6. lectosttic ou-lue Polems
6. Poisso s Lplce s qutio D v (6.) (6.) ( ) v v (Poisso's equtio) (6.3) (6.4) v (Lplce's equtio) (6.5)
(6.8) (6.7) (6.6) cooites o pheicl Cliicl, Lplce's equtio i Ctesi,
6.3 Uiqueess Theoem If solutio to Lplce s equtio c e fou tht stisfies the ou coitios, the the solutio is uique. 3
6.4 Geel Poceue fo olvig Poisso s o Lplce qutio 6.4 ectio 에서문제풀이할때사용하는방정식. D I D J R / I 4
X 6. 그림 6. 과같이 HD Pump 에서 ρ =5 mc/m 3 인 Chge 로채워져있고 = k 일때 Pump 의압력을구하라. Figue 6. electohomic pump. 5
6 ) ( ) ( Figue 6. electohomic pump.
7 Figue 6. electohomic pump. v 3 F F v F
8 3 3 v 3 m N / 55 5 F p F F v F Figue 6. electohomic pump.
X 6. 그림 () 에서전극사이의전장을구하라. ε ε ρ s 광전도체 빛 재결합 () () (c) 복사기원리. * 광전도체 : 빛을받으면전기전도성이생기는물질. 9
v D D ) ( D D ) ( ) ( ) ( ) ( ) ( ) ( ρ s ε ε
t / ) ( / t /,,, ukows : 4 & eqs 4 of set cosistet elf ρ s ε ε
X 6.3 φ= 평면의전위가 volt, φ=π/6 평면의전위가 volt 일때작은간격으로분리된두면사이의전위와전장을구하라. 6 (3.9) 6 6/ / 6 / 6) ( ) ( (6.7) φ = π/6 volt volt
3 (3.3) (3.39) (3.8)
4 (3.3) (6.8) X 6.4 θ=π/ 면의전위가 volt, θ=π/6 면의전위가 5 volt 일때작은간격으로분리된두면사이의전위와전장을구하라. θ θ Gp
cos cos( / )( / ) cos ( / )( / ) / cos( / ) (/ )sec ( / ) t( / ) t( / ) t( / ) l t( / ) t cot sec csc ( ) cos cos () cos cos( ) coscos cos() cos t t t( ) t t cos cos t sec 5
6 ) / l t( ) / t( ) / t( /) t( 5/ l ) / t( /) t( 5/ l /) l t( 5 ) / l t( 5 / 6) ( /) ( ) / l t(
7 ) / t( /) t( / l 5 ) / t( ) / t( /) t( 5/ l )) / l(t( ) / t( /) t( 5/ l )) / l(t( ) / t( ) / t( /) t( 5/ l ) / t( /) t( 5/ l (3.3) (6.8)
X 6.5 길이가무한대인사각형관내의전위분포를구하라. (, ) X()Y() XY YX X Y X Y X X Y Y (, ) X()Y() X() (, ) X()Y() X() (,) X()Y() Y() (,) X()Y() 8
Cse : X X X X X() X() X() : Meigless 9
X X X X X X XX X X X X X X X X X X X X : Cse i i cos icos i ) (i X X X X X X
참고 (cos t i t) t icos t t (cos t i t) (cos t i t)it (cos t i t) it cos t i t l l(cos t i t) it C cos t i t e e cos t i t e itc it Ce it e C it
) ( ) ( ) ( ) ( ) ( i e e e e e e X e Xe Xe e X X e X X X X l i X ) i l(cos le i ) ( i
3 Meigless : X() h ) X( ) X( h cosh e e h e e cosh e e X
Cse X X X X() C X( ) g g X( ) g X Y Y () g h 3: X g e cos g Y i h C e i,,, 3,... 여러해의합 (, ) g (, ) X X Y Y - β= 는해의더하기에서의미가없음. - 이 Mius 인경우 Plus 인경우의 e 함수의상수배이기때문에해를고려할필요가없음. h c h h 4
5 짝수홀수,, h 4 c..,4,6,. m,..,3,5,. m, m 4 ) cosm ( m m h c ) cosm ( m m h c m m m h c m h c m (,) h c (,) h c ) (, m m m,3,5,.. h h 4 ) (,
