Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e

7 -a 7 -a February 4, 2007 1. 2. 3. 4. 1. 2. 3. 1

Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e z (2) Jacobian J = r 2 sin θ dv = dxdydz = J drdθdφ = r 2 sin θdrdθdφ x 2 + y 2 = r 2 sin θ Gauss Q = r 3 sin 2 ddφ = r 3 dr sin 2 θdθ dφ = r4 4 ɛ 0 E ds = 4πɛr 2 E ( π 2 + 1 ) 2π = π (2π + 1) r4 4 8 E = 2π + 1 32ɛ 0 r 2 e r 2

A = G M e r 2 r r M e, R e A(r, θ, φ) = 1 r 2 (r 2 A r ) r + 1 (sin θa θ ) + 1 (A φ ) r sin θ θ r sin θ φ r M = (r/r e ) 3 M e A = G r 3 R 3 e M e r 2 e r = G rm e Re 3 e r A = GM e r 3 Rer 3 2 r = 3GM e Re 3 G = (4πτ) 1 3GM e R 3 e = 3M e 4πR 3 eτ = M e 4 3 πr3 eτ = ρ e τ ρ 0 τ ɛ 1 Gauss r = R e τ A ds = 4πτRe 2 M e 4πτRe 2 = M e 1 (graviton) 3

Ampère Ampère ( ) E B ds = µ 0 i + ɛ 0 ds t (1) (2)i. a ii. (1) xy x x z y = B(z) Ampère a ABCD{ AB CD y BC DA z AB CD 0 Ampère BC DA {B(z 1 ) B(z 2 )}a (B(z 1 ) B(z 2 ))a = 0 B(z 1 ) = B(z 2 ) = B i (B(z 1 ) B(z 2 ))a = µ 0 ia 2aB = µ 0 ia B = µ 0 i/2 (2)i. r r B(r) I Ampère 2πrB(r) = µ 0 I... B(r) = µ 0I 2πr { µ0i 2πr 0 B = e θ r a 0 r < a 2 ii. 0 I/r 2... B = { µ0 I 2πr e θ µ 0 I 2πr e 3 θ r a r < a 2 4

S h dq V dq h S V (1) (2) Q (3) (2) (1) K = dqv (2) E Gauss 3... E ds = ES = Q E = Q S e z Φ V Φ = E ds = Qh S = V... Q = SV h (3) q ( K = V qh ) dq S 3 5

q = Q U lost = Q 0 ( V qh ) dq = QV Q2 h S 2S = 1 2 QV W = QV U c U c = QV 1 2 QV = 1 2 QV = 1 2 CV 2 4 4 C Q/V 6

a b(b > a) +q, q, a < r < b r, S E nda = 4πr 2 E, q V ρ ɛ 0 dv = q ɛ 0 2 4πr 2 E = q ɛ 0...E = q 4πr 2 ɛ 0,ab V, b, C V = = = V = b q a 4πr 2 dr ɛ 0 [ ] b q 1 4πɛ 0 r a ( q 1 4πɛ 0 a 1 ) b q 4πɛ 0 ( ) 1 a C = q V = 4πɛ 0 a 7

Q a ɛ C = 4πɛ 0 a C = 4πɛa U = Q2 2C, U = Q2 2C ( U U = Q2 2C Q2 2C = Q2 1 1 ) 8πa ɛ 0 ɛ 8

d S ±Q ɛ F = 1 2 ɛe2 S E = Q ɛs F = 1 2 ɛ ( Q ɛs ) 2 S = 1 Q 2 2 ɛs ɛ ɛ 0 F 0 ɛ > ɛ 0 F < F 0 Q 2 F 0 = 1 2 ɛ 0 S 9

6.4 10 6 [m] C = 4πɛ 0 a = 4π (8.854 10 12 ) (6.4 10 6 ) = 7.1 10 4 [F] 10

ρ(r) a b c 4 3 πa3 ρ(r) 4πa e 0 k 3 ρ(r)e 0 0 4πa v B mcv B = k 3 e 0 0 ρ(r) 3c 2 πb 2 I = k 4πa 3 e 0 0 3c 2 4πa 3 e 0 v B = k 0 3mc 3 ρ(r) 3mc 3 e 0 πb 2 2πa = 2e2 0 k 0a 3 3πmab 2 c = 2e2 3 0 k 0a 2 3πmb 2 c 3 H = I 4πc 2 ds = I 2c = e2 0k 0 a 2 3πmb 2 c 4 e 2 0k 0 a 2 3πmb 2 c 4 I 11

