5 (Green) δ

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1 (Laplace)

2 5 (Green) δ ϕ D = ρ, E = 0 E dl = 0 P P 2 E dl E dl = 0 E dl = E dl C 2 C C C 2 P P 2 P P 2 P2 ϕ(p 2 ) ϕ(p ) = E dl P 2

3 .2 E ϕ dl dϕ = ϕ x dϕ = E dl dx + ϕ y ϕ dy + dz = ϕ dl z E = ϕ ϕ r ϕ(p ) = P E dl E ϕ.3 ϕe.4 E = ϕ = Q r r 3 Q r ϕ(p ) = i Q i r i ρ(x ) d ϕ(x) = r 3

4 .5 Poisson 2 ϕ = ρ ε 0 n (E 2 E ) = 0 n (D 2 D ) = σ f D 2 = ε 2 E = ε 2 ϕ ϕ = ϕ 2 n D 2 = ε 2 (n ϕ) ϕ ε 2 n ε ϕ n = σ f ϕ = ϕ 2 ϕ = ϕ 2 n (E 2 E ) = 0 () n (E 2 E ) = 0? ϕ = ϕ 2 (2) () P P 2 P P 2 ϕ P = ϕ P2, ϕ P = ϕ P 2 P P P 2 P 2 l ϕ P ϕ P = ϕ P 2 ϕ P2 E l = E 2 l l E = E 2.6 γ ϕ = const ε ϕ n = σ f 4

5 .7 ω = (E D + H B) 2 W = 2 E D d E = ϕ D = ρ E D = ϕ D = (ϕd) + ϕ D = (ϕd) + ρϕ W = 2 = 2 = 2 ( (ϕd) + ρϕ) d ρϕ d 2 (ϕd) d ρϕ d 2 ϕd ds ϕ r D r 2 r ϕd ds 0 r 2 W = ρϕ d (3) 2 ρ(x ) d ϕ(x) = 4πεr W = d d ρ(x)ρ(x ) 8πε r (4) ρϕ 2 ρϕ 2 ρϕ 5

6 .8 ϕe ϕ W = E D d 2 Page 93:,7 2. ρ(x) S (i) ϕ S (ii) ϕ n S (Dirichlet) (Neumann) 2.2 Neumann 6

7 2.3 i ε i ρ(x) ϕ i i j 2 ϕ = ρ ε i ϕ i = ϕ j ε i ( ϕ n ) i = ε j ( ϕ n ) j S ϕ S ϕ n S ϕ ϕ ϕ ϕ = ϕ ϕ 2 ϕ = ρ ε i 2 ϕ = ρ ε i i 2 ϕ = 0 i i S i ε i ϕ ϕ ds = S i ϕ i = ϕ j ε i ( ϕ n ) i = ε j ( ϕ n ) j ϕ S = ϕ S ϕ S = 0 ϕ n S = ϕ n S ϕ n S = 0 i (ε i ϕ ϕ) d = ε i ( ϕ) 2 d + i i = ε i ( ϕ) 2 d ε i ϕ 2 ϕ d i i ε i ( ϕ) 2 d = i i i S i ε i ϕ ϕ ds 7

8 i j ϕ ε ϕ ds i = ds j S ε i ϕ ϕ ds = ε i ϕ ϕ ds i S i S Sϕ S = 0 ϕ n S = 0 ε i ( ϕ) 2 d = 0 i i ε i ( ϕ) 2 0 ϕ = 0 ϕ = const 2.4 ϕ i Q i ρ(x) S (i) ϕ S (ii) ϕ n S 2.5 S S S SS i S i ϕ i S ϕ S ϕ n S S ϕ S ϕ n S 8

9 2.6 ρ(x) ϕ 2 ϕ = ρ ε i Q i i ϕ n ds = Q i ε S i ϕ Si = ϕ i = const S ϕ S ϕ n S ϕ ϕ ϕ ϕ = ϕ ϕ ϕ 2 ϕ = 0 S i ϕ n ds = 0 ϕ Si = ϕ i = const ε i ϕ ϕ ds = ϕ S = ϕ S ϕ S = 0 ϕ n S = ϕ n S ϕ n S = 0 (ε i ϕ ϕ) d = ε i ( ϕ) 2 d + = ε i ( ϕ) 2 d ε i ϕ 2 ϕ d S S i S i ns i ϕ ϕ ϕ ds = ϕ i n ds = 0 S i 9 S i

