Ejercicios Resueltos Electricidad y Magnetismo (Garrido - Narrias)

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@' 7-,05' *!0.'$ N~ ", 4-,(, ,"+0%#%0 -"'*(! "-" +!.4!*,*&," 0,+&'*1-$'0," +!.!? ~ = N cos(π/3)î + N sin(π/3)ˆj N

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B

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X

kq 2 =0 R2

(1)

FY = N sin(π/3) − mg = 0

(2)

FX = N cos(π/3) −

X

N=

mg sin(π/3)

@**(,1%+%.3# *&/% *A,$*&2#. *. 7

mg kq 2 cos(π/3) − 2 = 0 sin(π/3) R ;*&,*C%.3# q

q = ±R

r

mg cot(π/3) k

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=⇒ x0 = q ·

s

k mg

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F =m

kq 2 kq 2 kq 2 x −2 ∼ x d2 x −mg + = −mg + = −mg + · (1 + ) · (1 − 2 ) = dt2 (x0 + x)2 x0 2 x0 x0 2 x0

K"/' &" LM= #"%")'( 0," mg =

kq 2 x0 2

=⇒ m =⇒ K"/' &" LM=A mx0 2 =

d2 x ∼ −2kq 2 x = dt2 x0 3

2kq 2 x d2 x + · =0 dt2 mx0 2 x0

kq 2 g

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q F~2 = k 2 x ˆ L 32 A.,(B2 (,%.#)2"), ,%

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r L L √ = 2π =⇒ r = sen( π6 ) sen( 3 ) 3 K$( )2")$; 2# *0.2#2( #$%
%$√

3·k

qQ q2 1 = k L 2 =⇒ Q = √ · q 2 L ( √3 ) 3

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kqQ

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=

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· (Rcos(θ)ˆ x + Rsen(θ)ˆ y − aˆ z + Rcos(θ)ˆ x + Rsen(θ)ˆ y + aˆ z) · (cos(θ)ˆ x + sen(θ)ˆ y)

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=⇒

1

df (R) (R2 + a2 ) 2 − 3R2 (R2 + a2 ) 2 = =0 dR (R2 + a2 )3

1 1 =⇒ (R2 + a2 ) 2 · ((R2 + a2 ) − 3R2 )) = 0 =⇒ R = √ · a 2

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T~ = −T sin(θ)î + T cos(θ)ˆj

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X

FX = qE − T sin(θ) = 0

(1)

X

FY = T cos(θ) − mg = 0

(2)

T = ?''+(,%.%0*# '0 < >7

qE −

mg cos(θ)

mg sin(θ) = 0 cos(θ)

;' '&1#@ $'&),1% 5)'7

q=

mg tan(θ) E

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q(t) =

mgθ(t) E

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dq mg dθ = dt E dt F)'E#@ /#+#

dq dt

= α@ *'&('G%0*#

dθ dt

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αE dθ = dt mg

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k(−q)Q(xî + d/2ˆj) F~1 = l3

0'$+*$" 1% F~2/

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(1)

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π 2π = T = w 2

s

md3 kqQ

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E=

kq d2 + r 2

! "#$%&'( $' )&*+, -$. /0$ cos(θ) =

Ey = (

√ r 1 d2 +r2

d2

2$ $'34 %45$.4 3$5$%6' /0$

r kq kqr ·√ )ˆj = 2 2 2 2 +r (d + r2 )3/2 d +r

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~y = − E

(d2

kqr ˆj + r2 )3/2

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~ =E ~y + E ~y + E ~1 E ?$$%9,4@45#6 ,6' -4,6.$' A4 6:3$5+#6'

2kqr ~ = ( 2kq − )ˆj E 2 2 r (d + r2 )3/2

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1 ~ = 2qk( 1 − E 2 r r2 (1 +

d2 32 ) r2

)

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(1 +

d2 − 3 3 d2 2 ≈ (1 − ) ) r2 2 r2

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!

d2 θ dt2 "#$% &'(%')*+ ,)-&.&+')%/ /% 01,&21# &#'.)3). '1214 −2qEaθ = I

d2 θ 2qEaθ + =0 I dt2 51+ /1 '(%/ 6&21# 7(& /% 1.)&+$%')*+ %+8(/%. ,&/ ,)01/1 #&8().9 (+ 216)2)&+$1 %.2*+)'1 #)20/& '1+ -.&'(&+')% %+8(/%.4 r 2qEa w= I : 01. /1 $%+$1 /% -.&'(%+')% #&.94 1 w = f= 2π 2π

r

2qEa I

!

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~ dq(~r − r‘) ~ 3 4πǫ0 |~r − r‘|

Z

Ω

9&$-(*&' (*+&' ()(*+,!' !$ !) !8! <2 %! "() 1&,*( 70! 0$& /0+,( %! x = 0 ( x = L : !) &",& %! x = L+d ( x = 2L+d4 !($ r~1 = x1 xˆ2 r~2 = x2 xˆ2 /&$ 0 ≤ x1 ≤ L : L+d ≤ x2 ≤ 2L+d2 )&' >!/"&,!' 3&'#/#=$ %! /(%( ()(*+,!4 ?!$!*&' 70! )( 10!,6( 70! !8!,/! @ '&+,! 0$ !)!*!$"& %< %! A !'

dF~21 = dq2 ·

Z

Ω1 L

dq1 (r~2 − r~1 ) 4πǫ0 |r~2 − r~1 |3

λ1 dx1 (x2 − x1 )ˆ x 3 4πǫ0 |x2 − x1 | 0 Z L dx1 λ1 = dq2 · ·x ˆ· 2 4πǫ0 0 (x2 − x1 ) L λ1 1 = dq2 · ·x ˆ· 4πǫ0 x2 − x1 = dq2 ·

Z

0

=

=

1 1 λ1 x ˆ · dq2 · ( − ) 4πǫ0 x2 − L x2 1 1 λ1 λ2 x ˆ · dx2 · ( − ) 4πǫ0 x2 − L x2

@B

!"#$%&' () *+$,-.!&,/ 0, '%&'12

!

=⇒ F~21 = = =

Z 2L+d 1 1 λ1 λ2 x ˆ dx2 · − 4πǫ0 x2 − L x2 L+d λ1 λ2 x ˆ 2L+d · (ln(x2 − L) − ln(x2 ))|L+d 4πǫ0 λ1 λ2 x ˆ x2 − L 2L+d · ln( ) 4πǫ0 x2 L+d

λ1 λ2 · ln =⇒ F~21 = 4πǫ0

(L + d)2 d(2L + d)

·x ˆ

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~1 = ( dE !"#$%&'"() ($*($ x = −3R +'*#' x = −R

~ 1 = k(λ1 E ,) -.$ &$*./#'0

Z

−R 1 1 î = (kλ1 ( − ))î dx) x2 x2 −3R ~ 1 = 2kλ1 î E 3R

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2q πR

λ3 = −

2q πR

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(-%/(' .'" #&)! %!"3! &#"@ &/+.$#+#*)#5

dE =

kdq R2

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!

