15 de va dap an truong dien tu 1227

Khoa cng ngh thng tin

Khoa cng ngh thng tin cng ho x hi ch ngha vit nam

B mn in t vin thng c lp - T do - Hnh phc

THI S 1

Mn: L thuyt trng in t

Thi gian : 90 pht

Hnh thc thi : VitCu 1 : (3 im)

Trnh by phng trnh 1 v 2 ca Maxwell v ngha vt l ca chng.

Cu 2 : (3 im)

Trinh bay v khai nim v mi trng khng ng hngCu 3 : (2 im)

Cho mt hinh cu tich in ban kinh la a. Gia s in tich phn ph u trn b mt cua no vi mt in tich mt s = Q/4a2. Tinh cng in trng tai nhng im ngoai va trong hinh cuCu 4 : (2 im)

t kh co . Hay tim gii han theo bc song t o xem t kh la dn in va in mi.

p n:

Cu 1 : (3 im)

1. Phng trnh Maxwell th nht.

Bng cch b sung thnh phn dng in dch vo v phi ca biu thc nh lut dng ton phn cng vi dng in dn phng trnh th nht nh sau:

(1)

Phng trnh (1.1.31) m t mi quan h gia cc vect ca trng in t () trong mt vng kn bt k cc dng in dn chy qua n, m t n trong khng gian:

(2)V mt S l tu nn ta nhn c phng trnh Maxwell th nht dng vi phn nh sau:

(3)(1 im)

Nu mi trng c dn in ring =0 th =>

EMBED Equation.3 nn phng trnh c dng:

(4)

Phng trnh ch ra : Dng in dich hay in trng bin thin cng to ra t trng xoy tng ng dng in dn2. Phng trnh Maxwell th hai:

Maxwell cho rng biu thc ca nh lut cm ng in t p dng khng ch cho mt vng dy dn kn m m cn ng cho bt k mt vng kn no( khng nht thit dn in) trong khng gian. Trong trng hp tng qut vng kn ny c th mt phn nm trong trn khng, phn khc nm trong in mi hay trong kim loi. Ta nhn c phng trnh sau:

(5)(2 im)

Nu p dng nh l Grin Stc cho v tri vi S l tu nhn c phng trnh sau:

(6)

Vy t trng bin thin to ra in trng xoy3. ngha vt l ca phng trnh th nht v th hai ca Maxwell:

Bt k s bin thin no ca in trng u gy nn t trng xoy(ng sc khp kn) v ngc li. in trng v t trng bin thin khng th tn ti c lp vi nhau, chng lun lin h mt thit vi nhau v lin tc chuyn t dng ny sang dng khc to nn sng in t truyn lan vi vn tc nh sng.

(3 im)

Cu 2 : (3 im)

Mi trng ng hng la mi trng ma tinh cht cua no moi im la nh nhau. Trong cac mi trng nay cac vec t va la song song vi nhau tng i:

Nu chiu cac phng trinh vec t trn xung cac truc toa ta c cac phng trinh v hng:

(1 im)

i vi cac mi trng bt ng hng mi quan h gia cac vec t trn c xac inh qua cac phng trinh:

Cac hng s co th c vit di dang nh sau:

EMBED Equation.3

(2 im)

goi la tenx t thm

goi la tenx in thm

Trong thc t khng tn tai cac mi trng ma ca va u mang tinh tenx.

Mi trng bt ng hng co tenx t thm in hinh la pherit c t hoa bi t trng khng i; con mi trng co tenx in thm in hinh la mi trng ion hoa( mi trng plasma).(3 im)

Cu 3 : (2 im)

Ap dung phng trinh 3 cua Maxwell dang tich phn:

Ly S la mt cu ban kinh a. Do tinh cht i xng nn D tai moi im trn hinh cu la nh nhau

(

(1 im)a) Xet trng hp th nht: im M ngoai hinh cu(r>a)

Ta co: q = Q

( D.4 r2 = Q = S.4 a2

( D = S.(a2/r2)b) Trng hp th hai: im M trong hinh cu(r (2/3).102 m thi t co tinh dn in.

Vi < (2/3).102 m thi t co tinh in mi.(2 im) Khoa cng ngh thng tin cng ho x hi ch ngha vit nam


THI S 2


Thi gian : 90 pht


Trnh by phng trnh 3 v 4 ca Maxwell v ngha vt l ca chng.Cu 2 : (3 im)

Hy trinh by v s phn cc cua song in tCu 3 : (2 im)

Mt in tich dong Q phn b u theo th tich qua cu co ban kinh la a, vi mi in thm t trong khng khi. Hay tim cng in trng E trong va ngoai qua cu oCu 4 : (2 im)

Song phng truyn trong mi trng in mi ng nht ng hng rng v han co tham s = 40; ; bin cng in trng cua song Em = 10-3 (V/m) va f = 106Hz. Lp biu thc gia tri tc thi cng t trng cua song va mt dong cng sut trung binh.p n:

Cu 1 : (3 im)

Maxwell coi nh lut Gauss v nguyn l lin tc ca t thng p dng cho c trng hp in trng v t trng l tnh, khng i cng nh vi trng hp tng qut ca in t trng bin thin theo thi gian. Ta c:

(1)

(2)V th tch V l tu nn nhn c cc phng trnh Maxwell th 3 v th 4 nh sau:

(3)

(4)(1 im)

tin cho vic theo di, ta vit thnh hai dng sau:

Dng tch phn:

(5)Dng vi phn:

(6)

(2 im) ngha vt l ca phng trnh 3 v 4 ca Maxwell:

0: ta thy ng sc ca in trng l nhng ng cong khng khp kn m c im u ti in tch +q, im cui ti q.