cos( ) coscos cos( ) coscos cos( ) cos( ) (cse : m ) m (m ) (m ) (m ) (m ) m (m ) (m ) cos cos 계산 cos t cot sec csc ( ) cos cos () cos cos( ) coscos cos() cos t t t( ) t t cos cos t sec 6
7 t cot cos csc sec sec t cos cos t t t t ) t( cos ) cos( cos cos ) cos( cos ) ( cos cos ) ( m m m m cos m ) m : (cse m m 계산 cos ) cos(
X 6.6 6.5 에서 가다음값을가질때전위분포를구하라. () (, ) (,) 3, c, h 3 c, 3 3 3 c3 h c3 /h 3 h 3 (, ) 3 h c (6.5.9) h (, ) 4,3,5,.. h h 8
9 5 h 5 h 5 h h ) (, 5 h c 5 h c 5 h c h c 5,, c h c 5 (,) (6.5.9) h c ) (,,, 5 () 5 5,3,5,.. h h 4 ) (,
3 X 6.7 전하가없는영역에서 (ρ, φ, ) 의분리된미분방정식을구하라. h c cosh c Z (6.7.7) Z Z Z Z Z Z R R Z R RZ R Z )Z() ( ) R( ft essel () : J R (6.7.) )R ( R R c cos c (6.7.) R R R R R R 4 3
6.5 Resistce Cpcitce R R l I (vli ol if the couctivit is uifom) L L (6.6) J. Choose suitle cooite sstem. ssume o s the potetil iffeece etwee coucto temils 3. olve Lplce s qutio to oti, the I i q. (6.6) 4. Oti R s o /I L (6.7) 3
C L (6.8). ssumig etemiig i tems of (Guss s Lw): ectio -C ) Choose suitle cooite sstem ) Let the two couctig pltes c chges + c) Detemie (Coulom s o Guss s Lw), the ( fuctio of ) ) Oti C = /. ssumig etemiig i tems of (Lplce s q.) : 6./4 ) Choose suitle cooite sstem ) ssume s the potetil iffeece etwee coucto temils c) olve Lplce s qutio to oti, the ) Oti s fuctio of e) Oti C = / 3
Figue 6. two-coucto cpcito. 33
34. Pllel-Plte Cpcito (6.) C (6.) L (6.) ) ( (6.9) Dielectic ε Plte e
Plte e Dielectic ε () Figue 6.3 () Pllel-plte cpcito. () Figig effect ue to pllel-plte cpcito. 35
36 (6.4) C C W C / C () v v W (6.3) C C C C v v
37. Coil Cpcito ρ L Dielectic ε (6.8) l L C (6.7) l L (6.7) ) ( L L (6.6) L (6.5) L
C. pheicl Cpcito 4 (6.9) Dielectic ε 4 L 4 4 ( (6.3) ) (6.3) C 4 (6.3) 38
39 (6.35) RC L C (6.7) L I R (6.6) (6.34) C C C (6.33) C C C C C C C C C C () C C ()
4 F 7.7 4 36 4 (6.39) 4 R, 4 C (6.38) 4 R, 4 C (6.37) L l R, l L C (6.36) C R RC R, C 4 7 9 지구
4 / ) l( l( / ) / l( / ) l / l( / ) l ) ( l ) ( l X 6.8 () 9 o 로구부러진도체의 ρ= 와 ρ= 사이의저항을구하라. t 전도율 =σ
4 t l( / ) I R l( / ) t l( / ) J I J l( / ) / ) l( l( / ) / t (3.3) (3.9) (3.8) t 전도율 =σ
43 X 6.8 () 9 o 로구부러진도체의 = 와 =t 사이의저항을구하라. ) ( 4t I R 4t ) / ( t J I t J t t t / t) ( ) ( / t 전도율 =σ
다른방법 R L t ( 4 4t ( ) ) 44