10 S ( ϕ) 2 d = 0 ϕ i ( ϕ) 2 0 ϕ = 0 ε ϕ n = σ Neumann Q i S i 2.7 ρ(x) ρ(x) ϕ i S Q i ϕ i Page 96 4,5 3. (Laplace) ρ = 0 (Laplace) 2 ϕ = 0 0

11 3.2 2 ϕ = r 2 (5) r2 RΘΦ d R dr ϕ (r2 r r ) + r 2 sin θ (r2 dr dr ) = Θ sin θ R(r) Θ(θ)Φ(φ) Θ sin θ r 2 d2 R dr 2 (6)sin 2 θ ϕ (sin θ θ θ ) + 2 ϕ r 2 sin 2 θ φ = 0 (5) 2 ϕ(r, θ, φ) = R(r)Θ(θ)Φ(φ) d dθ (sin θ dθ dθ ) + Φ sin 2 θ dr + 2r n(n + )R = 0 dr R(r) = A n r n + B n r n+ d dθ (sin θ dθ dθ ) + Φ sin 2 θ d 2 Φ = n(n + ) dφ2 d 2 Φ = n(n + ) (6) dφ2 sin θ d dθ (sin θ Θ(θ) dθ dθ ) + n(n + ) sin2 θ = d 2 Φ Φ dφ = 2 m2 Φ(φ) d 2 Φ Φ dφ = 2 m2 Φ(φ) = C m sin(mϕ) + D m cos(mϕ) Θ(θ)(Legendre) d 2 Θ dθ + ] dθ [n(n tan θ dθ + + ) m2 sin 2 Θ = 0 θ 3.3 ϕ(r, θ, φ) = n,m + n,m Θ(θ) = k Pn m (cos θ) (a nm r n + b nm r )P m n+ n (cos θ) cos(mφ) (c nm r n + d nm r )P m n+ n (cos θ) sin(mφ)

12 Pn m (cos θ)(legendre) a nm b nm c nm d nm ϕ φ ϕ(r, θ) = n (a n r n + b n r n+ )P n(cos θ) 0 θ < α < π ν P n (cos θ) a n b n 3.4 ( ζ = cos θ) d dζ [ ( ζ 2 ) dθ dζ ] + n(n + )Θ = 0 P 0 (cos θ) = P (cos θ) = cos θ P 2 (cos θ) = 2 (3 cos2 θ ) P 3 (cos θ) = 2 (5 cos3 θ 3 cos θ) (Rodrigues) P n (cos θ) = d n [ (cos 2 θ ) ] n 2 n n! d(cos θ) n 3.5 ε E 0 R 0 E 0 ϕ φ ϕ ϕ 2 ϕ (r, θ) = n (a n r n + b n r n+ )P n(cos θ) (7) ϕ 2 (r, θ) = n (c n r n + d n r n+ )P n(cos θ) (8) a n b n c n d n 2

13 () r E E 0 ϕ r E r = E 0 r cos θ = E 0 rp (cos θ) b n a = E 0, a n = 0 (n ) (2) r = 0 ϕ 2 d n = 0 (3) r = r 0 ϕ = ϕ 2 (9) ϕ ε 0 r = ε ϕ 2 r (0) ϕ ϕ a r 0 P (cos θ) + n b n r n+ 0 P n (cos θ) = n c n r n 0 P n(cos θ) P 0 P b 0 r 0 = c 0 () a r 0 + b r 2 0 = c r 0 (2) P n (n > ) b n r n+ 0 = c n r n 0 b n = r 2n+ 0 c n (3) ϕ ϕ ϕ r = [ ] na n r n ( n + )b n P r n n+2 n (cos θ) ϕ 2 r = n a P (cos θ) n [ ] nc n r n (n + )d n P r n+2 n (cos θ) (n + )b n r n+2 0 P n (cos θ) = ε ε 0 n nc n r n 0 P n (cos θ) P 0 b 0 r 2 0 = ε ε 0 0 b 0 = 0 (4) P a 2b r 3 0 = ε ε 0 c (5) 3

