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3 Contents (Cartesian ) grad div Gauss (rotation) rot Stokes

4 4 CONTENTS grad div rot Gauss Gauss ( 2) Poisson Coulomb Gauss Ampere Biot-Savart

5 CONTENTS 5 7 ( 2) Poisson Poisson Poisson Poisson Poisson A = µ 0 i E D H B Coulomb H rot H = i D= t rot E = B= t

6 6 CONTENTS Maxwell A= t Poisson A 67 A A A A A A A A A A A A A A A A A A.4.3 Stokes A A.5.1 ( ) A.5.2 ( ) A.5.3 dω ds A A A.6.2 (D = ε 0 E) A A A A

7 CONTENTS 7 A A A A A A A A A A A A A A A

9 Chapter φ (r) A(x;y;z) 1.2 (Cartesian ) x y z 1 e x ;e y ;e z ffl A =(A x ;A y ;A z ) ffl A = A x e x + A y e y + A z e z A = jaj = q A 2 x + A2 y + A2 z, 9

10 10 CHAPTER 1. z e z e x e y y x z A x e x r A z e z A A y e y A( r ) r x y A ± B =(A x ± B x )e x +(A y ± B y )e y +(A z ± B z )e z, aa = aa x e x + aa y e y + aa z e z 1.3 A ± B = B ± A A ± (B ± C) =(A ± B) ± C α(βa) =β (αa) =(αβ)a (α ± β )A = αa ± βb α(a ± B) =αa ± αb

11 A B θ A B = jajjbjcosθ A B B O θ A ffl A B = B A ffl A A = jaj ffl A B = 0 $ A? B ffl cosθ = A B jajjbj ffl (A + B) C = A C + B C ffl (aa) B = a(a B) =A (ab) ffl e x e y = e y e y = e z e z = 1 ffl e x e y = e y e z = e z e x = 0 A B = A x B x + A y B y + A z B z

12 12 CHAPTER 1. B A = Proj: B ona = jbj cos(ab) = jbj A B jajjbj = A B jaj = e A B B O ΑΒ B A A 1.5 A B θ A B e A B =(jajjbjsinθ)e A B A B S = jajjbjsinθ = ja Bj ffl A B = B A ffl A A = 0 ffl A B = 0 $ A k B

13 A B A e θ B ffl (A + B) C = A C + A C ffl (aa) B = a(a B) =A (ab) ffl e x e x = e y e y = e z e z = 0 ffl e x e y = e z ; e y e z = e x ; e z e x = e y A B = (A y B z A z B y )e x +(A z B x A x B z )e y +(A x B y A y B x )e fi fi z fi fi e x e y e z fi fi = fi fi A x A y A z fi fi fi B x B y B z fi (r;φ;z) r 0 0» φ» 2π e r e φ e z

14 14 CHAPTER 1. z e z r e φ e r φ y x Cartesian x = r cosφ y = r sinφ z = z p = re r + ze z (r;θ;φ) r 0 0» φ» 2φ 0» θ» π z e r θ r e φ e θ y φ x

15 e r e θ e φ Cartesian x = r sinθ cosφ y = r sinθ sinφ z = r cosθ p = re r

17 Chapter d A(t + t) A(t) A(t) = lim dt t!0 t d dt A(t) = d dt A x(t)e x + d dt A y(t)e y + d dt A z(t)e z d dt fa(t) ± B(t)g = d dt A(t) ± d dt B(t) d dt fc(t)a(t)g = d dt c(t)a(t)+c(t) d dt A(t) d dt fa(t) B(t)g d d = A(t) B(t)+A(t) dt dt B(t) d dt fa(t) B(t)g = d dt A(t) B(t)+A(t) d dt B(t) 2.2 A(x;y;z) =A x (x;y;z)e x + A y (x;y;z)e y + A z (x;y;z)e z 17

18 18 CHAPTER 2. C r = r(t) =x(t)e x + y(t)e y + z(t)e z C t = a t = b u(t) C A u dt = b A C a A(x(t);y(t);z(t)) u(t) dt A ( r(t) ) = A ( x(t), y(t), z(t) ) u A u C t=b t=a r (t) = x(t) e x + y(t) e y + z(t) e z O u(t) C u(t) = d dt x(t)e x + d dt y(t)e y + d dt z(t)e z A u dt = = b a xb x a A x dx dt + A dy y dt + A dz z dt dt A x dx+ yb y a A y dy+ zb z a A z dz x a ;x b ;y a ;y b ;z a ;z b t = a;t = b x;y;z dr = dxe x + dye y + dze z C A dr

19 A S S n A A n = A n S i A lim A n S i = lim A n S i S i!0 i S i!0 i S A n ds S A ds A S S (i) A n n A S i S A n S i = A x e x n S i + A y e y n S i + A z e z n S i A z je z j = jnj = 1 A z e z n S i = A z je z jjnjcosγ S i = A z cosγ S (i) γ z n ( e z n ) S i cosγ S i x-y S i cosγ = x y