"#$%&#' %()#(*%' %'*+,-,+ %. */&0# *#&#

kλ2 Rdθ kλ2 dθ = R2 R

dE2 =

12%3#4 ./ *#&0#(%()% %( x $%. */&0# dE2 '%+/

dEx2 = dE2 cos(θ) =

kλ2 cos(θ)dθ R

5()%3+/($# $%'$% θ = 0 6/')/ θ = π2 7

Ex2

kλ2 = R

8%*)#+,/.&%()%

Z

π 2

cos(θ)dθ =

0

kλ2 R

~ x2 = kλ2 î E R 9# %' $,:*,. ;%+ ./ *#&0#(%()% %( x $%. */&0# 0+#$2*,$# 0#+ %')% /+*# '%+< ,32/. /. /()%+,#+4 $% %')/ &/(%+/7

!"# $%&

~ x3 = kλ2 î E R ~2 + E ~3 = =', '% )%($+/ >2% E

'!#(# %)&

2kλ2 ˆ R i

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λ4 =

q′ R

12%3#4 0+#*%$,%($# $% ,32/. C#+&/ >2% 0/+/ %. )+#@# AB ;%&#' >2%

~ 4 = (kλ4 E

Z

2R

R

−kλ4 ˆ 1 dx)î = ( )i 2 x 2R

?2&/($# )#$#' .#' */&0#' */.*2./$#' % ,32/./($# / *%+#7

~1 + E ~2 + E ~3 + E ~ 4 = 2kλ1 î + 2kλ2 î + ( −kλ4 )î = 0 E 3R R 2R

!

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"#$%#&'()* λ4 +

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q′ =

10 q 3

!

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)"$*+,-./

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~ r) = =⇒ E(~

Z

+∞ Z +∞

−∞

=

σ 4πǫ0

Z

σdwdv(wˆ x + v yˆ + z zˆ) 3

−∞ 4πǫ0 (w2 +∞ Z +∞

−∞

y1 = y − v 5 @$

+ v2 + z2) 2 dwdv · (z zˆ)

−∞

3

(w2 + v 2 + z 2 ) 2

B&'<#('0,' C#3#(*2 %#(:.* /' <#-.#:$'25

w = rcos(θ)

~ r) = =⇒ E(~ = = = = =

v = rsen(θ) wr wθ cos(θ) −rsen(θ) =r = J = vr vθ sen(θ) rcos(θ)

σ 4πǫ0

Z

0

2π

Z

+∞

rdrdθ · (z zˆ) 3

(r2 + z 2 ) 2 Z +∞ rdr

0

σ · 2πz zˆ 3 4πǫ0 0 (r2 + z 2 ) 2 Z +∞ du σ · z zˆ 3 4ǫ0 z2 (u) 2 2 1 z σ · z zˆ 2u− 2 4ǫ0 +∞ σ z · zˆ 2ǫ0 |z| σ · sign(z)ˆ z 2ǫ0

pero

u = r2 + z 2 → du = 2rdr

!

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!

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σ E~1 (~r) = · sign(x)ˆ x 2ǫ0 > (, 0-3+# (,80/)&0# 1($()-'# +#) (, '&%0# (%

−σ x E~2 (~r) = ·x ˆ sign(x) − √ 2ǫ0 R2 + x2 6

~ = E~1 + E~2 = σ · =⇒ E 2ǫ0

x √ 2 R + x2

x ˆ

;7 H($(3#% 9*( ,- 2*()C- (,80/)&0- 9*( (A+()&3($/- *$ (,(3($/# dx '(, -,-3;)( '(;&'# -,

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!

~0 "#$%& '&% (# &)*+'*& (,-. /$/$ "&) dF~ = dq E =⇒ F~

=

Z

= = =

~ dq E

con dq = λdx Z d+a σ x λdx x ˆ · √ 2ǫ0 R2 + x2 d Z d+a λσ xdx √ · ·x ˆ 2ǫ0 d R2 + x2 d+a λσ p 2 ·x ˆ R + x2 · 2ǫ0 d p λσ p 2 · ˆ R + (d + a)2 − R2 + d2 · x 2ǫ0 d

=

d+a

!

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~ ~0) = E(

Z

kdq(~r − r~′ ) k~r − r~′ k3

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dq = σR2 sin(γ)dγdθ

!"#$%&' () *+$,-.!&,/ 0, '%&'12

!

"#$ #%$# &'(#) (*+*,#- (*%*$,./'$ &#- 0*1%#$*- r~′ 2 ~r3 4&'$',*/%* r~′ = ~0 2 5#$ #%$# &'6 (#7,.$'/(# &' 89:$'; 0*$*,#- <:*

~r = R sin(γ) cos(θ)î + R sin(γ) sin(θ)ˆj + R cos(γ)kˆ =# *- (.>?1.& 0*$ <:* k~r − r~′ k = R @* *-%' ,'/*$' *-5*1.81'/(# &#- $*1#$$.(#- (* &'- 0'$.'+&*- 5'$' <:* $*1#$$'/ *& $*1.5.*/%* -* %.*/* <:*

~ ~0) = E(

Z

2π

0

Z

3π 2 π 2

ˆ kσ(−R sin(γ) cos(θ)î − R sin(γ) sin(θ)ˆj − R cos(γ)k) R2 sin(γ)dγdθ 3 R

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~ ~0) = E(

Z

0

2π

Z

π π 2

ˆ sin(γ)dγdθ = kσ kσ(− cos(γ)k)

Z

0

2π

dθ

Z

π π 2

ˆ sin(γ)dγ = πkσ (− cos(γ)k)

Z

π π 2

ˆ (− sin(2θ)k))dγ

@* *-%' ,'/*$'

π 1 1 ~ E(~0) = πkσ( cos(2θ) )kˆ = πkσ (cos(2π) − cos(π))kˆ = (πkσ)kˆ 2 2 π 2

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!

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dE =

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d cos(θ) = p 2 (x + d2 )

!"#$%&' () *+$,-.!&,/ 0, '%&'12

! "# #$%& '&(#)&

dEy =

kλ1 ddx 3

(x2 + d2 ) 2

*(%#+)&(,- ,#$,# x = −L .&$%& x = L/

Ey = kλ1 d

Z

L x ) = √2kλ1 L √ 3 dx = kλ1 d( 2 2 2 2 2 (d x + d −L d L2 + d2 (x + d ) 2

L

1

−L

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λ2 =

Q L

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*(%#+)&(,- ,#$,# y = L .&$%& y = 2L

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L

−L

E# %8#(# 42#

F-) 0- %&(%-

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1 p

p

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dy

p L2 + y 2 − L ) y

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G##'90&3&(,- 0-$ ;&0-)#$ ,# λ1 > λ2 #( 12(68H( ,# Q/

√ kQ2 5−1 ) F = 2 ln( √ L 2( 2 − 1)

L

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dF~21 =⇒ dx2

=

=

=

=

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Z

+∞

1

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~ = dE

kdq(~r − r~′ ) kdq(−xî + dˆj) = 3 |~r − r~′ |3 (x2 + d2 ) 2

6?#)+D %( 0&($( >*( dq = λdx A ;#) ,# 0+$0#B

~ = dE

kλdx(−xî + dˆj) 3

(x2 + d2 ) 2

E$0(4)+$'# '(%'( −∞ + +∞B

~ = E

Z

+∞

−∞

kλdx(−xî + dˆj) (x2

+

3

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= kλ((−

Z

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−∞

x (x2

+

Z ˆ ) i + ( 3

d2 ) 2

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0

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~ = 2kdλ E

Z

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0

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R

1 3

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=

1 +

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ˆj

x 1

d2 (x2 +d2 ) 2

∞ ˆ 2kλ ˆ ~ = 2kdλ( )j = j E 3 d d2 (x2 + d2 ) 2 x

0

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dF = dqE = λRdθ ·

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!

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F = 4kλ2

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~ · dS ~= q E ǫ0 S

(1)

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Z

Se

~ · dS ~ E

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HI

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~ · dS ~= E

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φe =

1 q q · 2πR2 = 2 4πǫ0 R 2ǫ0

(3)

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φc =

q 2ǫ0

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!