r 1, r 2,..., r n 1 ɛ 1, ɛ 2,..., ɛ n a Q r ɛ i A a r D r i 1 < r < r i D(r) = E i = Q 4πr 2 Q 4πɛ i r 2 r V = Edr ( r = E i dr + r i = Q ( 1 4πɛ i r 1 r i ri E i+1 dr + + r i+1 ( 1 1 r i r i+1 ) + Q 4πɛ i+1 rn 1 E n dr ) ) + + Q 4πɛ n 1 ( 1 1 ) + Q 1 r n 1 r n 4πɛ n r n 12

a a b ɛ 0 Q O r(> a) E D = Q 4πr 2 E = D (b < r) D (a < r < b) ɛ 0 ɛ V D D V = dr + ɛ 0 ɛ sdr = Q ( 1 4πɛ 0 b + s ( 1 ɛ a 1 )) b C C = Q V = 4πɛ ( 0 1 a ) 1 b 1 b + s ɛ = 4πsɛ 0 ɛ ab (1 ɛs) b a 13

z ν a n xy V 0 B 0 Φ Φ nb(πa 2 ) B = B 0 sin 2πνt V i V i = dφ dt = 2nπ2 a 2 νb 0 cos 2πνt V 0 V 0 = 2nπ 2 a 2 νb 0 B 0 = V 0 2nπ 2 a 2 ν 14

ρ A A B A B c r E A ρ B ρ B r B r r = c + r A r E A B r E B E A = 1 4 4πɛ 0 3 π r 3 ρ r r 3 = ρ r 3ɛ 0 r E E A = 1 4πɛ 0 4 3 π r 3 ( ρ) r r 3 = ρ 3ɛ 0 r E = E A + E B = ρ 3ɛ 0 (r r ) = ρ 3ɛ 0 c 15

q (1) E = 4πɛ 0 r 2 E(r, θ, z) (2) (1) ( ) (1) z E E = cos θ = λ z 4πɛ 0 (r 2 + z 2 ) r r, sin θ = z 2 +z 2 r 2 +z 2 rλ z E r = E cos θ = 4πɛ 0 (r 2 + z 2 ) 3 2 zλ z E z = E sin θ = 4πɛ 0 (r 2 + z 2 ) 3 2 E r = λ 4πɛ 0 A B r (r 2 + z 2 ) 3 2 dz, E z = λ 4πɛ 0 A B z dz (r 2 + z 2 ) 3 2 z = r tan θ OA = r tan α, OB = r tan β r 2 + z 2 = r 2 (1 + tan 2 θ) = r2 cos 2, dz = r θ cos 2 θ dθ E r = λ α r 4πɛ 0 β ( r 2 cos 2 θ ) 3 2 r cos 2 θ dθ = λ 1 4πɛ 0 r 16 α β cos θdθ

E z = λ 4πɛ 0 α β = λ 1 (sin α sin β) 4πɛ 0 r r tan θ ( r2 cos 2 θ ) 3 2 r cos 2 θ dθ = = λ 1 (cos β cos α) 4πɛ 0 r α π 2, β π 2 λ 1 4πɛ 0 r α β sin θdθ E r λ 2πɛ 0 r, E z 0 E = λ 2πɛ 0 r e r (2) ɛ 0 E ds = λ 1 S E//S E(r) ds = λ S ɛ 0 E(r) 2πr = λ ɛ 0 E(r) = E = λ 2πɛ 0 r λ 2πɛ 0 r e r 17

-e -e exωb E i : A B ee i = exωb E i = xωb V = = = C 0 i ds l 0 l 0 i dx xωbdx ( x 2 = ωb 2 = 1 2 Bl2 ω ) l 0 18

2 ( j) 2 d l a B dr = µ 0 jl C 2lB = µ 0 jl B = µ 0j 2 2 B = µ 0 j F = B Il = B jal = 1 2 µ 0jS ( )(S = al) t d V V = dφ dt = Bl d t (E = B d x ) j j a El = jes t d ( ) (B = µ 0 j) u = 1 2 µ 0 B 2 U = jes t 1 2 BjS d = (jb 12 ) Bj S d = 1 2 BjS d = 1 2 µ 0j 2 S d = 1 B 2 S d 2 µ 0 19