20 20 CHAPTER 2. z n γ e z S i γ x y x y dscosγ = dxdy A n S i! A x dydz+ A y dzdx+ A z dxdy A S A n ds = A x dydz+ A y dzdx+ A z dxdy A 2.4 A(x;y;z) V A dv = A x (x;y;z) dv e x + A y (x;y;z) dv e y + A z (x;y;z) dv e z V dv dv = dxdydz A x ;A y ;A z

21 ds (x;y;z) (r;φ;z) (r;θ;φ) ds = dxdy; dydz; dzdx ds = rdφdz ds = r 2 sinθdφdθ r sin θ dφ r dθ dz 1 dφ dz r dφ dφ θ r dθ r sin θ dy dx r ds = dxdy, dydz, dzdx ds = r dφ dz ds = r 2 sin θ dθ dφ 2.6 dv (x;y;z) (r;φ;z) (r;θ;φ) dv = dxdydz dv = rdrdφdz dv = r 2 sinθdrdθdφ

22 22 CHAPTER 2. r dφ d r r sin θ dφ dz r dθ θ r dr dθ dz 1 dφ dφ r sin θ dy dx r dv = dx dy dz dv = r dr dφ dz dv = r 2 sin θ dr dθ dφ

23 Chapter grad φ (x;y;z) φ grad φ = φ = φ x e x + φ y e y + φ z e z = x e x + y e y + z e z φ φ φ A A = grad φ (or φ ) φ φ A E ( )φ 23

24 24 CHAPTER 3. A φ A φ b A dr = φ (a) φ (b) div a A A div A = A A x A = x + y A y + z z A div A Gauss S V A div A A S div A dv = A n ds 3.3 (rotation) rot V A A rot A = A A z = ( y A y z )e A x x +( z A z x )e A y y +( x A x y )e z fi fi fi fi e x e y e z fi fi fi = fi fi fi fi fi x y z fi fi fi A x A y A z fi S

25 A rot A Stokes C S A rot A A C 3.4 S (rot A) n ds = C A dr 4 = = 2 x y z 2 div grad 4φ 2 φ 2 φ 2 φ = x 2 + y 2 + z 2 div grad 4A =(4A x )e x +(4A y )e y +(4A z )e z 4A 2 A 2 i A 2 i = x 2 + i A y 2 + i z grad ffl grad ( f + g) =grad f + grad g ( f + g) = f + g

26 26 CHAPTER 3. ffl grad (cf)=c grad f (cf)=c f ffl grad ( fg)=(grad f ) g + f grad g ( fg)=( f )g + f g div ffl div (A + B) =div A + div B (A + B) = A + B ffl div (φa) =grad φ A + φ div A (φa) =( φ ) A + φ A rot ffl rot (A + B) =rot A + rot B (A + B) = A + B ffl rot (φa) =grad φ A + φ rot A (φa) =( φ ) A + φ A ffl grad (A B) =(A grad) B +(B grad) A + A rot B + B rot A ffl div (A B) =B rot A A rot B ffl rot rot A = grad div A 4A ffl rot (A B) =A div B B div A +(B grad) A (A grad) B

27 Chapter q(c) Q(C) r(m) F(N) F 1 qq = 4πε 0 r 2 e 1 qq r r = 4πε 0 r 2 r where e r = r r e r ε 0 (F/m=CV 1 m 1 ) Q q r e r F E F=qE Q r Q r q 27

28 28 CHAPTER q(c) F(N) E(V/m) F = qe Q(C) r E(V/m) E 1 Q r 1 Q = 4πε 0 r 2 r = 4πε 0 r 3 r 4.3 Gauss Q(C) ffl E(V/m) Q=ε 0 (Vm) ffl D(C/m 2 ) Q(C) D E [ ] D = ε 0 E S E n ds = 1 ε 0 Q S D n ds = Q S E n ds 1 = ε Q i D n ds = 0 i S i Q i ( ρ(r) S E n ds = 1 ε 0 V ρ(r) dv S D n ds = ρ(r) dv V

29 4.4. GAUSS Gauss Gauss Gauss S D n ds = Gauss V div D dv = V div D dv ρ(r) dv V dv Gauss div D = ρ(r) F = GmM r r 3 $ φ (r) = GmM r F = grad φ E = Q r 4πε 0 r 3 $ φ Q 1 (r) = 4πε 0 r E = grad φ ffl B A E dr = φ (A) φ (B) ffl I E dr = 0 $ rot E = 0

30 30 CHAPTER dq(c) de(v/m) dφ(v) ( ) ds dφ r de dq = λ ds ds dφ r de dq = σ ds dv dφ r de dq = ρ dv de = 1 4πε 0 dq r 3 r dφ 1 dq = 4πε 0 r λ(c/m) ds(m) dq = λ ds E = 1 4πε 0 λ r 3 r ds φ = 1 4πε 0 λ r ds σ(c/m 2 ) ds(m 2 ) dq = σ ds E = 1 4πε 0 σ r 3 r ds φ = 1 4πε 0 σ r ds