Qint ~ ·n E ˆ dS = ǫ0 Ω

I

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I

C #2I$; I

Ω

~ ·n E ˆ dS =

Z

manto

= E(r)

E(r)ˆ r·n ˆ dS +

Z

Z

tapas

E(r)ˆ r·n ˆ dS

dS

manto

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A!( 0&"0!;

I

~ E(r) =

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σ ǫ0

~0

R r

R r

r
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I

Ω

I

Ω

~ ·n E ˆ dS =

E(r)ˆ r·n ˆ dS = E(r)4πr2 = E(r)4πr2 = E(r)4πr2 = =⇒

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I

~ ·n E ˆ dS =

! "#$ %&'%#( %)')*#+ ,-)

~ E(r) =

(

α 2 ˆ 4ǫ0 r r R 2 2 α R rˆ 4ǫ0 r

r
./#$& ,-) 0#'#0)*#+ )1 0&*2# )130%$40# )' %#5# )1 )+2&04#( 2#5)*#+ 0&10-1&$ 1& )')$67& 2#%)'04&1 &0-*-1&5& )' 1& 54+%$48-049' 5) 0&$6& Θ:

U

= = = = = = =

Z ǫ0 ~ 2 d3 x E 2 R3 Z Z Z ǫ0 2π π +∞ E(r)2 r2 sen(ϕ)drdϕdθ 2 0 0 0 Z 4πǫ0 +∞ E(r)2 r2 dr 2 0 Z R 2 Z +∞ 2 8 α R 4πǫ0 α 6 r dr + dr 2 2 16ǫ20 r2 R 0 16ǫ0 ! R +∞ " r7 1 πα2 8 · − R 8ǫ0 7 0 r R πα2 R7 · + R7 8ǫ0 7 πα2 7 R 7ǫ0

!"#$%&' () &*+ ,* -!%..

!

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*"$+,-./0

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ρ · ~r para |~r| < R 3ǫ0 ρ ~ para |~r − d| ~
8#% 2(-/#)(% ($ 3, &$/()%(--&?$ '( ,*9,% (%+(),% %,/&%+,-($ %&*43/:$(,*($/( ~ < R 6 |~r| < R7 G%C0 =#) (3 =)&$-&=&# '( %4=()=#%&-&?$0 (3 -,*=# (3.-/)&-# ($ 3, |~r − d| &$/()%(--&?$ '( ,*9,% (%+(),% (%

~ = ρ · d~ ~ r) = E~1 (~r) + E~2 (~r) = ρ · ~r − ρ · (~r − d) E(~ 3ǫ0 3ǫ0 3ǫ0 (3 -4,3 (% -#$%/,$/(0 /,3 -#*# %( ;4()C, =)#9,)7

!

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*"$+,-./0 $; M1#/$31' .2 1)&/.# ,11)%.#$%1 .# .2 ,.#()1 %. 2$ .'7.)$: K.2 &(43 $#(.)&1)A $2 +'$) 2$ 2.> %. ?$+''A '$*.31' D+. .2 ,$381 .24,()&,1 /.#.)$%1 81) 2$ .'7.)$ .' ( ρr ˆ r
~ ~0) = ~0A 81) 21 ,+$2 .2 ,.#()1 .' *; N.#.31' D+. .2 ,$381 .# .2 ,.#()1 %. 2$ .'7.)$ .' E( +# 8+#(1 %. .D+&2&*)&1: @& 310.31' +# 81D+&(1 2$ ,$)/$A .'($ .G8.)&3.#($ +#$ 7+.)F$ ρr F~ = −Q 3ǫ · rÂ 2$ ,+$2 $8+#($ O$,&$ .2 ,.#()1 %. 2$ .'7.)$ ='&# &381)($) .# D+4 %&).,,&-# 0 '. O$>$ 310&%1 2$ ,$)/$;A 81) 21 ,+$2 .' +# 8+#(1 %. .D+&2&*)&1 .'($*2.: @& 310.31' 2$ ,$)/$ D+.%$#%1 '&.38). %.#()1 %. 2$ .'7.)$A (.#.31' D+. '+ .,+$,&-# %. 310&3&.#(1 .' ρr · rˆ 3ǫ0 ρr · rˆ = ~0 ~a + Q 3mǫ0 ρ =⇒ r¨ + Q ·r =0 3mǫ0 r¨ + w2 · r = 0

m~a = F~

= −Q

M1) ($#(1A .2 310&3&.#(1 %. q 2$ ,$)/$ .' +# 310&3&.#(1 $)3-#&,1 '&382. > '+ 8.)I1%1 %. 2π 0 1',&2$,&-# .' T = w = 2π 3mǫ Qρ

,; M1) .2 8)&#,&8&1 %. '+8.)81'&,&-#A .2 ,$381 .24,()&,1 ).'+2($#(. .' ( ρr ˆ + E0 x ˆ r
!"#$%&' () &*+ ,* -!%.. !""#$%&'&"()* "+ $,-.* "+/$0%&$* , +* +,%1* )"+ "2"  ρx ˆ   ( 3ǫ30 + E0 ) · x R ρ ~ ( 3ǫ0 x2 + E0 ) · x ˆ E(r) =   R3 ρ ˆ (E0 − 3ǫ0 x2 ) · x

34 %"#5+0,

−R < x < R R≤x

x ≤ −R

6($*(0%"-*# +,# (5"7,# .*#&$&*("# )" "85&+&'%&*9 &: −R < x < R ;5"%"-*# 85"

ρx −3ǫ0 E0 + E0 = 0 =⇒ x1 = 3ǫ0 ρ <,%, 85" x1 "3�"4 #" )"'" $5-.+&% 85"

x1 =

Rρ −3ǫ0 E0 > −R =⇒ E0 < ρ 3ǫ0

6#0, $*()&$&=( )"'" $5-.+&% E0 .,%, 85" >,?, 5( .5(0* )" "85&+&'%&* )"(0%* )" +, "#@"%,9 A,'"-*# 85" #& "+ +,.+,$&,(* )" +, "("%1B, .*0"($&,+ "+/$0%&$, ∇2 U "( "+ .5(0* )" "85&+&'%&* "# -,?*% 85" $"%*4 "(0*($"# 0,+ .5(0* "# "#0,'+"C #& "# -"(*% 85" $"%*4 "(0*($"# 0,+ .5(0* "# &("#0,'+"4 ? 85" U = −qV 4 )*()" D "# +, )&@"%"($&, )" .*0"($&,+ "+/$0%&$*9 E#B4 0"("-*# 85"

Z

~ E(x) ·x ˆdx + c Z ρx = − + E0 dx + c 3ǫ0 ρx2 +c = −E0 x − 6ǫ0

V (x) = −

U (x) = −QV (x) = QE0 x + Q =⇒ ∇2 U =

d2 U dx2

=

ρx2 − Qc 6ǫ0

Qρ >0 3ǫ0

<*% +* 0,(0*4 "# .5(0* )" "85&+&'%&* "#0,'+"9 &&: R ≤ x

;5"%"-*# 85"