= 0: in trng sinh ra ch do s bin thin ca t trng. ng sc ca n hoc khp kn hoc tin ra v cc. ( ng sc ca t trng va khp kn va tin xa v cc.(3 im)

Cu 2 : (3 im)

Ta c cc loi sng phn cc c bn c s dng :

Pht hnh : Sng phn cc ngang

Pht thanh: Sng phn cc ng hoc ngang

Sng ngn : Sng phn cc ngang

Sng FM: Sng phn ng hoc ngang

S ph thuc hng ca vect E vo thi gian v khng gian gi l s phn cc phn cc.

Sng in t khi truyn lan vect cng in trng v t trng c th thay i c v ch s v hng. V vy khi sng truyn lan nu quan st im cui ca vect E th ta thy n v ln mt qu o no

Xt ti mt im c nh trong khng gian cng vi thi gian im cui ca vect E thc hin mt chuyn ng tnh tin dc theo mt ng thng th ta ni sng in t phn cc thng(phn cc tuyn tnh). Tng t nu im cui ca vect E v nn mt hnh elip ta c phn cc elip, cn v nn ng trn ta c phn cc trn. Nu nhn theo hng truyn sng vect E quay theo chiu kim ng h ta c phn cc trn quay phi, ngc li c phn cc trn quay tri.

Gi s c hai sng phng phn cc tuyn tinh vung gc vi nhau ta c:

(1 im)

y Emx, v Emy l bin cc sng thnh phn, l gc lch pha ban u ca hai sng phng

Suy ra

Phng trinh ny biu din mt hnh elip

(2 im)

Elip c trc ln lam mt gc vi trc ox

; vi Emx > Emy Khi Emx =Emy; = /2 thi phn cc luc nay l phn cc trn.

Khi = n (n = 1, 2,...) thi la phn cc thng.

Nh vy khi t thay i vec t E se quay cung v phia ngc chiu kim ng h, vi chu ky: , u nut cua no vach thanh ng elip. Chiu quay cua la chiu quay v phia thanh phn trng chm pha.(3 im)

Cu 3 : (2 im)



(

a) Xet trng hp th nht: im M ngoai hinh cu(r>a)

Ta co: q = Q

( D.4 r2 = Q (

Mi trng la khng khi nn = 0

Ma D = .E = ( E =

(1 im)

b) Trng hp th hai: im M trong hinh cu(rb.p n:

Cu 1 : (3 im)

xt iu kin b i vi E1n v Hn ta xt hnh tr c y S1v S2 nh v cao h .

- i vi En

Ta xt phng trnh Maxwell dng vi phn sau y:

c th vit nh sau vi v tri ca phng trnh:

Khi h 0 ta c :(1)

S12* l giao tuyn gia mt phn cch v hnh tr

(1 im)

T v phi ca biu thc (1) ta c :

t ta c t v tri v v phi ta c biu thc sau y:

D1n - D2n =s1E1n - 2E2n =ss l mt in tch mt

Nh vy thnh phn php tuyn ca vect in cm D khi chuyn quab mt phn cch 2 mi trng thay i mt lng bng mt in tch mt s.

(2 im)

- iu kin b i vi HnXt phng trnh Maxwell

1H1n - 2H2n=0

Nh vy thnh phn php tuyn ca vect t cm Bn lin tc khi i qua b mt phn cch hai mi trng.(3 im)

Cu 2 : (3 im)

Ta khao st sng phng

Dang phu thuc vao thi gian(dang tc thi) nh sau:

Trong mi trng in mi l tng = 0 thi k l s thc:

Phng trinh mt ng pha cua song:

(3.5)

Suy ra: z = 1/k(t cosnt)(1 im)

mi thi im t mt ng pha cua song la mt phng z = const.Vn tc pha la vn tc di chuyn cua mt ng pha, ky hiu la

Trong khng gian t do:

Va

(2 im)

(3 im)

Cu 3 : (2 im)

Mt dong in dich chay qua hai ban tu la:

EMBED Equation.3

Ma . T y ta suy ra c:

Ma

Ap dung inh lut dong in toan phn cua Ampe

(1 im)

Ly L la chu vi cua ng tron ban kinh r = 1cm. Do tinh cht i xng nn tai moi im trn ng cong L la nh nhau.

(

Con tng ai s cac dong in xuyn qua ng cong L la:

(S la din tich cua ng tron ban kinh r).

( (

(2 im)Cu 4 : (2 im)


Xet L la ng tron co tm nm trn truc cua cap ng truc ban kinh r. L vung goc vi truc cua cap.

Do tinh cht i xng cua cap nn H tai moi im trn L la nh nhau:

(1 im)

Trng hp r>b thi

Trng hp a r b thi

(2 im) Khoa cng ngh thng tin cng ho x hi ch ngha vit nam


THI S 5


Thi gian : 90 pht


Hy chng minh v pht biu inh lut bao toan nng lng i vi trng in t. Vec t Poynting.Cu 2 : (3 im)

Hy trnh by v hiu ng b mt vt dn.Cu 3 : (2 im)

Co dong in khng i I chay theo dy dn hinh tru trong ban kinh a. Hay tim cng trng tai im bt ky cach truc dy dn 1 khoang r cho hai trng hp r>a va r 0 la mi trng dn in, cu th la kim loai ng co dn in ring , theo phng truc z truyn mt song thng ng nht vi tn s f = 105Hz. Hay xac inh vn tc pha, bc song, tr khang song, h s suy giam va thm su cua trng() trong kim loai ng cua song. Bin cng trng se giam i bao nhiu ln so vi b mt kim loai khi song i su vao c mt khoang d = 1mm.p n:

Cu 1 : (3 im)

Trng in t l mt dng c bit ca vt cht, nn n cng tun theo nh lut bo ton nng lng.