31 ρ(c/m 3 ) dv(m 3 ) dq = ρ dv E = 1 4πε 0 ρ r 3 r dv φ = 1 4πε 0 ρ r dv

33 Chapter 5 ( 2) 5.1 Poisson div E = ρ=ε 0 E = grad φ 4φ = ρ ε s s p = qs r 1 ;r 2 φ q ρ 1 = 1 ff 4πε 0 r 1 r 2 ( ) E = grad φ E 1 ρ = p ff 3r(r p) + 4πε 0 r3 r 5 33

34 34 CHAPTER 5. ( 2) V ([F]=[C]/[V]) C = Q V Q = CV a C = 4πε 0 a S d a b C = ε 0 S d ab C = 4πε 0 b a a b L(L fl a) L C = 2πε 0 ln(b=a) a d L (a fi d;a fi L) C = πε 0 L ln(d=a) U e = 1 C Q2 1 2 CV QV Q(C)

35 E E U e = Q2 (ε 2C = 0 SE) 2 ε 0 2( d S) = ε 0 2 E2 Sd = ε 0 2 E2 v U e = 1 2 ε 0 V E 2 dv

37 Chapter Coulomb q m Q m r(m) F(N) F 1 q m Q m r = 4πµ 0 r 2 r r(m) I 1 I 2 (A) F µ 0 I 1 I 2 r = 2π r r I 1 (A) F(N) B(T) ([T( )]=[Wb( )/m 2 ]=[10 4 Gauss] Wb Φ ) F = I 1 B 37

38 38 CHAPTER 6. I 2 (A) r(m) B(T) B µ 0 I 2 r = 2π r r 6.2 Gauss S 0 B n ds = 0 div B = Ampere I B I B µ 0 I C 0 B dr = µ 0 I rot B = µ 0 i E φ E = gradφ

39 6.5. BIOT-SAVART 39 rot E = rot B = 0 B A B = rot A div B = 0 E φ 6.5 Biot-Savart I ds( ) r P dh db µ 0 I ds r = 4π r 3 ds r θ db µ 0 I ds sinθ = 4π r 2 ds r Ampere

40 40 CHAPTER 6. I ds θ r db

41 Chapter 7 ( 2) 7.1 Poisson Poisson ρ(c/m 3 ) E(V/m) φ(v) Poisson 4φ = ρ ε 0 φ E = grad φ E Poisson i(a/m 2 ) B(Wb/m 2 ) A(Wb) Poisson 4A = µ 0 i A B = rot A 41

42 42 CHAPTER 7. ( 2) B ψ 1 A z rot A = r φ A! φ Ar e r + z z A ψ! z 1 (ra φ ) e r φ + A r e z r r φ Poisson dq(c) de(v/m) dφ(v) ( 4 6 ) E = V de 1 dq r = 4πε 0 V 1 dv ρ r r 3 = 4πε 0 V r 3 φ = V dφ 1 dq = 4πε 0 V 1 dv ρ r = 4πε 0 V r ρ(c/m 3 ) dq = dv ρ Poisson I ds(am) db(wb/m 2 ) da(wb) (Biot-Savart ) B = V A = db = µ 0 4π V C ds I r µ 0 dv i r r 3 = 4π V r 3 da µ 0 ds I µ 0 dv i = = 4π C r 4π V r

43 7.2. 4A = µ 0 I 43 i(a/m 2 ) ds I = ds ds i = dv i I ds ds I = Ids dv 7.2 4A = µ 0 i A B = rot A B Ampere rot B = µ 0 i rot rot A = grad div A 4A Ampere (a) grad div A 4A = µ 0 i B = rot A u grad rot B = µ 0 i rot grad = 0 A 0 B A 0 = A + grad u φ grad E u χ 4χ = div A 0

44 44 CHAPTER 7. ( 2) A 0 (a) χ A = A 0 + grad χ (a) χ div A = div A 0 + div grad χ = div A 0 + 4χ = 0 (a) Poisson 4A = µ 0 i

45 Chapter E D D = ε 0 E D = εe D B E D 1 n D 2 n = 0 ε 1 E 1 n ε 2 E 2 n = 0 E 1 t + E 2 t = 0 D D 1 ε 1 t + D 2 ε 2 t = 0 45

46 46 CHAPTER 8. E 1 t D 1 n ε 1 ε 2 D 2 D 1 n S n D 2 n ε 1 ε 2 E 2 S t t E 1 E 2 t H B B = µ 0 H B = µh B 1 n B 2 n = 0 µ 1 H 1 n µ 2 H 2 n = 0 H 1 t + H 2 t = 0 B 1 µ 1 t + B 2 µ 2 t = 0

47 H 1 t B 1 n µ 1 µ 2 B 2 B 1 n S n B 2 n µ 1 µ 2 H 2 S t t H 1 H 2 t

49 Chapter I 2 B I 1 F = I 1 B I B Ids df = Ids B I 1 I 2 I ds B F = I 1 B B θ F F = I ds B = I ds B sin θ 49

50 50 CHAPTER v(m/s) q(c) F = qe F = q v B F = q(e + v B) S(m 2 ) ([A]=[C]/[s]) v(m/s) S S L(m) =v(m=s) 1(s) nsl = nsv( ) n( /m 3 ) q(c) I = qnvs ds ds df = Ids B I df = qnsvds B = qnsdsv B ds nsds F = q v B N S