R3 ρ + E0 = 0 =⇒ x2 3ǫ0 x2

imaginario

.*% +* 85" (* >,? .*#&$&=( )" "85&+&'%&* .,%, R ≤ x9

! """# x ≤ −R

$%&'&()* +%&

R3 ρ E0 − = 0 =⇒ x3 = −R 3ǫ0 x2

r

ρR 3ǫ0 E0

,-'- +%& x1 &."*/&0 *& 1&2& 3%(45"' +%& r r Rρ ρR ρR < −R =⇒ > 1 =⇒ E0 < x3 = −R 3ǫ0 E0 3ǫ0 E0 3ǫ0 +%& &* 5- ("*(- 3)61"3"76 +%& &63)6/'-()* -6/&'")'(&6/& 4-'- +%& 8%2"&'- &+%"9 5"2'"): ;*/& 4%6/) 1& &+%"5"2'") &* "6&*/-25&0 *& 1&<- -5 5&3/)' =&'">3-'5): Rρ ?- 4'%&2- 1& 5) *"@%"&6/& *& 1&<- -5 5&3/)': A&()* ="*/) +%& 4-'- E0 < 3ǫ &."*/&6 1)* 4%6/)* 0 1& &+%"5"2'")0 %6) &*/-25& B &5 )/') "6&*/-25&: ,-'- E0 = 0 &."*/& %6- *)5- 4)*"3"76 1& &+%"5"2'")0 Rρ &."*/& %6- *)5- 4)*"3"76 1& &+%"5"2'")0 &* &6 &5 3&6/') 1& 5- &*C&'- B &* &*/-25&: ,-'- E0 = 3ǫ 0 Rρ &6 x = −R B &* "6&*/-25&: D 4-'- E0 > 3ǫ0 6) 8-B 4)*"3")6&* 1& &+%"5"2'"): E)1) &*/) *& 4%&1& =&' 35-'-(&6/& 8-3"&61) %6 @'F>3) 1& ;G.# =&'*%* . G*& 1&<- -5 5&3/)'#:

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!

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"#$%&'()( *$ +&,&$')# -*. ,/)0# '( )/'&# R 1*( %( +/)0/ ($ %* &$2()&#) +#$ *$/ '($%&'/' ρ = ρ0 (1 − Rr )3 '#$'( ρ0 (% *$/ +#$%2/$2( 4#%&2&5/3 %&($'# r ,/ '&%2/$+&/ -('&'/ '(%'( (, (6( '(, +&,&$')#7 8$+*($2)( / 1*( '&%2/$+&/ '(, (6( (, +/-4# (,9+2)&+# (% -:;&-# . +/,+*,( (%2/ -/0$&2*' -:;&-/7

)"$*+,-./

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~ = E rˆ3 '#$'( rˆ (% (, 5(+2#) *$&2/)&# A#'(-#% ($2#$+(% (%+)&B&) (, +/-4# (,9+2)&+# +#-# E )/'&/, ($ +##)'($/'/% +&,>$')&+/%7 A#) ,/ ,(. '( 0/*%% %( 2($'): 1*(@ Z Z Z qinterior ~ ·n ~ ~ E ˆ dS = (1) E·n ˆ dS + E·n ˆ dS = ǫ0 manto tapas S 8, 5(+2#) $#)-/, n ˆ '( ,/% 2/4/% (% 4()4($'&+*,/) / rˆ7 "#$ (%2# %( 2($'): 1*( rˆ · n ˆ = 0 . 4#) ,# 2/$2# Z Z ~ ·n E rˆ · n ˆ dS = 0 (2) E ˆ dS = tapas

tapas

8, 5(+2#) n ˆ 3 $#)-/, /, -/$2#3 %(): 4/)/,(,# / rˆ . 4#) ,# 2/$2# rˆ · n ˆ = 1 ,# 1*( &-4,&+/ 1*(@ Z Z Z ~ ·n E ˆ dS = E rˆ · n ˆ dS = E · dS manto

manto

manto

R C/ &$2(0)/, manto dS +#))(%4#$'( /, :)(/ '(, -/$2#@ manto dS = 2πrL7 A#) ,# 2/$2#@ Z ~ ·n E ˆ dS = E · 2πrL (3) R

manto

D((-4,/E/$'# FGH . FIH ($ FJH )(%*,2/ 1*(@ qinterior E · 2πrL = ǫ0

(4)

! "# $%& '()*( +(,( ,&-#).&, &-*& +,#/)&0( &- 1()1%)(, )( 1(,2( $%& 3(4 &5 &) 65*&,6#, 7& )( -%+&,8 916& 7& 2(%--: ;&5&0#- %5( 7&5-67(7 .#)%0<*,61( 7& 1(,2( &)<1*,61( 7&5*,# 7& &-*( -%+&,916&: =&0#- $%& -& 1%0+)6,> )( -62%6&5*& ,&)(16?5@

dq = ρ(r′ ) · dV

(5)

A(,( 1()1%)(, dV 1#5-67&,(0#- &) .#)%0&5 &5*,& 7#- 16)657,#- 7& )( )(,2# " 4 06-0# &B& $%& &) 16)657,# 7&) +,#/)&0(C 1#5 ,(76#- r′ 4 r′ + dr′ 1#0# 0%&-*,( )( 92%,(@

"%&2#C dV -&,> -60+)&0&5*& &) .#)%0&5 &5*,& )#- 7#- 16)657,#-C )# $%& ,&-%)*(@

dV = 2πr′ dr′ L D&&0+)(E(57# FGH &5 FIH@

(6)

dq = 2πr′ · dr′ · L · p(r′ ) dq = 2πr′ · dr′ · L · ρ0 (1 − =⇒ dq = 2πL · ρ0 (r′ −

r′ ) R

r′ 2 )dr′ R

J5*&2,(57#@

q = 2πLρ0

Z

0

r

Z r ′2 Z r r′ 2 r ′ ′ ′ (r − r dr − )dr = 2πLρ0 · ( dr′ ) R 0 R 0 ′

r3 3Rr2 − 2r3 r2 − ) = 2πLρ0 · ( ) 2 3R 6R πLρ0 (3Rr2 − 2r3 ) (7) =⇒ q = 3R

=⇒ q = 2πLρ0 · (

K& F H 4 F!H@

E · 2πrL =

πLρ0 (3Rr2 − 2r3 ) 3Rǫ0

K&-+&B(57# E @

E=

ρ0 (3Rr − 2r2 ) 6Rǫ0

(8)

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ρ0 dE = (3R − 4r) dr 6Rǫ0 89#).):/' ) ; ($ '*23$:$ ,#$ $. -).'% /$ r *#(0)/' $(7

r=

3R 4

<$ $(2) &):$%)

|E|max =

3ρ0 R 16ǫ0

!

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S

9( (1-# )#/('#= #*$.%#/0+ $# $(4 0( 2#&11= 1( -(/0'8 ;&( ($ >&7+ -+-#$ 1+A'( $# %#7# 1('8 ($ >&7+ # -'#?,1 0( $#1 %#'#1 $#-('#$(15 Z Z qint (1) dS + E dS = EA + EA = 2EA = φ=E ǫ0 S2 S1 9+/0( S1 4 S2 1+/ $#1 1&*('6%.(1 $#-('#$(1 ./0.%#0#1 #/-('.+')(/-(3 C+0()+1 %#$%&$#' $# %#'2# ./-('.+' qint 4# ;&( 1#A()+1 $# 0.)(/1.+/(1 0( $# %#7# 4 $# 0(/1.0#0 ?+$&),-'.%# 0( %#'2# ρ3 D# %#'2# ./-('.+' 1('8 1.)*$()(/-(

qint = ρV = ρ2xA E(()*$#F#/0+ (1-( ?#$+' (/ GHI +A-(/()+15

ρ2xA ǫ0 ρx E= ǫ0

2EA =

J 0#0# $# 1.)(-'K# ?()+1 ;&( ($ %#)*+ (1-# (/ 0.'(%%.:/ î= *+' $+ -#/-+ 1( -.(/( ;&(5

~ ρ = ρx î, si 0 < x < a E ǫ0

!