T vt l nng lng in t trong mt n v th tch :

H v E thay i theo thi gian v khng gian, suy ra W cng thay i

p dng phng trnh 1 v 2 c Maxwell :

;

(1 im)Vy ta c :

Ta c

Ly tch phn theo th tch V 2 v ri p dng nh lut L tstrgratski Gauss.

(1)

(2)

(2 im)

Xt ngha:

l cng sut tiu hao di dng nhit trong th tch V k hiu Pt

l Cng sut do ngun ngoi sinh ra trong th tch V k hiu Pe

l tc bin thin nng lng in t trong V

l cng sut chy ngoi V qua din tch S.

Tm li: Cng sut do ngun ngoi sinh ra trong th tch V bng tng cng sut tiu hao di tc dng nhit trong V, cng sut chy ra ngoi V qua din tch S v cng sut lm thay i in t trng trong V. chnh l nh lut bo ton nng lng i vi trng in t.

(3 im)

Cu 2 : (3 im)

Trong vt dn in tt ( >>1) ta c :

(1)

Khi rt ln th cng rt ln dn n suy gim cng nhiu, ta thy bin cng trng suy gim rt nhanh khi truyn vo trong vt dn. Ngha l sng in t ch tn ti mt lp rt mng st b mt ca vt dn in tt. Khi cho dng in cao tn chy trong vt dn in tt ngi ta cng thy dng in ny ch tn ti trn mt lp theo nh lut m.

Jd = E

i vi dng kho st:

E=Eme-ze-jzJd = Eme-ze-jz =J0e-ze-jz (2)

J0 l mt dng chn b mt vt cht J0 = Em (1 im)

Mt dng in s gim dn khi i vo su trong vt dn theo quy lut ging nh bin cng in trng

Hin tng sng in t hay sng in cao tn khi truyn trong vt dn in tt ch tp chung mt lp rt mng trn b mt ca n gi l hiu ng b mt, hay hiu ng Skin

c trng cho hiu ng b mt ngi ta a vo khai nim thm su ca trng hay su thm nhp ca trng , l khong cch m ng vi n bin cng trng suy gim i e ln: e 2,718

Ta c :

e = e suy ra = 1/

(2 im)

Hiu ng b mt c p dng trong thc t (m vng, bc), khi lm gim tiu hao khi truyn sng in t ngi ta ch m mt lp mng vng hoc bc ln b mt kim loi.

Khi tnh ton cc bi ton ngi ta thy khi nim tr khng mt ca kim loi: ZS = RS + XSRS l tr c trng cho cng sut tiu hao

XS l cm khng ca mt ring ZS

Vn tc pha:

(3 im)Cu 3 : (2 im)


Ly L la chu vi cua ng tron ban kinh r. Do tinh cht i xng nn tai moi im trn ng cong L la nh nhau.

(1 im)

Trng hp r>a: ( (

Trng hp r0 la mi trng dn in, cu th la kim loai ng co dn in ring , theo phng truc z truyn mt song thng ng nht vi tn s f = 105Hz. Hay xac inh vn tc pha, bc song, tr khang song, h s suy giam va thm su cua trng() trong kim loai ng cua song. Bin cng trng se giam i bao nhiu ln so vi b mt kim loai khi song i su vao c mt khoang d = 1mm.p n:

Cu 1 : (3 im)

T phng trinh 1 cua Maxwell dang bin phc:

Va ta co (

(

Trong o : la hng s in mi phc tuyt i cua mi trng

la hng s in mi phc tng i cua mi trng.

Co th chng minh rng ty s gia phn ao va phn thc cua la : la ty s in dn va in dich, no c trng cho tiu hao trong mi trng in mi.

t tg( =; ( la goc tiu hao in mi

Nu la cht in mi: tg( < 0,001

Dn in: tg( > 100

Ban dn 0,01 < tg( a)

Ta co: q = Q

( D.4 r2 = Q (

Mi trng la khng khi nn = 0

Ma D = .E = ( E =

(1 im)

d) Trng hp th hai: im M trong hinh cu(rb.p n:

Cu 1 : (3 im)

Noi chung khi giai cac bai toan v in t trng la mt cng vic phc tap. Vi vy trong mt s trng hp giai cac bai toan mt cach n gian va nhanh chong hn ngi ta ap dung mt s nguyn ly c ban cua trng in t.

1. Nguyn ly xp chng

i vi mi trng tuyn tinh phng trinh vi phn vit cho cac vec t in t trng cung la tuyn tinh. T giao trinh toan hoc ta a bit tng cac nghim ring cua bt ky phng trinh vi phn tuyn tinh nao cung la nghim ring cua phng trinh nay. Do o ta co th thy rng: trng do mt s ngun nao o se la tng vec t cac trng cua tng ngun. y chinh la ni dung cua nguyn ly xp chng. Nhng c mt iu l nguyn l ny khng c ap dung i vi cng sut hoc nng lng.

Thi du: Gia s co n ngun dong

Goi la trng do gy ra khi

Tng t: la trng do gy ra khi

la trng do gy ra khi

T y ta suy ra trng do gy ra la:

(1 im)

2. Nguyn ly i ln

a, Nguyn ly i ln c suy ra t tinh cht i xng cua cac phng trinh Maxwell. Xet phng trinh Maxwell o khng co ngun ngoai.