51 L[m] = v[m/s] 1[s] v S n [ /m 3 ] Coulomb q m (Wb) Q m (Wb) r(m) F(N) H F 1 q m Q m r = 4πµ 0 r 2 r H(N/Wb A/m) q m (Wb) F(N) F = q m H Q m (Wb) H(A/m) H 1 Q m r = 4πµ 0 r 2 r E(V/m) φ e (V) φ m (A) φ 1 Q m m = 4πµ 0 r

52 52 CHAPTER s s m = q m s r fl l φ m = q m 1 1 4πµ 0 r 2 r 1 φ m = q2l cosθ 4πµ 0 r 2 H 1» = m 3r(r m) + 4πµ 0 r3 r 5 grad H r = φ m 2ql 2cosθ r = 4πµ 0 r 3 H θ = 1 r φ m θ = 2ql sinθ 4πµ 0 r P I(A) ω(str) φ m (A) φ m = Iω 4π

53 a fi r ω ß π a2 r 2 cosθ φ m = Ia2 4r 2 cosθ l fi r φ m = 2ql 4πµ 0 r 2 cosθ m = 2ql I m = µ 0 IS S = πa 2 S

55 Chapter Gauss Gauss 10.2 rot H = i S S C D n ds = Q $ div D = ρ C B n ds = 0 $ div B = 0 E u ds = 0 $ rot E = 0 H u ds = I $ rot H = i rot H = i rot H = i + D t 55

56 56 CHAPTER D= t rot H = i div div rot H = div i div rot div i = 0 div i = 0 OK div i 6= 0 Q dt n i I out = dq dt S Q dq I out = i n ds (S S ) dq dt = S i n ds dv (Gauss ) dρ dt = div i ( ) div D = ρ div i + t div D = 0

57 10.3. ROT E = 0 57 rot H = i + D t div 10.3 rot E = 0 rot E = 0 rot E = B t B= t Φ S A C V Φ ( Wb 1 ) V Φ V = dφ dt Φ B Φ = S B n ds

58 58 CHAPTER 10. E V = C E dr = S (rot E) n ds rot E = B t 10.4 C Φ υ C v dt ds dr C v C dr dt ds ds = j(vdt) drj dφ dφ = = = dφ dt S S S = dt = I B nds B j(vdt) drj I B (vdt dr) C C (v B) dr (v B) dr dφ dt = I C E dr

59 E = v B B C B V = S t IC + (v B) dr 10.5 V dφ=dt Φ Φ I I L (t) V L (t) V L (t) = L d dt I(t) L(H) I(t) C V(t) L R RI = V (t)+ Q C LdI dt V (t) =V 0 cosωt L d2 I dt 2 + RdI dt + 1 C I = ωv 0 sinωt

61 Chapter Maxwell Maxwell div D(x; t) = ρ(x; t); (11.1) div B(x;t) = 0; (11.2) rot E(x;t) = B(x;t) ; t (11.3) rot D H(x;t) = + i(x;t): t (11.4) D = ε 0 E; (11.5) B = µ 0 H: (11.6) m dv dt = e E + e v B (11.7)

62 62 CHAPTER 11. d dt ψ! 1 i 2 m i v2 i +W = S n ds (11.8) 1 i 2 m i v2 i W = 1 (E D + B H)dV v v (v B) S = E H m dv dt = e E + e v B d 1 dt 2 mv2 = e ve+ e v (v B) =e ve= i E dv V i Maxwell rot H D = i t d 1 dt 2 mv2 = rot H D E dv V t div (E H) =H rot E E rot H

63 d dt d dt ρ 1 2 mv2 = H rot E D ff + E + div (E H) dv V t rot E = B t» 1 2 mv2 = H B V t + E D dv + div (E H) dv t V 1 2 t D (E D + B H) =E t + H B t (D = ε 0 E B = µ 0 H ) Gauss d dt d dt 1 2 mv2 = d dt 1 (E D + B H) dv + (E H) n ds 2 V S mv2 + (E D + B H) dv = (E H) n ds 2 V S ψ! d 1 dt 2 m 1 i v2 i + (E D + B H) dv 2 i 11.3 V E = grad φ B = rot A = S (E H) n ds

64 64 CHAPTER 11. Poisson 4φ = ρ ε 0 4A = µ 0 i E = A t grad φ B = rot A Poisson 4 1c 2 2 t 2 φ = ρ ε 0 4 1c 2 2 t 2 A = µ 0 i A= t div A + 1 c 2 φ t = 0 rot E = B= t grad φ rot E = 0 B = rot A rot E = B= t rot E A + = 0 t rot E = 0 E E + A= t E = grad φ A t

65 Poisson φ A E = grad φ A t B = rot A Maxwell div B = 0 div D = ρ rot E = B t rot H D t = i B E rot B ε 0 µ 0 E t = µ 0 i E B rot rot A ε 0 µ 0 A t t grad φ = µ 0 i 4 1c 2 2 t 2 rot rot A = grad div A 4A A grad div A 1 φ + c 2 = µ t 0 i A Poisson div D = ρ E 4 1c 2 2 t 2 φ div A 1 φ + t c 2 = ρ t ε 0 φ Poisson