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ρ2aA ǫ0 ρa E= ǫ0

φ = 2EA =

B- ',/$& #53, #2 ()%#'')12 î $&% -& 3,23&9

Eρ =

ρa ˆ i, si x > a ǫ0

C&3,% 0+# #- ',/$& #2 #53, %#*)12 #5 '&253,23#? !"#$% &'()*+,)% $+%-.),-% $%+ '" )"/)"+" &/0(+,)" 1Eσ 2 2 "#$,%,%#/&5 #- #5$,')& #2 D %#*)&2#5A #- )23#%)&% (# -, ',5',%, #5.7%)', E%FGH 4 #- #;3#%)&% E%IGH 4 ',-'+-,/&5 #- ',/$& #-7'3%)'& #2 ',(, +2& +3)-)J,2(& -, -#4 (# *,+559

• r
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!

R ~ ·n "# $%&' ( )*(+,- ./ /-)( -%0/*123/ -/*4 φ = S E ˆ dS 5 0/*' /# +/2)'* n ˆ /- 0(*(#/#' (# 2(67 R ~ ·n 0'8(69'- *(.3(#/-: ; 0'* #' <%/ E ˆ = E ; 0'* #' )(=)' φ = E S dS = E4πr2 > ?'* ')*' #(.'5 #( 2(*@( 3=)/*3'* /- =%#( ;( <%/ -'#' A(; 2(*@( /= #( 2(-2(*(B qint = 0> ?'* #( #/; ./ @(%-- -/ )/=.*4 <%/ qint φ = E4πr2 = =0 ǫ0 E = 0, si r < R • r>R C'=-3./*(6'-5 (# 3@%(# <%/ /= /# 2(-' (=)/*3'*5 %=( 2(-2(*( /-D,*32( ./ *(.3' r > R 2'6' -%0/*123/ @(%--3(=(> E(&' #'- 63-6'- (*@%6/=)'- ./ -36/)*F( (=)/*3'* +/6'- <%/ /# 2(60' /#,2)*32' -/*4 *(.3(# ; 0'* #' )(=)'5 0(*(#/#' (# +/2)'* ='*6(# ; ./ /-)( 6(=/*(B Z Z ~ dS = E4πr2 E·n ˆ dS = E φ= S

S

G( 2(*@( 3=)/*=( qint -/*4 #( 2(*@( 2'=)/=3.( /= )'.( #( -%0/*123/5 #( 2%(# -/ '9)3/=/ 6%#)30#37 2(=.' /# 4*/( )')(# 0'* #( ./=-3.(. -%0/*123(# ./ 2(*@(B qint = σ4πR2 > ?'* #( #/; ./ @(%--5 -/ )/=.*4 <%/

φ = E4πr2 =

qint σ4πR2 = ǫ0 ǫ0

H/-0/&(=.' /# 2(60' /#,2)*32' -/ )3/=/ <%/

E=

σR2 ǫ0 r2

"-)/ 2(60' /-)(*4 /= .3*/223I= *(.3(#5 0'* #' )(=)'B 2 ~ σ = σR rˆ, si r > R E ǫ0 r2

H'=./ rˆ /- /# +/2)'* %=3)(*3' /= .3*/223I= *(.3(#> JA'*( <%/ /=2'=)*(6'- #'- .'- 2(60'- /#,2)*32'- 0'./6'- /=2'=)*(* /# 2(60' */-%#)(=)/ -%0/*0'=3,=.'#'-> ?(*( /-)'5 2'=-3./*(*/ K */@3'=/-B L: 0 < x < a LL: a < x < a + 2R LLL: x > a + 2R

~ σ = σR22 rˆ> JA'*( )/=/6'~ ρ = ρx î ; E L: 0 < x < aB "= /-)( */@3I= #'- 2(60'- /#,2)*32'- -'=B E ǫ0 ǫ0 r <%/ /M0*/-(* r ; rˆ ./ D'*6( 2'=+/=3/=)/ 0(*( 0'./* -%6(* (69'- 2(60'-> C'=-3./*/6'- #( -3@%3/=)/ 1@%*(B N/6'- <%/ -/ 2%60#3*4 <%/ a + R = x + r 2'= #' <%/ */-%#)( r = a + R − x> J./64-5 +/6'<%/ rˆ = −î ; 0'* #' )(=)' 0'./6'- /-2*393* /# 2(60' /#,2)*32' 2'6'B ~σ = E

−σR2 î ǫ0 (a + R − x)2

!

~ =E ~ρ + E ~ σ * #$ 4#,0) "# #$%& '&(#)&* #+ ,&'-. #+/,%)0,. )#$1+%&(%# #( #$%& )#203( #$ E σR2 ~ = ( ρx − )î E ǫ0 ǫ0 (a + R − x)2 556 a < x < a + 2R7 8( #$%& )#203( #+ ,&'-. +. &-.)%& $.+. Eρ 9& :1# #+ ,&'-. -).41,04. -.) +& ,&$,&& #$;#)0,& #$ (1+. #( #+ 0(%#)0.) #++&< =.) +. %&(%.7

~ =E ~ ρ = ρa î, si a < x < a + 2R E ǫ0 5556 x > a + 2R 8( #$%& )#20.* &'>.$ ,&'-.$ #$%&)?( #( 40)#,,03( 9 $#(%04. î< =&)& .>%#(#) ~ σ #( ;1(,03( 4# @* ,.($04#)&'.$ +& $0210#(%# A21)&7 +& #@-)#$03( 4# E

B#'.$ :1# $# ,1'-+# +& )#+&,03( x = a + R + r* ,.( +. :1# )#$1+%& :1# r = x − a − R< C4#'?$* rˆ = î< "# #$%& ;.)'&* %#(4)#'.$ :1#7

~σ = E

σR2 î ǫ0 (x − a − R)2

D1#2.* -.) -)0(,0-0. 4# $1-#)-.$0,03(* #+ ,&'-. #+/,%)0,. #( #$%& )#203( $#)?7

σR2 ~ = ( ρa + )î, si x > a + 2R E ǫ0 ǫ0 (x − a − R)2

E0(&+'#(%# #+ ,&'-. #+/,%)0,. )#$1+%&(%# -.) &'>&$ 40$%)0>1,0.(#$ 4# ,&)2& $#)?7

!"#$%&' () &*+ ,* -!%..

!