(a)

(b)

Nu iu kin phep i ln: thi h (a) tr thanh h (b) va ngc lai. iu nay co nghia la: Nu nh co hai bai toan in ng ma tt ca cac iu kin i vi (hoc) cua bai toan nay se tr thanh cac iu kin i vi (hoc cua bai toan kia. Khi thc hin phep i ln thi nu bit nghim cua bai toan th nht ta co th suy ra c nghim cua bai toan th 2.(2 im)

Thi du: Nu nh ta tim c cac biu thc i vi cac h s tich phn c xac inh t iu kin thanh phn tip tuyn cua vec t = 0. Trn b mt bin gii cua 2 mi trng thi khi thc hin phep i ln: ta se c biu thc i vi ma thanh phn tip tuyn cua cung bng khng trn b mt phn cach.

b, Nguyn ly i ln cung ap dung c i vi khng gian o co ngun in t trng. Luc nay h phng trinh Maxwell co dang:

EMBED Equation.3

EMBED Equation.3 va i xng ta phai a vao cac ai lng quy c

: mt dong t

: mt khi t tich.

Cac ai lng trong t nhin khng co, ta a vao chi nhm muc ich lam cho h phng trinh Maxwell i xng vi ,. Do o co th ap dung nguyn ly i ln: ;

H phng trinh Maxwell la:

(3 im)Cu 2 : (3 im)

Mun tim , ta phai giai cac phng trinh song almbe. So sanh vi cac phng trinh almbe ta thy chung u co dang ging nhau. Do vy, chi cn tim nghim cua 1 phng trinh co dang sau:

(1)

Trong o: ai din cho

g ai din cho phng trinh v phai cua cac phng trinh song.

Co th tim c nghim cua phng trinh song almbe(1) co dang nh sau:

(3)(1 im)

V: la th tich cha ngun

r: la khoang cach t im tinh trng M ti cac im ngun(vi phn th tich dV)

la vn tc truyn song trong mi trngT phng trinh (3) ta thy rng trng tai im khao sat thi im t c xac inh khng phai bi gia tri ngun tai thi im t ma c xac inh bi gia tri cua ngun thi im sm hn t mt khoang thi gian la , chinh la khoang thi gian truyn t ngun n im quan sat vi vn tc v hu han. Nh vy trng im quan sat chm pha so vi ngun mt khoang thi gian . Do o nghim (3) ung la th chm cua trng in t.(2 im)

Nu trng la iu hoa thi:

Vi : la s song trong mi trng.

Do o th chm cua trng in t co dang:

(4)

Trng tai im khao sat chm pha so vi trng tai im ngun mt lng la kr.(3 im)

Cu 3 : (2 im)



(

(1 im)

c) Xet trng hp th nht: im M ngoai hinh cu(r>a)

Ta co: q = Q

( D.4 r2 = Q = S.4 a2

( D = S.(a2/r2)d) Trng hp th hai: im M trong hinh cu(rb thi

Trng hp a r b thi



THI S 9


Thi gian : 90 pht


Hy xc nh iu kin b i vi thanh phn phap tuyn cua vec t cng in trng va t trng trn b mt phn cach gia hai mi trng

Cu 2 : (3 im)

Hay trinh bay v song in t phng trong mi trng dn inCu 3 : (2 im)

Co dong in khng i I chay theo dy dn hinh tru trong ban kinh a. Hay tim cng trng tai im bt ky cach truc dy dn 1 khoang r cho hai trng hp r>a va r a: ( (

Trng hp r (2/3).102 m thi t co tinh dn in.



THI S 10


Thi gian : 90 pht


Hy trnh by v nh lut bo ton in tch.Cu 2 : (3 im)

Hy trinh bay v trng tinh in.Cu 3 : (2 im)

Co 1 tu phng khng khi tao thanh t hai ban tron ban kinh r1 =2cm, va khoang cach gia chung d = 0,5 cm. Tu in nay la mt phn cua mach dong in. Trn hai ban tu co mt in ap bin i um =sint; Um = 500V; = (2/7).106 rad/s. Hay tim dong in dich toan phn chay qua hai ban tu vi ban kinh r = 1cm.Cu 4 : (2 im)

Cho tham s in cua t kh:

Hng s in mi tng i

dn in ring = 10-3 1/m.

Chng to rng i vi song cc dai( = 104 105 m) thi mt t co tinh dn in tt hn, con i vi song cc ngn( = 10-3 10 m)thi mt t co tinh dn in kem.p n:

Cu 1 : (3 im)

in tch c th phn b gin on hay lin tc. N khng t nhin sinh ra v cng khng t nhin mt i. in tch tun theo nh lut bo ton. nh lut bo ton in tch (do Faray tm ra qua thc nghim nm 1843) c pht biu nh sau:

Lng in tch i ra khi mt mt kn S bao quanh th tch V trong mt khong thi gian no bng lng in tch trong th tch ny b gim i trong khong thi gian y.

i xc nh dng ton hc ca nh lut:

Gi s trong th tch V ty ca mi trng vt cht c bao bi mt kn S ti thi im t cha mi lng in tch l Q vi mt khi :

(1)

(1 im)

Sau mt khong thi gian dt lng in tch trong th tch V gim i 1 lng l dQ. Theo nh lut bo ton in tch lng in tch gim i trong V bng lng in tch i ra khi V qua mt S trong khong thi gian dt to ra dng in dn I.

(2)

V th tch V ng yn nn chng ta c hn thc sau y:

(3)(2 im)

nh lut trn l dng tch phn, nu p dng biu thc nh l tstrogratski Gauss cho v tri ta c:

(4)

V th tch V l ty nn suy ra:

(5)

Biu thc trn y l dng vi phn ca nh lut bo ton in tch hay gi l phng trnh lin tc.(3 im)

Cu 2 : (3 im)

Trng tinh in la trng c tao ra bi cac in tich ng yn va khng i theo thi gian.

Hay J = 0; ( (1)

Trng tinh in la 1 trng th ngoai co nn co th biu din qua 1 bin mi

(2) ( vi th ).