66 66 CHAPTER 11. u A L = A 0 + grad u φ L = φ 0 u t A 0 φ 0 u χ 4 1c 2 2 t 2 χ = div A 1 φ c 2 t χ () div A c 2 φ 0 t = div A + 1 c 2 φ t + 4χ 1 c 2 2 χ t 2 = 0 4 1c 2 2 t 2 φ = ρ ε 0 4 1c 2 2 t 2 A = µ 0 i () χ A φ div A + 1 c 2 φ t = 0

67 Appendix A A.1 A.1.1 y = f (x) f (x + x) f (x) lim x!0 x lim x!0 y x df(x) df dy dx ; dx ; dx ; f 0 (x); f 0 d dx f (x); d dx f d dx grad div rot 4 67

68 68 APPENDIX A. y f(x+ x) y x x 0 dy dx y f(x) x x x+ x A.1.2 d df(x) (α f (x)+βg(x)) = α + β dg(x) dx dx dx d df(x) ( f (x)g(x)) = dx dx g(x)+ f (x)dg(x) dx d dx ( f (x) g(x) )= d dx df(x) dx g(x) f (x)dg(x) dx g(x) 2 df(g(x)) dg(x) f (g(x)) = dg(x) dx dx 1 dy = dy dx x = x(t) y = y(t) dy dx = dy dt dx dt

69 A d df dx dx = d2 dx 2 f = d2 f dx 2 A.1.3 dy/dx > 0 d 2 y/dx 2 > 0 dy/dx < 0 d 2 y/dx 2 < 0 y = f (x) x dy dy > 0(yincreases)! dx dx < 0(ydecreases) dy dx d 2 y dx 2 < 0 y = f (x) x dy dy < 0(ydecreases)! dx dx > 0(yincreases) d 2 y dx 2 > 0 dy dx = 0

70 70 APPENDIX A. d 2 y dx 2 > 0! d2 y dx 2 < 0 d 2 y dx 2 < 0! d2 y dx 2 > 0 A.1.4 z = f (x;y) f (x + x;y) f (x;y) lim x!0 x x f (x;y); f (x;y) ; x z x ; f x ; x f A.2 A.2.1 f g g f g(x) = f (x) dx

71 A A.2.2 A.2.3 faf(x)+bg(x)g dx = a f (x) dx = f dx (x(t)) dt dt + c f 0 (x)g(x) dx = f (x)g(x) f (x) dx+ b g(x) dx+ c f (x)g 0 (x) dx+ c p x 2 + a 2 x = atanθ; p x 2 + a 2 = a cosθ ; dx a dθ = cos 2 θ p t 2 a 2 x 2 + a 2 = t x; x = 2t ; p x 2 + a 2 = t 2 + a 2 2t ; dx dt = t2 + a 2 2t 2 A.2.4 f (x) a» x» b n I 1 ;I 2 ;:::;I n x 1 ; x 2 ;:::; x n I i x i S n = n f (x i ) x i i=1 S n b S = a f (x) dx f (x) x = a x = b

72 72 APPENDIX A. y y S a x b x a b x A.2.5 f (x;y) x;y D D D f (x;y) dxdy f (x;y) dxdy f (x;y;z) x;y;z D D D f (x;y;z) dxdydz f (x;y;z) dxdydz A.2.6 ρ ff f (x;y) dxdy = f (x;y) dx dy = ρ ff f (x;y) dy dx

73 A fg dy dx f (x;y) dx f (x;y) dy z A.2.7 f (x)g(y) f (x)g(y)h(z) ρ ffρ ff f (x)g(y) dxdy = f (x) dx g(y) dy f (x)g(y)h(z) dxdydz = ρ ffρ f (x) dx ffρ g(y) dy ff h(z) dz A.3 A φ A φ b a A dr = b a dφ = φ (a) φ (b) φ dr = ( φ x e x + φ y e y + φ z e z) (dxe x + dye y + dze z ) = φ x dx+ φ y φ dy+ dz = dφ z

74 74 APPENDIX A. A.4 A.4.1 A div A S V A A nds S (V ) V! 0 diva 1 diva = lim V!0 V S A nds z n 1 S 1 S2 δz x δy δx n 2 y V = δxδyδz y S 1 y S 1 A nds = fa ngδxδz = A y (x;y;z)n 1 δxδz S 2 y S 2 A nds = fa ngδxδz = A y (x;y + δy;z)n 2 δxδz

75 A A V y ( n 1 = n 2 jn 1 j = jn 2 j = 1 ) S 1 A n 1 ds 1 + S 2 A n 2 ds 2 + = A y (x;y;z)n 1 δxδz + A y (x;y + δy;z)n 2 δxδz = fa y (x;y + δy;z) A y (x;y;z)gδxδz A y(x;y + δy;z) A y (x;y;z) = δxδyδz δy A y(x;y;z) = V y x;z A.4.2 S S S (A n)xds = A x x V (A n)yds = A y y V (A n)zds = A z z V 1 lim V V!0 = lim V!0 A nds S ρ Ax A x + y A y + z z A x A = x + y A y + z z V i = δx i δy i δz i S i A i n i ds = δx i δy i δz i div A i V V i i ff A i n i ds = S δx i δy i δz i div A i i i