1[m] 1. 5[m] 1 +20[C] 1 20[C] 2. 40[m] 4 1[C/m] 1. 4 2 0[V/m] 2. a = 20[m] λ h = 1[m] R y x[m] dx[m] λx[c] de y = λdx cos θ 4πɛ 0 (x 2 + R 2 ) tan θ = x R x2 + R 2 = R 2 sec 2 θ[m] dx = R sec 2 θdθ E = E y = λ θ0 cos θdθ = λ sin θ 0 4πɛ 0 R θ 0 2πɛ 0 R = R = h 2 + a 2 4 E all = 1 πɛ 0 200[V/m] 2ahλ (h 2 + a 2 ) h 2 + 2a 2 λ ah 2πɛ 0 R R2 + a 2 20

3 m q 3 3 a t = 0 t = 3 v 3 3 3 5 3a = d W 3q 2 4πɛ 0 d q2 4πɛ 0d 1 2 mv2 = v = q 2 4πɛ 0 3a q 2 3πɛ0 am 5 21

r A, r B 2 A, B q A, q B 1. A, B 2. AB E A, E B 1. ρ A, ρ B ρ A = q A 4, ρ B = q B 3 πr3 4 A 3 πr3 B E A, E B ɛ 0 ɛ 0 E A ds = ɛ 0 E A E B ds = ɛ 0 E B ɛ 0 E A ds = ρ A 4 3 πr3 A ds ds ɛ 0 E A 4πr 2 A = ρ A 4 3 πr3 A V A = r A q A E A = 4πɛ 0 ra 2 q B E B = 4πɛ 0 rb 2 E A ds = E A dr = q A r A 4πɛ 0 r A q B V B = 4πɛ 0 r B 2. AB A, B q A 4πɛ 0 r A = q B 4πɛ 0 r B q A, q B q A, q B q A + q B = q A + q B r A q A = (q A + q B ) r A + r B q B r B = (q A + q B ) r A + r B 22

E A = q A q A + q B = 4πɛ 0 r A 4πɛ 0 r A (r A + r B ) E B = q B q A + q B = 4πɛ 0 r B 4πɛ 0 r B (r A + r B ) 23

ρ EdS = Q E ds A ɛ 0 EdS = 2EA = Q ɛ 0 2EA = Aρ ɛ 0 E = ρ 2ɛ 0 24

Q a x ds dq P de 1 E y 0 x dq de = 4πɛ 0 (x 2 + a 2 ), cos θ = x x2 + a 2 xdq de x = de cos θ = 4πɛ 0 (x 2 + a 2 ) 3 2 E x = = 1 4πɛ 0 1 xdq 4πɛ 0 (x 2 + a 2 ) 3 2 xq (x 2 + a 2 ) 3 2 25

a[m] n 1 l[m] n 2 L 1, L 2 L 1 I = I 0 sin ωt L 2 L 1 I = I 0 sin ωt Φ n2 = B ds L 1 L 1 Φ n1 = CDEF B ds = = B CD D C B ds + DE ds = µ 0 n 1 l I 2πa2 B ds + B ds + B ds EF F C CD 1 L 1 Φ n1 = 2πa2 µ 0 n 1 I 0 sin ωt l L 2 1 L 2 Φ n1 V (i) 2 = Φ n 2 t = 2πa2 µ 0 n 1 n 2 I 0 sin ωt l = 2πωa2 µ 0 n 1 n 2 I 0 cos ωt l 2πωa 2 µ 0 n 1 n 2 I 0 l 26

2450MHz 141V( 50Hz) S = E H ( ) 2 = 1.41 V V max ω V = V max sin ωt ω = 2πf V = V max sin 2πft D V = D 0 Edx = ED ( ) E = V D = V max sin 2πft D C ( ) E H ds = j + ɛ 0 ds t (j: ) E H ds = ɛ 0 t ds C H = Sɛ 0V max 2πf cos 2πft D C dmathbfs S = E H C = E H( E H) = V max sin 2πft D S = πfɛ 0SV 2 max sin 4πft D 2 C ds S = 1 2 πfɛ 0 SV 2 max D 2 C ds = 1.71 10 17 [J/s m 2 ] S Sɛ 0V max 2πf cos 2πft D C ds 27