1 atm) PYL1_216_L5 1

11 Linea dielectics Note that in any linea dielectic, the volume bound chage density is always popotional to the fee chage density. f e e e e e e e b D D E P 1 1 ) ) (1 ( ) ( 11 f e e b 1 PYL1_216_L5 ] ) (1 [ E D e ] [ f D

12 Linea dielectics Fom the foegoing, we summaize: We can show that even though the dielectic is linea and the electic field E is always deivable fom a potential, the electic displacement and polaization ae not descibed by a potential fo a finite dielectic This is most easily seen by taking any closed path which passes though the mateial and the vacuum outside. (The integal of the electic field ound the closed path is zeo as it is a consevative field (= -ve gadient of a potential)) The polaization is popotional to the field but vanishes on those pats of the path lying in vacuum, so the integal ound the closed path fo P is nonzeo. PYL1_216_L5 12

13 Solving dielectic poblems Dielectic poblems may be divided into two types and solved accodingly. 1. Poblems fo which the Polaization is specified Fo these we fist find the bound chage densities, and P n b P b We then solve the equivalent poblem of these bound chages placed in vacuum. 2. Poblems fo linea dielectics having fee chage Hee, we use Gauss s law fo the electic displacement D in tems of the fee chages. We then use the known pemittivity and susceptibility to find E and P. PYL1_216_L5 13

14 (da in xy-plane and =/2) PYL1_216_L5 14

15 Ex A long cylinde caies a chage density (s) that is popotional to the distance fom the axis such that, ( s) ks, fo some constant k. Find the electic field inside this cylinde. Daw a Gaussian Cylinde of length l and adius s, and applying Gauss Law on it: The enclosed chage is : (Pime coodinates ae used fo Gaussian suface ; d uns fom 2 & dz: l Contibution to suface integal fom cuved suface: No contibution fom sides (as da E) PYL1_216_L5 15

16 Ex. 2.4 An infinite plane caies a unifom suface chage. Find its electic field. PYL1_216_L5 16

17 Pob Use Gauss s law to find the electic field inside a unifomly chaged sphee (chaged density ) Gaussian Suface S E da E / Qenc E. 4 3 ˆ 4 Q R 3 3 E 1 4 Q R 3 PYL1_216_L5 17

18 Pob A thick spheical shell (inne adius=a, oute adius=b) is made of dielectic mateial with a fozen-in polaization, k P( ) ˆ whee k is a constant and is the distance fom cente. Find the electic field in all 3 egions by two methods: (a) Locate all the bound chage, and use Gauss s law to calculate field it poduces. (b) Use D da Q f to find D, and then find E fom (a) enc bound 1 P 2 k P ˆ P ˆ b b k P ˆ a 2 k k 2 at at b, a PYL1_216_L5 18 D E P

19 Pob k enc P( ) ˆ Gauss' s Law E ˆ 2 Fo Fo a a, Q enc b, Q,so enc 4ka 4k ( a) E k a 4k Fo k ' b, Q enc 2 4 a 2 4 ' a 2 Q,so d' E 1 Qenc 1 4k k E ˆ ˆ ˆ k [Due to volume bound chages, b ] 2 PYL1_216_L5 19

20 Pob PYL1_216_L5 2 eveywhee. (b) D Q da D enc f P E P E D 1 k P ˆ ( ) k k E ˆ ˆ 1 b, a Fo b) a & (fo So, E

21 The Loentz Foce Thus chaged paticles ae subject to two additional foces due to the electic and the magnetic fields; in addition to the foces that all othe unchaged paticles can be subjected to. These can be combined to give a single foce equation as, F loentz ee e v Vey often, it is this combined foce which is efeed to as the Loentz foce. The magnetic pat of this foce is most easily peceived in the case of cuents in conductos. We emind ouselves that within a conducto, we have positively chaged nuclei which ae nomally stationay and an almost equal numbe of electons which move feely. B PYL1_216_L5 21

22 Effects of the magnetic field on cuents Using these ules along with the ule that the cuent poduces a magnetic field, shows 1. Paallel cuents attact each othe, 2. Anti paallel cuents epel each othe. PYL1_216_L5 22

23 The cuent in Wie 1, poduce magnetic field at Wie 2. Each moving chage in Wie 2 expeiences a Loentz Foce, as F Q vb Thus, the two cuent caying wies inteact though (i) geneation of magnetic field, & (ii) Loentz foce due to B on moving chage PYL1_216_L5 23

24 SELF STUDY (H.W.) Line Cuents We can define cuents of vaious kinds, the fist is a line cuent. A line chage density tavelling down a wie at speed v I q t l t ( vt) t v I v Applying the Loentz foce ule to ou moving chage density, magnetic foce on a segment of cuent caying wie is F mag v B dq v B dl I B dl As cuent is constant and lie along the wie, F mag I B dl I dl B PYL1_216_L5 24

25 SELF STUDY (H.W.) Suface cuents Fo a chage flow ove a suface; we define a suface cuent density K Cuent pe unit width pependicula to flow K di dl If is the mobile suface chage density, K I l 1 l q t 1 l ( vt l t We can use the Loentz foce law to compute the foce on this suface cuent K, K v F mag v B dq v B da F mag K B da PYL1_216_L5 ) v 25

26 Volume cuent density, J J cuent pe unit aea pependicula to flow If is mobile volume chage density (with velocity v) Foce on this volume cuent is, : di J J da J v Volume cuent density v t t v t l a a t q a a I J ) ( ) ( 1 1 d B v d B v B dq v F mag ) ( d B J F mag PYL1_216_L5 26 SELF STUDY (H.W.)

27 SELF STUDY (H.W.) Consevation of chage Cuent cossing a suface S, I Jda J da S S Total chage pe unit time leaving a volume V is, S J da V J d (using Div. theoem) Whateve flows out though the suface must come at the expense of that emaining inside: V dq dt d dt J d d d V V t PYL1_216_L5 27

28 SELF STUDY (H.W.) V dq dt d dt J d d d -ve sign means: the cuent flow deceases the chage left in V J Continuity equation mathematical statement of local chage consevation V t V t PYL1_216_L5 28

29 SELF STUDY (H.W.) Magneto-statics STEADY CURRENTS: We define these by analogy Steady cuents Continuous flow (So, no change and without chage piling up!) Thus, a point chage cannot constitute a steady cuent Steady cuent in a wie implies, I must be the same all along the wie, which equies that t J PYL1_216_L5 Slide No.29

30 SELF STUDY (H.W.) Biot Savat Law Fo a steady line cuent, the Biot Savat Law states B( ) I ˆ ˆ dl ' dl' I =41-7 N/A 2 Pemeability of space Units of B ae Tesla (=1N/(A-m)); 1 T = 1, Gauss Biot Savat law has the same impotance fo Magneto statics as Coulomb s law has fo Electostatics. Fo finite cuent loops, it has a 1 2 dependence PYL1_216_L5 Slide No.3