14 P n (n > ) 2 5 (n + )b n r n+2 0 = ε ε 0 nc n r n 0 b n = ε ε 0 n n + r2n+ 0 c n (6) a r 0 + b r 2 0 = c r 0 a 2b r 3 0 = ε ε 0 c b = ε ε 0 ε + 2ε 0 E 0 r 3 0, c = 3ε 0 ε + 2ε 0 E b n = c n = 0, n ϕ = E 0 r cos θ + ε ε 0 ε + 2ε 0 E 0 r 3 0 cos θ r 2 (7) ϕ 2 = 3ε 0 ε + 2ε 0 E 0 r cos θ E = 3ε 0 ε + 2ε 0 E 0 3ε0 ε+2ε 0 < E 0 P = χ e ε 0 E = (ε ε 0 )E = 3(ε ε 0)ε 0 ε + 2ε 0 E 0 p p = P = 4π 3 r3 0 P = (ε ε 0) r 3 0 ε + 2ε 0 E 0 7 p r = ε ε 0 E 0 r0 3 cos θ r 3 ε + 2ε 0 r 2 4

15 3.6 Z α ϕz ( r ϕ ) + 2 ϕ r r r r 2 θ = 0 2 ϕϕ = R(r)Θ(θ) r 2 d2 R dr 2 + r dr dr = ν2 R d 2 Θ dθ 2 + ν2 Θ = 0 ν 0 ϕ ϕ = (A 0 + B 0 ln r)(c 0 + D 0 θ) + ν (A ν r ν + B ν r ν )(C ν cos νθ + D ν sin νθ) ν θ = 0 ϕ = r A 0 C 0 =, B 0 = 0 C ν = 0 (ν 0) r 0 ϕ B 0 = B ν = 0 θ = 2π αϕ = r D 0 = 0 sin ν(2π α) = 0 ν ν n = n 2 α π, (n =, 2 ) ϕ ϕ = + n A n r νn sin ν n θ (8) A r 0 ϕ (8) r n = ϕ = + A r ν sin ν θ 5

16 E r = ϕ r ν A r ν sin ν θ E θ = ϕ θ ν A r ν cos ν θ σ = ε 0 E n = ε 0 E θ (θ = 0) σ = ε 0 E n = ε 0 E θ (θ = 2π α) σ = ε 0 E n ε 0 ν A r ν αν 2 r Page 93 2,3,6,7,8 4. 6

17 4.2 Q ϕ(x, y, z) = [ ] Q x2 + y 2 + (z a) Q 2 x2 + y 2 + (z + a) 2 7 ϕ = const F s = n T = n ε 0 ( 2 E2 I EE ) = ε0 2 E2 n = 2ε 0 σ 2 n 4.3 R 0 a(a > R 0 )Q Q b Q r + Q = 0 r P r r = Q Q = const [x x 0 (b, Q )] 2 + y 2 + z 2 = R 2 (b, Q ) b Q x 0 (b, Q ) = 0 R(b, Q ) = R 0 b = R 0 a R 0 Q = R 0 a Q ϕ = [ Q r R ] 0Q ar ϕ = 0 7

18 σds = ε ϕ n ds S S S ε ϕ R0 ds = n a Q = Q Q Q < Q 4.4 R 0 Q 0 a(a > R 0 ) QQ Q 0 R = 0 Q = Q 0 Q = Q 0 + R 0 a Q [ ϕ = Q r R 0Q ar + Q 0 + R0 R a Q ] Q F = [ ] QQ + QQ a 2 (a b) 2 = [ QQ0 Q2 R0 3(2a2 R0 2) a 2 a 3 (a 2 R0 2)2 ] a R 0 8

19 4.5 R 0 a(a > R 0 ) Q 0 R = 0 Q = Q = R 0 a Q ϕ = [ Q r R 0Q + R ] 0Q ar ar R = R 0 ϕ = Q a a > R r = 0 r = r = r 0 ε ( ϕ n ϕ = ϕ 2 ) = ε 2 ( ϕ n ) 2 ϕ = ϕ 0 ϕ = const ε ϕ n ds = Q σ = ε ϕ n 9