Th cua trng in tinh theo (2) co th xac inh bng biu thc:

(3)(1 im)

Cng A cua trng tinh in theo 1.18 c thc hin khi di chuyn 1 in tich im(+) q = 1C t im M1 n im M2 la:

(4)

Trong o la th cua trng tinh in tai M1, M2.

; ;

( Tich phn theo ng cong L khep kin)

Ta co:

(5)

Ta co:

(6)

Phng trinh (6) goi la phng trinh Poison. Phng trinh nay lin h th va in tich tai mt im bt ky cua trng.

- Tai nhng im ma o mt in trng bng khng thi ta co , phng trinh (6) tr thanh phng trinh Laplas.

Giai phng trinh Poison co nghim: (7)

r la khoang cach t im tinh trng n vi phn th tich dV.(2 im)

- i vi in tich im ta co:

(8)

Ta co:

Ta c: (9)

Trng hp in tich im ta co:

(10)

Nu a vao trng cua in tich q mt in tich th q1, ta co:

(11)

Biu thc (11) chinh la inh lut Culng. Vi vy inh lut Culng la h qua cua phng trinh Maxwell i vi trng tinh in.(3 im)

Cu 3 : (2 im)

Mt dong in dich chay qua hai ban tu la:

EMBED Equation.3

Ma . T y ta suy ra c:

Ma


(1 im)

Ly L la chu vi cua ng tron ban kinh r = 1cm. Do tinh cht i xng nn tai moi im trn ng cong L la nh nhau.

(

Con tng ai s cac dong in xuyn qua ng cong L la:

(S la din tich cua ng tron ban kinh r).

( (

(2 im)

Cu 4 : (2 im)

Ta co hng s in mi phc tng i cua t c biu din nh sau:

c trng cho tinh cht in mi

c trng cho tinh cht dn in.

Ta thycang ln thi t cang co tinh dn in hn.(1 im)

- Vi = 104m thi . T y ta suy ra c t co tinh dn in tt.

Vy vi song cc dai = 104 105 m thi t co tinh dn in tt.

Vi = 10 m ( ( t co tinh dn in kem

Vy vi = 10-3 10 m t co tinh dn in kem.(2 im)



THI S 11


Thi gian : 90 pht


Hy chng minh v pht biu inh lut bao toan nng lng i vi trng in t nh l Ump-PoyntingCu 2 : (3 im)

Trinh bay v t trng cua dong in khng i.Cu 3 : (2 im)

Tinh trng va th tao ra bi mt truc tich in co mt in tich dai la L, tai im cach truc 1 khoang r.Cu 4 : (2 im)

Song phng truyn trong mi trng in mi ng nht ng hng rng v han co tham s = 40; ; bin cng in trng cua song Em = 10-3 (V/m) va f = 106Hz. Lp biu thc gia tri tc thi cng t trng cua song va mt dong cng sut trung binh.p n:

Cu 1 : (3 im)

Trng in t l mt dng c bit ca vt cht, nn n cng tun theo nh lut bo ton nng lng.

T vt l nng lng in t trong mt n v th tch :

H v E thay i theo thi gian v khng gian, suy ra W cng thay i

p dng phng trnh 1 v 2 c Maxwell :

;

(1 im)Vy ta c :

Ta c

Ly tch phn theo th tch V 2 v ri p dng nh lut L tstrgratski Gauss.

(1)

(2)

(2 im)

Xt ngha:

l cng sut tiu hao di dng nhit trong th tch V k hiu Pt

l Cng sut do ngun ngoi sinh ra trong th tch V k hiu Pe

l tc bin thin nng lng in t trong V

l cng sut chy ngoi V qua din tch S.

Tm li: Cng sut do ngun ngoi sinh ra trong th tch V bng tng cng sut tiu hao di tc dng nhit trong V, cng sut chy ra ngoi V qua din tch S v cng sut lm thay i in t trng trong V. chnh l nh lut bo ton nng lng i vi trng in t.

(3 im)

Cu 2 : (3 im)

Trang thai ring quan trong th 2 la t trng do dong in khng i tao ra. y la trang thai dng cua trng in t: J ( 0,

(1)

(2)

Tng t nh trng tinh in, i vi trng dng ta cung co nhng nhn xet sau:

in trng va t trng dng khng c lp vi nhau na ma lin tuc vi nhau thng qua .

Trng dng co nn in trng dng la mt trng th co th t

T trng dng co tinh cht xoay vi nn khng th dung th va co th biu din qua bin mi (3)

Vi trong o la vec t th, ta co:

xac inhn tri thi ta thm mt iu kin tuy y. n gian ta ly ( (4) . Biu thc (4) goi la phng trinh Poison cho . Phng trinh vec t nay tng ng vi (5) phng trinh sau:

EMBED Equation.3 (5)(1 im)

Nghim cua phng trinh (5) la (6)

Dang vec t cua no la: (7)

Biu thc xac inh va la:

(8)

(9)

Trng hp dong in chay trong dy dn co thit din ngang nho co th bo qua so vi chiu dai dy dn va khoang cach t dy n im quan sat. Vec t th luc nay co dang:

(10)

Va (11)hay

(12)

(2 im)

Ap dung hng ng thc vec t:

ly ta co:

Bi vi trng c tinh im quan sat M vi toa x, y, z ma khng phu thuc vao im M nn ( , va thay vao biu thc (12) ta co:

(10)

Biu thc (10) la biu thc dang tich phn cua inh lut Bixava. Con dang vi phn nh sau:

(11)

Kt lun: inh lut Bixava la h qua cua phng trinh Maxwell i vi trng dng.(3 im)

Cu 3 : (2 im)


Ly S la hinh tru thng dai v han ( D tai moi im trn din tich xung quanh(Sxq) cua hinh tru nh nhau.