76 76 APPENDIX A. i j ni ds A nj V i V j n i = n j A i = A j A i n i + A j n j = 0 V i V j V A.4.3 Stokes V div A dv = S A n ds S S i S i z x;y S i C i C i A dr = A x (x;y;z)dx+ A y (x + dx;y;z)dy A x (x;y + dy;z)dx A y (x;y;z)dy = A y x dxdy+ A x(x;y;z)dx+ A y (x;y;z)dy = = A x y dxdy A y(x;y;z)dy A x (x;y;z)dx Ax x A y dxdy y S i (rot A) n ds

77 A y (x,y+dy) S i x (x,y) (x+dx,y+dy) C i (x+dx,y) S i S j Stokes A.5 S (rot A) n ds = C A dr A.5.1 ( ) 1 θ(radian) ffl r rθ A 1 O r θ rθ B A.5.2 ( ) 1 Ω(steradian) ffl r r 2 Ω

78 78 APPENDIX A. r 1 O Ω r 2 Ω A.5.3 dω ds ds dω dω r n = r 3 ds ds r ds dω θ n 1 O r dω r 2 dω dω r ds n θ r 2 dω r 2 dω θ ds r 2 dω ds dscosθ = r 2 dω ds n(jnj = 1) ds r(jrj = 1) r n = r cosθ dω = r n r 3 ds A.6 E nds = jejcosθds

79 A Q R n E dω ds 1 R dω R 2 dω n θ ds E dω dscosθ = R 2 dω E nds = jejr 2 dω = 1 Q 4πε 0 R 2 R2 dω = Q 4πε 0 dω S E n ds = Q 4πε 0 S dω = Q ε 0 R S dω = 4π A.6.1 ( Ψ ) ffl 1C 1 ffl 1C 1

80 80 APPENDIX A. +Q Q +Q C D(C/m 2 ) n ds(m 2 ) dψ(c) D = dψ ds n A.6.2 (D = ε 0 E) Gauss D = ε 0 E Q(C) r D(C/m 2 ) D Q Q = S D n ds = 4πr 2 jdj = 4πr 2 D D = Q=(4πr 2 ) D r=r D Q r = 4πr 2 r E = Q r 4πε 0 r 2 r D = ε 0 E

81 A A.7 A.7.1 S Gauss S D n ds = V S ρ dv ε 1 S 1 ε 2 S 2 S S d S 1 D 1 n 1 ds + S 2 D 2 n 2 ds + S d D n ds = V S ρ dv S 1 S 2 D 1 D 2 ( ) ( ) S d! 0 S 1 = S 2 = S n 1 = n 2 = n D 1 n S D 2 n S = ρ S (D 1 D 2 ) n = ρ D ρ D D i = ε i E i E ε D 1 ε 1 ε2 n 1 S S 1 d D 2 n 2 S d S 2

82 82 APPENDIX A. A.7.2 C rot E = 0 I C E uds = 0 ε 1 L 1 ε 2 L 2 L d ( ) E 1 u 1 ds+ E 2 u 2 ds+ E uds = 0 L 1 L 2 L d L 1 L 2 E 1 E 2 ( ) ( ) L d d! 0 L 1 = L 2 = L u 1 = u 2 = u E 1 u L E 2 u L = 0 (E 1 E 2 ) u = 0 E E i = D i =ε i D ε E 1 C ε 1 ε2 L 1 u 1 L 2 L d L d d u 2 E 2 A.8 A.8.1 S Gauss S B n ds = 0

83 A µ 1 S 1 µ 2 S 2 S S d S 1 B 1 n 1 ds + S 2 B 2 n 2 ds + S d B n ds = 0 S 1 S 2 B 1 B 2 ( ) ( ) S d! 0 S 1 = S 2 = S n 1 = n 2 = n B 1 n S B 2 n S = 0 (B 1 B 2 ) n = 0 B B i = µ i H i H µ B 1 µ 1 µ2 n 1 S S 1 d B 2 n 2 S d S 2 A.8.2 C rot H = i i I H u ds = i n ds C µ 1 L 1 µ 2 L 2 L d ( ) L 1 H 1 u 1 ds+ S L 2 H 2 u 2 ds+ L d H u ds = 0

84 84 APPENDIX A. L 1 L 2 H 1 H 2 ( ) ( ) L d d! 0 L 1 = L 2 = L u 1 = u 2 = u H 1 u L H 2 u L = 0 (H 1 H 2 ) u = 0 E H i = B i =µ i B µ H 1 C µ 1 µ2 L 1 u 1 L 2 L d L d d u 2 H 2 A.8.3 e h C(ABCDA) e t C 0 (A BCD ) i = i h e h + i t e t C rot H = i d! 0 I C H e t ds = S i ( e h ) ds H 1 e t L H 2 ( e t ) L = i h L (H 1 H 2 ) e t = i h