~ = E

 ρx   ( ǫ0 −   ( ρa + ǫ0

σR2 )î ǫ0 (a+R−x)2 ρa ˆ ǫ0 i σR2 )î ǫ0 (x−a−R)2

0 a + 2R

!"#$%&' ()

!"#$%&'(')"$ &"$ '$*'(+$ #" ,"#,-#.(%,+$ &' (+&%" R/ ,"# &'#$%&+&'$ &' ,+(0+ 1"23)-.(%,+$ ρ 4 −ρ 3#%*"()'$5 6"$ ,'#.("$ &' +)7+$ '$*'(+$ '$.8# + 3#+ &%$.+#,%+ )'#"( 93' 2R5 :'+ d~ '2 1',."( 93' 1+ &'2 ,'#.(" &' 2+ '$*'(+ ;"$%.%1+ +2 ,'#.(" &' 2+ '$*'(+ #'0+.%1+5 <(3'7' 93' '2 ,+);" '2-,.(%," '# 2+ %#.'($',,%=# &' 2+$ '$*'(+$ '$ ,"#$.+#.' 4 '#,3'#.(' $3 1+2"(5

*"$+,-./0

>'2 ;("72')+ ?@/ $+7')"$ 93' '2 ,+);" '2-,.(%," '# '2 %#.'(%"( &' 3#+ '$*'(+ )+,%A+ &' &'#$%&+& ρr ~ &' ,+(0+ 3#%*"()' ρ '$ E(r) = 3ǫ · rˆ5 0 >' 2+ $%)'.(B+ &' 2+ &%$.(%73,%=#/ + ;(%"(% $+7')"$ 93' +&')8$ ;+(+ R ≤ r '2 ,+);" ,3);2' ~ r) = E(r)ˆ E(~ r5 :% 7%'# #" '$ #','$+(%" ;+(+ 2+ ('$"23,%=# &'2 ;("72')+ ,"#",'( '2 ,+);" '2-,.(%," *3'(+ &' 2+ '$*'(+/ '#,"#.(-)"$2" 3$+#&" 2+ 2'4 &' C+3$$5 !"#$%&'(')"$ ,")" $3;'(D,%' &' %#.'0(+,%=# 3#+ ,8$,+(+ '$*-(%,+ &' (+&%" r > R5 E'#')"$ 93' I Qint ~ ·n E ˆ dS = ǫ0 Ω 4πR3 ρ E(r)4πr2 = 3ǫ0 3 ~ r) = R ρ rˆ =⇒ E(~ 3ǫ0 r2 <"( .+#."/ .'#')"$ 93' '2 ,+);" '2-,.(%," 0'#'(+&" ;"( 2+ '$*'(+ )+,%A+ '$ ( ρr ˆ r
ρ · ~r para |~r| < R 3ǫ0 ρ ~ para |~r − d| ~
6"$ 1',."('$ '# 2+ %#.'($',,%=# &' +)7+$ '$*'(+$ $+.%$*+,'# $%)32.8#'+)'#.' ~ < R 4 |~r| < R5 F$B/ ;"( '2 ;(%#,%;%" &' $3;'(;"$%,%=#/ '2 ,+);" '2-,.(%," '# 2+ %#.'($',,%=# |~r −d| &' +)7+$ '$*'(+$ '$

~ = ρ · d~ ~ r) = E~1 (~r) + E~2 (~r) = ρ · ~r − ρ · (~r − d) E(~ 3ǫ0 3ǫ0 3ǫ0 '2 ,3+2 '$ ,"#$.+#.'/ .+2 ,")" $' 93'(B+ ;("7+(5

!

!"#$%&' () &*+ ,* -!%..

!"#$%&' (

!"#$%&'( )(#%"*!#+"'"&%! !"#$%&' ()

!"#$%&'& (" )*+"! $","$-! %& %&"#$%+% %& .+'/+ #()&',.$+* ("$0!'1& σ > 0 "!'1+* +* &2& 3 %& &.(+.$4" x = 05 6" aˆx #& &".(&"-'+ ("+ .+'/+ )("-(+* −q < 05 +7 6".(&"-'& &* )!-&".$+* &*8.-'$.! #!9'& &* &2& 3 : &"-'& *+ .+'/+ −q < 0 : &* !'$/&" .!!'%&"+%! ;5 97 <"+ )+'-=.(*+ %& 1+#+ m : .+'/+ −e < 0 #& (9$.+ &" &* )("-! 1&%$! &"-'& −q : ; : #& %&2+ *$9'&5 > !" ?(8 &"&'/=+ .$"8-$.+ **&/+ *+ .+'/+ +* )*+"!@ A$/"!'& &0&.-!# /'+B$-+.$!"+*K

*"$+,-./0

+7 !"!.&1!# &* .+1)! &*8.-'$.! /&"&'+%! )!' &* )*+"! : *+ .+'/+ −q #!9'& &* &2& 3 ?(& *!# ("&5 C+*&# .+1)!# #!" σ E~1 =

E~2 =

2ǫ0

x ˆ

−q 1 q 1 · −ˆ x= ·x ˆ 2 4πǫ0 (a − x) 4πǫ0 (a − x)2

D#=E &* .+1)! &*8.-'$.! &"-'& ; : −q A?(& &#-FE %$/+1!#E &" &* )("-! D7 &# ~ = E~1 + E~2 = E

σ 1 q + 2ǫ0 4πǫ0 (a − x)2

·x ˆ

~ E &"-!".&# &" Ω A*+ '&/$4" )&%$%+7 G&'! :+ ?(& E~ = −∇V ∂V q 1 σ =0 − = + ∂x 2ǫ0 4πǫ0 (a − x)2 q 1 σx +C + =⇒ V (x) = − 2ǫ0 4πǫ0 (a − x)

∂V =0 ∂z

∂V ∂y

97 6* .+1)! &*&.-'!#-F-$.! &# .!"#&'B+-$B!E +#= ?(& )!%&1!# (#+' .!"#&'B+.$4" %& &"&'/=+ &"-'& &"&'/=+ )!-&".$+* : .$"8-$.+5 H+ .+'/+ −e )+'-& %&* '&)!#! %&#%& a/2ˆx I+.$+ &* )*+"!5 G!' .!"#&'B+.$4" %& &"&'/=+E -&"&1!# ?(& JK

!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

!

−eV (a/2) = =⇒ K =

mv 2 2

mv 2 − eV (0) 2

= e (V (0) − V (a/2)) 1 q 1 2q σa = e − +C + + −C 4πǫ0 a 4πǫ0 a 4ǫ0 σa 1 q = e + 4πǫ0 a 4ǫ0

!

!"#$%&' ()

"# $%#&#& '() #)*+,-) .(&'/.$(,-) '# ,-'%() r1 , r2 0 .-,1-) q1 , q2 )#2-,-'-) 2(, /&- 1,-& '%)3 $-&.%- d >> r1 , r2 4 "% -56-) )# .(&#.$-& - $,-7+) '# /& .-68# .(&'/.$(, 9'#)2,#.%-68#: ;/# )%,7# )<8( 2-,- $,-&)2(,$-, .-,1- '# /&- - ($,-=: #&./#&$,# 8-) '#&)%'-'#) '# .-,1- )/2#,>.%-8#) '# .-'- /&- 9#& */&.%<& '# 8-) 7-,%-68#) .(&(.%'-)= /&- 7#? ;/# #8 )%)$#5- -8.-&?- #8 #;/%8%6,%(4

*"$+,-./0

@-'( ;/# #8 )%)$#5- #)$A #& /&- ,#1%<& -.($-'- '#8 #)2-.%(: 2('#5() $(5-, .(5( 2/&$( '# ,#*#,#&.%- '#8 2($#&.%-8 #8 %&>&%$( # %1/-8-, #8 2($#&.%-8 - .#,( -88B: #) '#.%,: V (+∞) = 04 C)B: #8 2($#&.%-8 )(6,# 8-) )/2#,>.%#) '# 8-) #)*#,-) .(&'/.$(,-) #)