Ta co: Sxq = 2r.l (l ) va q = L.l ( D.Sxq = q va D.2rl = L.l (

Ma D = E (

(1 im)

Th tai im cach truc mt khoang r la:

Tng t tai im cach truc mt khoang x:

in trng:

Th:

(2 im)

Cu 4 : (2 im)

Ta co ma

(

Biu thc gia tri tc thi cua cng t trng:

(1)(1 im)

Vi va (

Biu thc tc thi cua mt dong cng sut trung binh la:

, vi Hm co biu thc nh (1).(2 im)



THI S 12


Thi gian : 90 pht


Hy xac inh iu kin b i vi thanh phn tip tuyn cua vec t cng in trng va t trng trn b mt phn cach gia hai mi trng.Cu 2 : (3 im)

Hay trinh bay nguyn ly Huy ghen Kic hp va nguyn ly dong tng ng v hin tng nhiu xa.Cu 3 : (2 im)

Ap dung inh lut Bixava tinh t trng tao bi mt dy dn manh co dong in I tai im cach truc mt khoang r.Cu 4 : (2 im)

t kh co . Hay tim gii han theo bc song t o xem t kh la dn in va in mi.p n:

Cu 1 : (3 im)

p dng phng trnh Maxwell dng tch phn cho mt hnh tr ta c kt qu sau:

xt iu kin b vi thnh phn tip tuyn ca E v H ta xt 1 khung ch nht nh ABCD vung gc vi mt phng S12. v ABCD nh nn giao tuyn ab ca S12 vi ABCD c th coi l ng thng v trng in t c th coi l khng i khi i t im A n im B v t im C n im D(0,5 im)

Cc vect n v : L vect tip tuyn vi mt phn cch S12

l php tuyn vi S12

Vung gc vi mt ABCD.

iu kin b i vi Ep dng mh lut hai dng tch phn:

Ly l lm chu vi ca ABCD v tri c th vit thnh :

Trong : E1 E2 l cc vect trong mi trng (MT) 1 v mi trng 2

E12 v E21 l cc vect va trong MT1 v MT2

Khi h 0 th AB ab v CD ba

BC v AD 0

(1 im)

Nh vy:

v phi :

Do :

Nh vy: Thnh phn tip tuyn ca vect cng in trng khng thay i khi i qua b mt phn cch gia hai mi trng. Thnh phn tip tuyn ca vect in cm thay i hng theo khi chuyn qua b mt phn cch :

D1 = (1/2)D2

(1,5 im)

iu kin b i vi thnh phn tip tuyn ca vect cng t trng :

Ta c :

(2 im)

Xt phng trnh sau:

Khi h 0 lm tngt nh phn trc ta c:

()

Khi v phi phng trnh trn l :

Do D hu hn nn: Cn ()JS l vect mt dng in mt t () v () ta c :

Nh vy vect cng t trng c thnh phn tip tuyn thay i mt lng mt dng in mt khi i qua b mt phn cch gia hai mi trng

(3 im)

Cu 2 : (3 im)

1. Nguyn ly HUYGHEN KICHP

Nguyn l cho ta c nghim ca phng trnh sng thun nht i vi mi hm v hng no hoc mt thnh phn vung gc bt k ca vect cng in trng. N c p dng gi bi ton nhiu x gn ng. Tm nghim ca phng trnh sau:

Ti im P bt k trong vng V c gii hn bi mt kn S. p dng nh l Grin cho hm v

y hm l tu , chn sao cho

y r l khong cch t im tnh trng P n im bt k trong vng V. tai y khi r 0 khi ta c:

Cc o hm theo php tuyn trn S v S0 hng ra ngoi vng V. Do trn mt S0 ta c:

(1 im)

2. Nguyn l dng tng ng

Gi s cc ngun q1, q2, q3, q4,qn t trong vng V gii hn bi mt kn S chng ta cn tim trng im P bt k trong khng gian V ngoi mt S.

Theo nguyn l Huyghen-Kichp ta c th tnh trng ti P trong V ca cc ngun cho qua cc ngun bc x nguyn t phn b trn mt S to ra. Cc ngun nguyn t phn b trn mt S c gi l cc ngun dng tng ng .

Cc thnh phn tip tuyn ca in trng v t trng st bn trong mt S bng khng.

(1)

Theo nguyn l nghim duy nht, mn ngun cho v trng ca ngun dng tng ng to ra im P trong vng V trung vi nhau phi c iu kin l: cc thnh phn tip tuyn ca cng in trng v t trng ca hai trng ny trn mt S pha bn ngoi phi bng nhau v chng khc khng:

Eng = Eng 0

Hng = Hng 0 (2)(2 im)

T biu thc (1)v (2)chng ta thy cc thnh phn tip tuyn ca cng trng ca ngun dng tng ng bin i nhy vt t 0 sang khc khng khi qua mt gii hn S. Tn ti dng in mt IS v t mt ISM:

(3)

y n0 l vect n v php tuyn ngoi ca mt gii hn S. p dng phng php th in ng chng ta tm c biu thc cho cc th chm vect in v t do cc ngun dng tng ng IS v ISM trn S to ra tai im P trong V lc ta c:

Trong cng thc (4) cc tham s nh , v s sng k phi tnh trong mi trng vng khng gian ngoi V

Cc biu thc (3) v (4) l biu thc nguyn l dng tng ng ca trng in t. Nguyn l ny c ng dng gii bi ton nhiu x sng in t rt li.