X 6.9 무한동축 Cle 의단위길이당 ρ=, ρ= 사이의저항과단위길이당 Couctce 를구하라. 와 사이는도전율 σ 인물질로채워져있다. 6.8과유사 l ( ) l / l( / ) ( ) l l / l( / ) l( / ) l( / ) 45
l( / ) l( / ) J I J l( / ) L l( / ) L l( / ) R G l( / ) I L RL l( / ) L L l( / ) l( / ) 46
47 / / ) ( ) ( (3.6) X 6. 도체이중구각에서 (=)= volt, (=)= volt 일때, 분포를구하라. 사이물질은 ε =.5 일때총전하와 Cpcitce 를구하라. = cm =3 cm
48 4 C 4 / 반지름 인축구공껍데기적분 (3.3) (3.9) (3.8) = cm =3 cm
ρ= θ θ θφ θ φ φ θ 는 Zeith gle 이고 φ 는 imuthl gle. 49
X 6. 를가정하고 를유도하여평면사이의 C=ε/ 를유도하라. 그림의상판을 하판을 volt 라고가정. volt. 라고가정 D C 5
X 6. Cpcitce 를구하라. =5mm, =3cm. () Cpcito의직렬연결 C / C / C C C C C 4 () 6 / / () Cpcito의병렬연결 C C C ( ) w/ w/ () 5
X 6.3 =cm, =.5cm 이고극판사이에 ε =(+ρ)/ρ 인유전체가있다. ρ 는 cm 단위이다. 단위길이당 Cpcitce 를구하라. C L L L L L l L l L 적분을위한 Close ufce L 방향 L L 5
6.6 Metho of Imge The Imge Theo sttes tht give chge cofigutio ove ifiite goue pefect couctig ple m e eplce the chge cofigutio itself, its imge, equipotetil sufce i the plce of the couctig ple. Figue 6. Imge sstem: () chge cofigutios ove pefectl couctig ple, () imge cofigutio with the couctig ple eplce equipotetil sufce. 53
=- (= )= (= )= v 4 k ( ) v k( ) ( ) 3 4 ( ) δ(- ) /Δ ( k f () ( Δ ) ( k +Δ ) ( ) f ( ) k ) ( k ) 54
(= )= (= )= =- v v 4 k ( ) k( ) k( ) ( ) ( ) 3 4 ( ) 4 ( ) k ( 4 ) 3 (= )= (= )= δ(- ) /Δ Metl Δ +Δ 55
(= )= (= )= =- (= )= (= )= Metl Metl 56
57. poit Chge ove Goue Couctig Ple (6.45) ] h) ( [ ] h) ( [ 4 4 4 (6.44) ] h) ( [ h) ( ] h) ( [ h) ( 4 (6.43) h), (, h) (,,,) (, (6.4) h), (, (,, h),) (, (6.4) 4 4 (6.4) / / 3 / 3 / 3 3 o
Figue 6. () Poit chge goue couctig ple. () Imge cofigutio fiel lies. 58
59 (6.49) ] h [ h ) ( ] h [ h (6.48) ] h [ h (6.47) ] h [ h (6.46) ] h [ h D ] h) ( [ h) ( ] h) ( [ h) ( 4 / 3/ i 3/ i 3/ i 3/ 3/ 3/
φ ρ ρ ρφ 6
6. Lie Chge ove Goue Couctig Ple P(,,) (,,h) (,,-h) h -h (6.55) l l l (6.54) h) ( h) ( h) ( h) ( (6.53) h) (,, h), (,,) (, (6.5) h) (,,,h) (,,) (, (6.5) (6.5) L L L L L L
6 (6.59) hsec h t h h (6.58) (chge pe uit legth) h h (6.57) ) h ( h D (6.56) h) ( h) ( l l (6.55) h) ( h) ( h) ( h) ( (6.54) L / / L L i L / L L L
X 6.4 그림 () 와같이두개의반무한평면사이에점전하 가 (,,) 에있다. 전위를구하고 에작용하는힘을구하라. 4 3 4 [( ) [( ) [( ) [( ) 3 4 ( ) ( ) ( ) ( ) ] ] ] ] () 3 () 4 F F 4 () 6 F 3 F ( 4 4 ) 3/ [() () ( 3/ ] 4 () ) 3/ 63
Nume of imge N o 36 Figue 6.5 Poit chge etwee two semi-ifiite couctig wlls iclie t 6 to ech othe. 64