85 A e t i h C 0 I C 0 H e h ds = S i ( e t ) ds (H 1 H 2 ) e h = i t e h i t (H 1 H 2 ) e n = i H i n h t A D B A' D' C A.9 (x;y;z) ffl grad V = V x e x + V y e y + V z e z ffl div A A x A = x + y A y + z z ffl rot A = Az y A y Ax e x + z z A z Ay e y + x x A x e z y (r;φ;z)

86 86 APPENDIX A. ffl grad V V = r e 1 V r + r φ e V φ + z e z ffl div A = 1 r ffl rot A = ψ (ra r ) r 1 r + 1 r A φ A φ + z z A z φ A! φ Ar e r + z z A ψ z e r φ + 1 r (ra φ ) r A r φ! e z (r;θ;φ) ffl grad V V = r e 1 V r + r θ e 1 V θ + r sinθ φ e φ ffl div A = 1 r 2 (r 2 A r ) r ψ ffl rot A 1 (Aφ sinθ) = r sinθ θ ψ 1 (A θ sinθ) 1 A φ + + r sinθ θ r sinθ φ + 1 r A φ φ 1 A r sinθ! e r φ (ra φ e r θ (raθ ) + 1 r )! r A r e θ φ

87 A A.10 15, Oct., A (B C) =(A C)B (A B)C 2. A = 3e x e y + 2e z, B = 2e x + 3e y e z, C = 4e y 3e z A (B C), A (B C), A (B C), (A B) C 3. A B A B A B A = 3e x + e y + p 3ez, B = 3e x + e y p 3e z 4. (r;φ;z) e r, e φ, e z (x;y;z) 5. (r;θ;φ) e r, e θ, e φ (x;y;z)

88 88 APPENDIX A. A.11 22, Oct., A = 4xye x 8ye y + 2e z (a) y = 2x, z = 0 (3;6;0) (0;0;0) (b) y = 2x, z = 2x (0;0;0) (3;6;6) (c) x 2 + y 2 = 4, z = 0 (2;0;0) (2;0;0) 2. (a) A = x 2 e x + xye y + ze z xy, yz, zy x = 2, y = 2, z = 2 (b) A = ze z x 2 + y 2 + z 2 = 1 r (c) r 3 x2 + y 2 + z 2 = a r r = jrj 3. (a) ρ(r;θ;φ) =r a 1 8 M (b) ρ(r;θ;z) =e r2 a L M 4. px (a) 2 + a 2 dx 1 n = x p p o x a 2 + a 2 lnjx + x 2 + a 2 j 1 (b) p dx x 2 + a 2 = lnjx p + x 2 + a 2 j

89 A (c) (d) x p dx p x 2 + a 2 = x 2 + a 2 x 2 p dx 1 n x 2 + a 2 = x p p o x a 2 a 2 lnjx + x 2 + a 2 j (e) (f) (g) 1 (x 2 + a 2 ) 3=2 dx x = a 2p x 2 + a 2 x (x 2 + a 2 ) 3=2 dx = 1 p x 2 + a 2 x 2 (x 2 + a 2 ) 3=2 dx = x p x 2 + a 2 + lnjx p + x 2 + a 2 j

90 90 APPENDIX A. A.12 29, Oct., A (i) grad r = r r (ii) grad 1 r = r r 3 (iii) div r r = 0 (iv) rot A A r r = r 3 2. (a) grad (A B) =(A grad) B +(B grad) A + A rot B + B rot A (b) div (A B) =B rot A A rot B (c) rot rot A = grad div A 4A (d) rot (A B) =A div B B div A +(B grad) A (A grad) B 3. Gauss Gauss S A n ds = V div A dv (a) A = x 2 e x + xye y + ze z x = 0;2 y = 0;2 z = 0;2 (b) A = x 2 e x + ze y + yze z x = 0;1 y = 0;1 z = 0;1 (c) A = 6ze x +(2x + y)e y xe z x 2 + y 2 = 3 2 x = 0 y = 0 z = 0 y = 8 1/4 (d) A = x 2 e x xye y + z 2 e z x 2 + y 2 = 1 0» z» 1

91 A Stokes Stokes C A dr = S rot A n ds (a) A = 2ye x + 3xe y z 2 e z S x 2 + y 2 + z 2 = 9 (z 0) C x 2 + y 2 = 9 (z = 0) (b) A = ye x +(x 2 + z)e y + ye z S x 2 +y 2» 9 (z = 0) C x 2 +y 2 = 9 (z = 0)

92 92 APPENDIX A. A.13 5, Nov., 1999 E(V/m) φ(v) 1. (a) λ(c/m) z ±L(m) z = 0 a(m) (b) λ(c/m) z a(m) z z(m) (c) σ(c/m 2 ) z a(m) z z(m) 2. Gauss (a) R 1 ;R 2 (m) (0 < R 1 < R 2 ) Q 1 ;Q 2 (C) r(m) (φ ( ) =0 ). (b) ρ 1 ;ρ 2 (C/m 2 ) z R 1 ;R 2 (m) (0 < R 1 < R 2 ) z r(m) (φ (R 2 )=0 ). (c) z a;b(m) (0 < a < b) ρ(r;φ) =r 1 cos 2 φ (C/m 3 ) z r(m) (φ (b) =0 ).