(3 im)

Cu 3 : (2 im)

Ap dung inh lut Bixava(inh lut Ampe)

Ma ta co. T y ra suy ra:

T hinh ve ta thy

(1 im)

(

(1)

( (

(2)

Khi l bin i t - + thi bin i t 0 n .

Thay (2) vao (1) ta co:

(2 im)

Cu 4 : (2 im)

Ta co: (hng s phc tuyt i)

EMBED Equation.3 ()

Ty s gia phn ao va phn thc cua p chinh la ty s gia dong in dn va dong in dich (dong in dich chay trong in mi, con dong in dn di chuyn trong kim loai).

Nu thi t co tinh cht cua cht in mi

Nu thi t co tinh cht dn in.

(1 im)

Gii han theo bc song t o xem t kh la dn in hay in mila:

Ma

( = (2/3).102(m)

cang ln thi t cang co tinh dn in hnT y ta co th kt lun la:

Vi > (2/3).102 m thi t co tinh dn in.



THI S 13


Thi gian : 90 pht


Hay trinh bay v h phng trinh Maxwell dang bin phc.Cu 2 : (3 im)

Trinh bay v trng tinh in.Cu 3 : (2 im)

Co dong in khng i I chay theo dy dn hinh tru trong ban kinh a. Hay tim cng trng tai im bt ky cach truc dy dn 1 khoang r cho hai trng hp r>a va r b.p n:

Cu 1 : (3 im)

Trong thc t thng gp cac dao ng iu hoa. Mt khac 1 dao ng iu hoa khng phai la iu hoa thi bng phep bin i Fourier bao gi cung co th phn tich thanh tng cua cac dao ng iu hoa. Vi vy vic nghin cu trng iu hoa nh mt tp hp ring cua trng in t la rt cn thit. Cho dao ng iu hoa:

Ta a bit phng phap bin phc lam cho phng trinh phn tich cac dao ng iu hoa tr nn n gian i rt nhiu. S dung phng phap bin phc ta a vao phng trinh trn nh sau:

(1 im)

Trong o:: Bin phc cua vec t

Ap dung inh cng thc le:

Ta co:

Nh vy i vi trng iu hoa tng ng vi cac vec t ,,,, ta co dang bin phc tng ng ,,,, .1. H phng trinh Maxwell dang bin phc

Phng trinh 1:

S dung phng phap bin phc ta a vao ky hiu:

(2 im)

Thay vao phng trinh 1 cua Maxwell ta co:

(

(

Tng t vi cac phng khac ta co c h phng trinh Maxwell dang bin phc:

(3 im)

Cu 2 : (3 im)

Trng tinh in la trng c tao ra bi cac in tich ng yn va khng i theo thi gian.

Hay J = 0; ( (1)

Trng tinh in la 1 trng th ngoai co nn co th biu din qua 1 bin mi

(2) ( vi th ).

Th cua trng in tinh theo (2) co th xac inh bng biu thc:

(3)(1 im)

Cng A cua trng tinh in theo 1.18 c thc hin khi di chuyn 1 in tich im(+) q = 1C t im M1 n im M2 la:

(4)

Trong o la th cua trng tinh in tai M1, M2.

; ;

( Tich phn theo ng cong L khep kin)

Ta co:

(5)

Ta co:

(6)

Phng trinh (6) goi la phng trinh Poison. Phng trinh nay lin h th va in tich tai mt im bt ky cua trng.

- Tai nhng im ma o mt in trng bng khng thi ta co , phng trinh (6) tr thanh phng trinh Laplas.

Giai phng trinh Poison co nghim: (7)

r la khoang cach t im tinh trng n vi phn th tich dV.(2 im)

- i vi in tich im ta co:

(8)

Ta co:

Ta c: (9)

Trng hp in tich im ta co:

(10)

Nu a vao trng cua in tich q mt in tich th q1, ta co:

(11)

Biu thc (11) chinh la inh lut Culng. Vi vy inh lut Culng la h qua cua phng trinh Maxwell i vi trng tinh in.(3 im)

Cu 3 : (2 im)


Ly L la chu vi cua ng tron ban kinh r. Do tinh cht i xng nn tai moi im trn ng cong L la nh nhau.

(1 im)

Trng hp r>a: ( (

Trng hp rb thi

Trng hp a r b thi



THI S 14


Thi gian : 90 pht


Trnh by v hng s in mi phc va goc tiu hao in mi trong trng iu hoa.Cu 2 : (3 im)

Hy trnh by v hiu ng b mt vt dn.Cu 3 : (2 im)

Cho mt hinh cu tich in ban kinh la a. Gia s in tich phn ph u trn b mt cua no vi mt in tich mt s = Q/4a2. Tinh cng in trng tai nhng im ngoai va trong hinh cu.Cu 4 : (2 im)

Cho tham s in cua t kh:

Hng s in mi tng i

dn in ring = 10-3 1/m.

Chng to rng i vi song cc dai( = 104 105 m) thi mt t co tinh dn in tt hn, con i vi song cc ngn( = 10-3 10 m)thi mt t co tinh dn in kem.p n:

Cu 1 : (3 im)

T phng trinh 1 cua Maxwell dang bin phc:

Va ta co (

(

Trong o : la hng s in mi phc tuyt i cua mi trng

la hng s in mi phc tng i cua mi trng.

Co th chng minh rng ty s gia phn ao va phn thc cua la : la ty s in dn va in dich, no c trng cho tiu hao trong mi trng in mi.

t tg( =; ( la goc tiu hao in mi

Nu la cht in mi: tg( < 0,001

Dn in: tg( > 100

Ban dn 0,01 < tg( >1) ta c :

(1)

Khi rt ln th cng rt ln dn n suy gim cng nhiu, ta thy bin cng trng suy gim rt nhanh khi truyn vo trong vt dn. Ngha l sng in t ch tn ti mt lp rt mng st b mt ca vt dn in tt. Khi cho dng in cao tn chy trong vt dn in tt ngi ta cng thy dng in ny ch tn ti trn mt lp theo nh lut m.