20 4.7 Page 95 9,0,,2,3 (Green) 5. δ δx ρ(x) = δ(x x ) δ(x x ) = 0, x x δ(x x ) dv =, x δ(x) = δ( x), δ(x x ) = δ(x x) δ(r r ) = δ(x x )δ(y y )δ(z z ) δ(r r ) = r δ(r 2 r )δ(φ φ )δ(cos θ cos θ ) 5.2 x G(x, x ) 2 G(x, x ) = ε 0 δ(x x ) (9) x S G S = 0 (20) G S = G(x, x 2 ) = G(x 2, x ) 20

21 S G n = S ε 0 S (2) n S92 x x 2 x r = 4πδ(r) G(x, x ) = r = (x x ) 2 + (y y ) 2 + (z z ) 2 G(x, x ) = (x x ) 2 + (y y ) 2 + (z z ) 2 (x x ) 2 + (y y ) 2 + (z + z ) 2 x R2 0 x R0 R R R r = x x = R 2 + R 2 2RR cos α r = x R2 0 x R 2 = R 2 R 2 + R 4 R 0 2R2 0 RR cos α G(x, x ) = [ r R ] 0 = [ R r R2 + R 2 2RR cos α ( RR R 0 ) 2 + R0 2 2RR cos α 2

22 5.4 ϕ(x) ψ(x) ( ψ 2 ϕ ϕ 2 ψ ) d = S ( ψ ϕ n ϕ ψ n ) ds (ψ ϕ) = ψ ϕ + ψ 2 ϕ (22) (ϕ ψ) = ϕ ψ + ϕ 2 ψ (23) ψ 2 ϕ ϕ 2 ψ = (ψ ϕ ϕ ψ) ( ψ 2 ϕ ϕ 2 ψ ) d = 5.5 ϕ S 2 ϕ = ρ(x) ε 0 ( ψ ϕ ) n ϕ ψ ds n ψ G(x, x ) 2 G(x, x ) = ε 0 δ(x x ) xx x x [ G(x, x) 2 ϕ(x ) ϕ(x ) 2 G(x, x) ] d = = S ϕ(x) = [ G(x, x) ρ(x ) + ϕ(x ) ] δ(x x) d ε 0 ε 0 [ G(x, x) ϕ(x ) ϕ(x ) ] n n G(x, x) ds G(x, x)ρ(x )d + ε 0 ϕ(x ) ] n G(x, x) ds G(x, x) x S = 0 S [ G(x, x) ϕ(x ) n G(x, x) ϕ S ϕ(x) ϕ(x) = G(x, x)ρ(x )d ε 0 ϕ(x ) n G(x, x)ds (24) S 22

23 5.6 (ρ(x ) = 0) 5.7 a 0 0 x (R cos φ, R sin φ, z) x (R cos φ, R sin φ, z )x (R cos φ, R sin φ, z ) G(x, x ) = [ r ] r r = x x = R 2 + R 2 2RR cos(φ φ ) + (z z ) 2 r = x x = R 2 + R 2 2RR cos(φ φ ) + (z + z ) 2 [ G(x, x ) = R2 + R 2 2RR cos(φ φ ) + (z z ) 2 ] R2 + R 2 2RR cos(φ φ ) + (z + z ) 2 ϕ(x) = ε 0 S ϕ(x ) n G(x, x)ds z G = G n z = 2πε 0 z =0 z [R 2 + R 2 + z 2 2RR cos(φ φ )] 3/2 23

24 S ε 0 ϕ(x ) G n ds = 0z 2π a 0 R dr 2π 0 [ dφ + R 2 2RR cos(φ φ ] 3 2 ) (R 2 + z 2 ) 3/2 R 2 + z 2 R 2 + z 2 a 2 ϕ(x) = 0a 2 [ z 3a 2 2 (R 2 + z 2 ) 3/2 4(R 2 + z 2 ) + 5R2 a 2 ] 8(R 2 + z 2 ) + 2 a2 R 2 +z 2 ϕ(x) 0a z (R 2 + z 2 ) = 0a 2 z 3/2 2 r 3 p r r 3 (δ(x)) Page 96 8,9 6. ϕ r = 0ρ(x ) ϕ(x) = r x x ρ(x ) r d (25) r x (25) f(x 0 + h, y 0 + k) f(x 0, y 0 ) + n m= 24 ( h m! x + k ) m f(x, y) y x=x 0 y=y 0 (26)