93 A

94 94 APPENDIX A. A.14 12, Nov., p = qs r 2. (a) ( a, Q, ) (b) ( a, 2L) C = C = 4πε 0 a 4πε 0 L ψ p! L + L 2 + a 2 ln a (c) ( S, σ) C = ε 0 S d (d) ( a, b (0 < a < b), Q) ab C = 4πε 0 b a (e) ( a, b, λ, L) C 2πε 0 L = ln(b=a) (f) ( a, d (d fl a), L (L fl a), λ) C = πε 0 ln(d=a)

95 A (a) a(m) ( Q(C)) (b) a(m) ( Q(C))

96 96 APPENDIX A. A.15 26, Nov., a(m) I(A) x(m) B(T) : B µ 0 Ix = (r» a) 2πa2 : B µ 0 I = (r a) 2πx 2. x(m) θ 1 θ 2 B µ 0 I = 4πx (cosθ 1 + cosθ 2 ) 3. a(m) I(A) z(m) B = a 2 µ 0 I 2(a 2 + z 2 ) 3=2 4. n θ 1 θ 2 B = nµ 0 I 2 (cosθ 2 cosθ 1 )

97 A

98 98 APPENDIX A. 5. 2a 2b(m) I(A) z(m) P ( 6. ( R(m)) O n I h(m) P B = nµ 0 I 2 ( ln ψ R + p R 2 + h 2 h R p R 2 + h 2!)

99 A A.16 3, Dec., x(m) A(Wb) B(Wb/m 2 ) θ 1 θ 2 0q A µ 0 I x 2 + L 2 = 4π ln + L 1 1 q Ae z x 2 + L 2 2 L d(m) I(A) A(Wb) B(Wb/m 2 ) A = µ 0 I 2π ln r 1 r 2 e z

100 100 APPENDIX A. A.17 3, Dec., a(m) b(m) ε 2. d(m) x(m) ε(x) =ε 1 +(ε 2 ε 1 ) x d 3. a b(m) r(m) k(m) ε(r) = k + r r 4. a b(m) r(m) ε(r) =ε ε 2 ε (r a) b a d S = 1 m 2 b O a ε ε 0 ε ε 2 ε 1 0 d x

101 A a b a b ε 1 + k/a 1 + k/b 1 0 a b r ε ε 2 ε 1 0 a b r

102 102 APPENDIX A. A.18 10, Dec., (a) a b(m) B(Wb/m 2 ) I(A) (b) L(m) ( d(m)) I 1 I 2 (A) (c) L(m) ( d(m)) I(A) B L A A' L θ 2 I 2 θ 1 r 2 df r 1 dx I 2 B x L C B' D D' d d C' (d) I 1 (A) d(m) a(m) I 2 (A) π 0 cosx bcosx + c dx π = b c π p (c b c 2 b 2 > b)

103 A ds α d a θ O I 2 I 1

104 104 APPENDIX A. 2. (a) B(Wb/m 2 ) m(kg) q(c) v(m/s) i. ii. r = mv=(eb) (m) iii. ω = eb=m (rad/s) (b) I(A) B(Wb/m 2 ) I B V H (V) ( q(c) n(m 3 )) V H V H = a E H z y E H = E H e y v B = B z e z x d b a i = i x e x I = ad i x

105 A A , Dec., B(Wb/m 2 ) S(m 2 ) ω(rad/s) 2. I = I 0 sinωt (A) 3. B(Wb/m 2 ) a(m) ω(rad/s) 4. LR LR I = 0(A) SW 1 ON SW 2 ON N B S S r B ω I = I 0 sin ωt L ds dz dr b a

106 106 APPENDIX A. B L dr r ds v a V SW 1 SW 2 V E R

107 A A.20 3, Dec., x(m) A(Wb) B(Wb/m 2 ) θ 1 θ 2 0q A µ 0 I x 2 + L 2 = 4π ln + L 1 1 q A x 2 + L L d(m) I(A) A(Wb) B(Wb/m 2 ) µ 0 I 2π ln r 1 r 2

120 9 I I 1 I 2 I 1 I 2 ( a) ( b) ( c ) I I 2 I 1 I ( d) ( e) ( f ) 9.1: Ampère (c) (d) (e) S I 1 I 2 B ds = µ 0 ( I 1 I 2 ) I 1 I 2 B ds =0. I 1 I 2

120 9 I I 1 I 2 I 1 I 2 ( a) ( b) ( c ) I I 2 I 1 I ( d) ( e) ( f ) 9.1: Ampère (c) (d) (e) S I 1 I 2 B ds = µ 0 ( I 1 I 2 ) I 1 I 2 B ds =0. I 1 I 2 9 E B 9.1 9.1.1 Ampère Ampère Ampère s law B S µ 0 B ds = µ 0 j ds (9.1) S rot B = µ 0 j (9.2) S Ampère Biot-Savart oulomb Gauss Ampère rot B 0 Ampère µ 0 9.1 (a) (b) I B ds = µ 0 I. I 1 I 2 B ds = µ 0