Jd = E

i vi dng kho st:

E=Eme-ze-jzJd = Eme-ze-jz =J0e-ze-jz (2)

J0 l mt dng chn b mt vt cht J0 = Em (1 im)

Mt dng in s gim dn khi i vo su trong vt dn theo quy lut ging nh bin cng in trng

Hin tng sng in t hay sng in cao tn khi truyn trong vt dn in tt ch tp chung mt lp rt mng trn b mt ca n gi l hiu ng b mt, hay hiu ng Skin

c trng cho hiu ng b mt ngi ta a vo khai nim thm su ca trng hay su thm nhp ca trng , l khong cch m ng vi n bin cng trng suy gim i e ln: e 2,718

Ta c :

e = e suy ra = 1/

(2 im)

Hiu ng b mt c p dng trong thc t (m vng, bc), khi lm gim tiu hao khi truyn sng in t ngi ta ch m mt lp mng vng hoc bc ln b mt kim loi.

Khi tnh ton cc bi ton ngi ta thy khi nim tr khng mt ca kim loi: ZS = RS + XSRS l tr c trng cho cng sut tiu hao

XS l cm khng ca mt ring ZS

Vn tc pha:

(3 im)Cu 3 : (2 im)



(

(1 im)

a) Xet trng hp th nht: im M ngoai hinh cu(r>a)

Ta co: q = Q

( D.4 r2 = Q = S.4 a2

( D = S.(a2/r2)b) Trng hp th hai: im M trong hinh cu(r0 la mi trng dn in, cu th la kim loai ng co dn in ring , theo phng truc z truyn mt song thng ng nht vi tn s f = 105Hz. Hay xac inh vn tc pha, bc song, tr khang song, h s suy giam va thm su cua trng() trong kim loai ng cua song. Bin cng trng se giam i bao nhiu ln so vi b mt kim loai khi song i su vao c mt khoang d = 1mm.p n:

Cu 1 : (3 im)

Maxwell coi nh lut Gauss v nguyn l lin tc ca t thng p dng cho c trng hp in trng v t trng l tnh, khng i cng nh vi trng hp tng qut ca in t trng bin thin theo thi gian. Ta c:

(1)

(2)V th tch V l tu nn nhn c cc phng trnh Maxwell th 3 v th 4 nh sau:

(3)

(4)(1 im)

tin cho vic theo di, ta vit phng trnh Maxwell thnh hai dng sau:

Dng tch phn:

(5)

Dng vi phn:

(6)(2 im)

ngha vt l ca phng trnh 3 v 4 ca Maxwell:

0: ta thy ng sc ca in trng l nhng ng cong khng khp kn m c im u ti in tch +q, im cui ti q.

= 0: in trng sinh ra ch do s bin thin ca t trng. ng sc ca n hoc khp kn hoc tin ra v cc. ( ng sc ca t trng va khp kn va tin xa v cc.(3 im)

Cu 2 : (3 im)

Trang thai ring quan trong th 2 la t trng do dong in khng i tao ra. y la trang thai dng cua trng in t: J ( 0,

(1)

(2)

Tng t nh trng tinh in, i vi trng dng ta cung co nhng nhn xet sau:

in trng va t trng dng khng c lp vi nhau na ma lin tuc vi nhau thng qua .

Trng dng co nn in trng dng la mt trng th co th t

T trng dng co tinh cht xoay vi nn khng th dung th va co th biu din qua bin mi (3)

Vi trong o la vec t th, ta co:

xac inhn tri thi ta thm mt iu kin tuy y. n gian ta ly ( (4) . Biu thc (4) goi la phng trinh Poison cho . Phng trinh vec t nay tng ng vi (5) phng trinh sau:

EMBED Equation.3 (5)

(1 im)

Nghim cua phng trinh (5) la (6)

Dang vec t cua no la: (7)

Biu thc xac inh va la:

(8)

(9)

Trng hp dong in chay trong dy dn co thit din ngang nho co th bo qua so vi chiu dai dy dn va khoang cach t dy n im quan sat. Vec t th luc nay co dang:

(10)

Va (11)hay

(12)

(2 im)

Ap dung hng ng thc vec t:

ly ta co:

Bi vi trng c tinh im quan sat M vi toa x, y, z ma khng phu thuc vao im M nn ( , va thay vao biu thc (12) ta co:

(10)

Biu thc (10) la biu thc dang tich phn cua inh lut Bixava. Con dang vi phn nh sau:

(11)

Kt lun: inh lut Bixava la h qua cua phng trinh Maxwell i vi trng dng.(3 im)

Cu 3 : (2 im)


Ly S la hinh tru thng dai v han ( D tai moi im trn din tich xung quanh(Sxq) cua hinh tru nh nhau.

Ta co: Sxq = 2r.l (l ) va q = L.l ( D.Sxq = q va D.2rl = L.l (

Ma D = E (

(1 im)

Th tai im cach truc mt khoang r la:

Tng t tai im cach truc mt khoang x:

in trng:

Th:

(2 im)

Cu 4 : (2 im)

Ta c ln nn

Vn tc pha:

Ma

Va

EMBED Equation.3

EMBED Equation.3

(1 im)

Ta co: (1) ma trn b mt thi z = 0 nn bin cua phng trinh (1) la Em.

Khi song i su mt oan d = 1mm, luc o bin cua phng trinh (1) la

( bin song suy giam: ln(2 im)

EMBED Visio.Drawing.11



(4)



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2b

I

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