V1 = k

q1 q2 , V2 = k r1 r2

'(&'# D#5() )/2/#)$( ;/# #8 2($#&.%-8 '# /&- #)*#,- &( #) -*#.$-'( 2(, #8 '# 8- ($,- 9( 5A) 6%#& #) -*#.$-'( '# *(,5- '#)2,#.%-68#=: '-'( ;/# #)$A& 5/0 5/0 )#2-,-'-: #) '#.%,: d >> r1 , r2 : 0 $-52(.( #8 .-52( '# /&- #)*#,- ,#'%)$,%6/0# 8- .-,1- '# 8- ($,-4 E(, .(&)#,7-.%<& '# .-,1-: )% (q1 )f , (q2 )f )(& 8-) .-,1-) #& .-'- #)*#,- /&- 7#? -8.-&?-'( #8 #;/%8%6,%(: $#() ;/#

q1 + q2 = (q1 )f + (q2 )f C8 -8.-&?-, #8 #;/%8%6,%(: #) '#.%,: ./-&'( '#F- '# D-6#, $,-&)*#,#&.%- '# .-,1-) #&$,# 8-) #)*#,-): $#() ;/# 8- '%*#,#&.%- '# 2($#&.%-8 #8+.$,%.( #&$,# -56-) #) &/8(: #) '#.%,:

V1 (q1 )f k r1 2 4π(r1 ) (σ1 )f r1 (σ2 )f =⇒ (σ1 )f

= V2 (q2 )f = k r2 4π(r2 )2 (σ2 )f = r2 r1 = r2

@# #)$( 7#5() ;/# #& 1#&#,-8: 8-) ,#1%(&#) #& 8- )/2#,>.%# '# /& .(&'/.$(, .(& 5#&(, ,-'%( '# ./,7-$/,- 92/&$-)= .(&.#&$,-& /&- 5-0(, '#&)%'-' )/2#,>.%-8 '# .-,1-: 2(, 8( ./-8 #8 .-52( #8+.$,%.( .#,.- '# #88-) 9#& )/ #G$#,%(,= #) 5A) */#,$# ;/# #& ,#1%(&#) .(& 5#&(, ,-'%( '# ./,7-$/,-4 H)-&'( 8- ,#8-.%<& #&.(&$,-'- #& 8- #./-.%<& '# .(&)#,7-.%<& '# .-,1-: (6$#()

(q1 )f + (q2 )f 2

2

4π(r1 ) (σ1 )f + 4π(r2 ) (σ2 )f

= q1 + q2 = q1 + q2

4πr1 (σ1 )f (r1 + r2 ) = q1 + q2 1 q1 + q2 =⇒ (σ1 )f = 4πr1 r1 + r2 1 q1 + q2 (σ2 )f = 4πr2 r1 + r2

!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

!

!"#$%&' () !"#!$%&! '#! () *+$,-.%+.#+*)* !. |E~⊥ (~r)| )( /)$)& *! #.) &!0+1. 2),3) ) -%&) 2),3) /-& #.) $#/!&4,+! ,)&0)*) *! *!.$+*)* $#/!&4,+)( σ(~r)5 ,-. ~r !. () $#/!&4,+!5 !$ σ(~r) ǫ0

6-.$+*!&).*- !$%-5 !.,#!.%&! !( ,)"/- !(7,%&+,- !. () $#/!&4,+! *! #. ,-.*#,%-& !. !'#+(+8&+!(!,%&-$%9%+,-:

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I

>!&-

Qint ∼ σ(~r)A Qint ∼ = = σ(~r)A =⇒ ǫ0 ǫ0

?!.!"-$ )*!"9$ I

Ω

~ r) · n E(~ ˆ dS =

Z

manto

~ r) · n E(~ ˆ dS +

Z

carasup

E~1 (~r) · nˆ1 dS +

Z

carainf

E~2 (~r) · nˆ2 dS

>-*!"-$ !@/&!$)& !( ,)"/- !(7,%&+,- ,-"- () $#") *! $# ,-"/-.!.%! %).0!.,+)( ; .-&")( ) () $#/!&4,+!5 !$ *!,+&5 E~ = E~⊥ + E~// : A*!"9$5 ,-"- !( 9&!) !$ "#; /!'#!<)5 !( "1*#(*!( ,)"/- !(7,%&+,- $-8&! 7( $! ").%+!.! ,)$+ ,-.$%).%!5 /-& (- '#! (- /-*!"-$ )/&-@+")& /-& ~ r)| !. %-*) !( 9&!): ?)"8+7.5 nˆ1 = −nˆ2 |E(~ 6-. !$%- !. "!.%!5 %!.!"-$ '#! Z

carasup

E~1 (~r) · nˆ1 dS +

Z

carainf

E~2 (~r) · nˆ2 dS = (E~1 )⊥ A − (E~2 )⊥ A = ((E~1 )⊥ − (E~2 )⊥ )A

!@/&!$+1. '#! .- *!/!.*! *! h B)( +0#)( '#! σ(~ǫr)A C: ?)"8+7. %!.!"-$ '#! 0

Z

manto

~ r) · n E(~ ˆ dS → 0

cuando

h→0

! "#$% &'(')*# +(,-)'(&'

σ(~r)A ǫ0 σ(~r)A △E⊥ A = ǫ0 σ(~r) =⇒ △E⊥ (~r) = + ǫ0

((E~1 )⊥ − (E~2 )⊥ )A

=

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!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

!

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*"$+,-./0 A= J3 :+(63#,* $%#&# )%,#$+D* #)./+%0* 0(& 0#&$+( '# 0((+'#&*'*) #3 0#&$+( '# 3*) 02)0*+*) #)./+%0*)1 '#6%'( * ;-# 3* $%#++* #)$2 %&<&%$*,#&$# 3#@*&* 7+#0(+'*+ ;-# 3* $%#++* )# :%#&)* 0(,( -& )-,%'#+( 5 .-#&$# %&<&%$* '# 0*+?*9= K('*) 3*) '#&)%'*'#) '# 0*+?* )#+2& -&%.(+,#)= H*'* 3* )%,#$+D* #)./+%0*1 #3 0*,:( 5 :($#&0%*3 #3/0$+%0() '#:#&'#+2& )F3( '# 3* '%)$*&0%* *3 0#&$+( '# 3*) 02)0*+*) 5 #3 0*,:( *:-&$*+2 #& #3 )#&$%'( '# rˆ1 #) '#0%+1 ~ r) = E(r)ˆ V = V (r) 5 E(~ r= *9 "#*& q1 , q2 , q3 , q4 3*) 0*+?*) '# 3*) )-:#+<0%#) %&$#+%(+ 5 #B$#+%(+ '# 3*) *+,*'-+*) %&$#+&* 5 #B$#+&*1 +#):#0$%C*,#&$#= K#&#,() ;-# q0 = q1 + q2 = L:3%0*&'( #3 $#(+#,* '# M*-)) 0(& -&* )-:#+<0%# #)./+%0* 0(&0/&$+%0* 5 #&$+# 3*) )-:#+<0%#) '# 3* :+%,#+* *+,*'-+*1 $#&#,() I q1 ~ ·n E ˆ dS = 0 = =⇒ q1 = 0 =⇒ q2 = q0 ǫ0

~ = ~0 :-#) #3 0*,:( #3/0$+%0( #& #3 %&$#+%(+ '# -& 0(&'-0$(+ #& #;-%3%6+%( '(&'# E #3#0$+()$2$%0( #) &-3(=

! "#$%&'()* '+*,' -$ .-*,-/' )- 0'122 &*( 1(' 21#-,3&%- -245,%&' &*(&5(.,%&' 6 -(.,$'2 21#-,3&%-2 )- $' 2-71()' ',/')1,'8 .-(-/*2 I q1 + q2 + q3 ~ ·n E ˆ dS = 0 = =⇒ q1 + q2 + q3 = 0 =⇒ q3 = −q2 = −q0 ǫ0