25 r 0 = x 2 + y 2 + z 2 r (26) r x r = x x = (x x ) 2 + (y y ) 2 + (z z ) 2 r = r 0 + n ( x m! x y y ) m z z r m= = r 0 x r 0 + 2! (x ) 2 r 0 + = r 0 x r 0 + 2! i,j x 2 i x j x i x j r 0 + r=r0 (27) (27) (25) ρ(x ) ϕ(x) = x + x 2 i r 0 r 0 2! x j + x i x j r 0 i,j d (28) ρ(x) r ρ(x ) r(x ) ρ(x ) x Q = ρ(x )d D = (28) ϕ(x) = p = x ρ(x )d 3x x ρ(x )d, D ij = i,j 3x i x j ρ(x )d Q p + 2 D ij + r 0 r 0 6 x i x j r 0 p D 6.2 (29) (29) Q = ρ(x )d 25

26 r 0 ϕ (0) = Q r 0 ϕ () ϕ (2) 6.3 ϕ () = p r 0 ϕ (2) = D : 6 r 0 p = x ρ(x )d r 2 0 ϕ () = p r 0 = p r 0 r 3 0 p = x [ Qδ(x x + ) Qδ(x x )] d = Q(x + x ) = Ql ϕ = Q ( Q ( r + r r 0 e r x + ) = Q ( ) r 0 x + r 0 x ) Q e r l r 0 e r x r0 2 ϕ Q l cos θ = Ql z r0 2 r0 3 ϕ (p z z ) r 0 = Ql z r 0 r r 2 26

27 6.4 D = 3x x ρ(x )d, D ij = 3x ix jρ(x )d r 3 0 ϕ (2) = D : 6 r 0 r r 2 D 33 = 3z z ρ(x )d = 3z z [Qδ(z b) + Qδ(z + b) Qδ(z a) Qδ(z + a)] d D 33 = 6Q(b 2 a 2 ) = 6Q(b a)(b + a) = 6pl ϕ p z = p z + p r + z ( ) r + r r p z = pl 2 = 2 z r 0 6 D z r 0 ( l z ) zz D x D 22 y D 2 xy 6.5 r 0 D = 3x x ρ(x )d, D ij = 3x ix jρ(x )d 27

28 2 r 0 = 0, r 0 0 ( ) 2 2 x y + 2 = 2 z r 0 i,j δ ij 2 x i x j r 0 = 0 6 r 2 ρ(x )d ϕ (2) = [ ] 3x i 6 x j ρ(x )d 2 x i x j r 0 = 6 [ (3x i x j r 2 δ ij )ρ(x )d ] 2 x i x j r 0 D ij = (3x i x j r 2 δ ij )ρ(x )d ϕ (2) = 6 i,j D ij 2 x i x j r 0 D + D 22 + D 33 = 3x 2 + 3y 2 + 3z 2 3r 2 = 0 D = (3x x r 2 I )ρ(x )d 6.6 ρ(x) ϕ e (x) W = ρϕ e d (30) ϕ e (x) ϕ e (x) = ϕ e (0) + 3 i= x i ϕ e (0) x i + 2! 3 i,j= x i x j 2 ϕ e (0) x i x j + (3) ρ(x) 28

29 (3) (30) W = Qϕ e (0) + p ϕ e (0) + 6 D : ϕe (0) + Q p D W (0) = Qϕ e (0) W () = p ϕ e (0) = p E e (0) F = W () = (p E e ) = p ( E e ) + (p )E e = (p )E e E p () W L θ = θ = θ (pe e cos θ) = pe e sin θ L = p E e W (2) = D : ϕe (0) = D : Ee (0) R n n 29

30 6.8 r 0 r 2 0 r 3 0 Qϕ e (0) p E e (0) 6 D : E e (0) Page 96 4,5,6 30

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84 / ! / ! 9 9 9!! 9 : ; < = 1 //< & >!! ? : ; <. 1 //< &! Α

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