"#$%&'()* )- (1-9* -$ .-*,-/' )- 0'122 &*( 1(' 21#-,3&%- -245,%&' &*(&5(.,%&' ),')%* r > b 6 :1- &*(.-(7' $' 2-71()' ',/')1,'8 .-(-/*2 I q4 rˆ q1 + q2 + q3 + q4 ~ ~ ·n =⇒ E(r) = E ˆ dS = E(r)4πr2 = ǫ0 4πǫ0 r2

;*, *.,' #',.-8 2'<-/*2 :1- $' )%4-,-(&%' -(.,- $' ',/')1,' -=.-,(' 6 $' .%-,,' >-( -$ %(3(%.*? -2 V0 8 -2.* -2 Z b ~ · d~r E V (b) − V (+∞) = V (b) = V0 = − +∞ b

V0 = − V0 =⇒ q4

Z

+∞

q4 dr 4πǫ0 r2

q4 1 = 4πǫ0 b = 4πǫ0 bV0

)*()- .*/'/*2 1( &'/%(* ,')%'$ 2*<,- $' %(.-7,'$ )- $@(-'A ,-7%D( %(.-,%*, 6 /-.'$ )- $' ',/')1,' %(.-,('? F% -$ ,')%* )- $' 21#-,3&%- )- %(.-7,'&%D( -2 r ≤ a I q1 ~ ·n ~ = ~0 E ˆ dS = E(r)4πr2 = = 0 =⇒ E ǫ0 %%? a < r ,-7%D( -(.,- ',/')1,'2? G2 -9%)-(.- :1- -=%2.- &'/#* -$5&.,%&* -( -2.' ,-7%D(8 #1-2 +'6 1(' )%4-,-(&%' )- #*.-(&%'$ -(.,- ',/')1,'2A C-(-/*2 -( -2.- &'2* :1I q0 q0 rˆ ~ ·n ~ E ˆ dS = E(r)4πr2 = =⇒ E(r) = ǫ0 4πǫ0 r2 %%%? r = b >/-.'$ )- $' ',/')1,' -=.-,('? ~ G( -2.- #1(.*8 E(b) = ~08 #1-2 -$ &'/#* -$5&.,%&* -( -$ %(.-,%*, )- 1( /-.'$ -2 2%-/#,- (1$*A %9? r > b >,-7%D( -=.-,%*, ' $'2 ',/')1,'2? C'/<%5( ':1@ -2 -9%)-(.- :1- -=%2.- 1( &'/#* -$5&.,%&*8 #1-2 +'6 1(' )%4-,H -(&%' )- #*.-(&%'$ -(.,- $' ',/')1,' -=.-,(' 6 -$ %(3(%.* >#1(.* )- ,-4-,-(&%' )-$ #*.-(&%'$?A "2@8 .-(-/*2 :1I q4 4πǫ0 bV0 bV0 ~ ~ ·n = =⇒ E(r) = 2 · rˆ E ˆ dS = E(r)4πr2 = ǫ0 ǫ0 r

!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

!

"#$%&'#()*+ #, -.&/* #,0-12'-* #$     ~ E(r) =   

~0 q0 4πǫ0

~0

bV0 r2

·

rˆ r2

· rˆ

r≤a a
-3 4.&5'0( )'6').&*$ #, #$/.-'* #( 6.2'.$ 2#7'*(#$8 '3 a ≤ r ≤ b 4#(#&*$

V (r) = −

~ · dˆ E r+C

q0 dr +C 4πǫ0 r2 q0 1 · +C 4πǫ0 r

= − =

Z

Z

)*()# #, -.&'(* )# '(1#72.-'9( %$.)* :%# 2.)'.,8 ;.2. )#1#2&'(.2 < '&/*(#&*$ ,. -*()'-'9( )# 5*2)#8 =.5#&*$ >%# V (b) = V0 + q0 /*2 1.(1* C = V0 − 4πǫ · 1b + #(1*(-#$ 0 1 1 q0 + V0 − V (r) = 4πǫ0 r b ;*2 1.(1*+ 1#(#&*$ >%# ,. )':#2#(-'. )# /*1#(-'., #(12# ,.$ .2&.)%2.$ #$ 1 1 q0 − V (a) − V (b) = 4πǫ0 a b

''3 r ≤ a ?, /*1#(-'., #( r = a @.2&.)%2. '(1#2(.3 #$ q0 1 1 V (a) = + V0 − 4πǫ0 a b

<*&* #, -.&/* #,0-12'-* #( #$1. 2#7'9( #$ (%,*+ #(1*(-#$ 1*). ,. 2#7'9( )#5# 1#(#2 #, &'$&* /*1#(-'., @)#5# $#2 #>%'/*1#(-'.,3+ /*2 ,* >%# q0 1 1 + V0 V (r ≤ a) = − 4πǫ0 a b

'''3 b ≤ r

V (r) = −

=

~ · dˆ E r + C‘

q4 dr + C‘ 4πǫ0 r2 q4 1 · + C‘ 4πǫ0 r bV0 + C‘ r

= − =

Z

Z

! "#$% V (b) = V0 =⇒ C‘ = 0

=⇒ V (r) =

bV0 r

&#'()*#+,%- #. /%0#+1*2. #.310$*1% #'

V (r) =

  

q0 4πǫ0 q0 4πǫ0

1 a 1 r

− 1b + V0 r ≤ a − 1b + V0 a < r (#$2 ,# .2 '#?(+,2 2$)2,($2~ = −∇V ~ 5 /(#' '@.% #:*'0# 12)/% 1(2+,% 92A (+2 ,*>#$#+1*2 ,# /%0#+1*2.5 (#$,# <(# E B%' 1;.1(.%' '%+ *,3+0*1%' 2 .%' 2+0#$*%$#'- 2'8 72'02 *+0$%,(1*$ V0 = 0 A 1%+'*,#$2$ .% ,*19% 2+0#$*%$)#+0#5 C# #'02 >%$)2- 0#+#)%' 12$?2D

q1 = 0 q2 = q0 q3 = −q0

q4 = 0

12)/%D

~ = ~0 r≤a : E

~ = q0 · rˆ a
1 1 q0 − r ≤ a : V (r) = 4πǫ0 a b q0 1 1 a < r #$#+1*2 ,# /%0#+1*2. #+0$# 2$)2,($2' #' V (a) − V (b) = )*')2 <(# #+ #. 12'% 2+0#$*%$5

q0 4πǫ0

1 a

−

1 b

- .2

4445 6. ,#'1%+#102$ .2 2$)2,($2 ,# .2 0*#$$2- '*+ 21#$12$ 2F+ .2 12$?2 q > 0- .2' 12$?2' #+ .2' 2$)2,($2' +% '(>$#+ )%,*G121*@+5

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I

q1 + q2 + q3 q0 + q3 ~ ·n E ˆ dS = 0 = = =⇒ q3 = −q0 =⇒ q4 = 0 ǫ0 ǫ0

9-( ,%2,-+ $%0 &%()%0 '2 $%0 0/3'(@&1'0 !' $%0 &%0&%(-2'0 2- &%.>1%2 ?$- */' '(% !' '03'(%(0'"7 '" B% !'201!%! !' &%()% !' $% 0/3'(@&1' !' $% %(.%!/(% 'C,'(2% 0' .-!1@&%+ 3/'0 $% &%()% q 12,(-!/&' /2% %01.',(D% '2 '$ '03%&1-+ : $%0 &%()%0 '2 $-0 .',%$'0 ?&-2E !/&,-('0" '0,F2 $1>('0 !' .-4'(0' ?$% !'201!%! !' &%()% !' $% 0/3'(@&1' 'C,'(2% 0' ('!10,(1>/:' !' ,%$ A-(.% !' %2/$%( '$ &%.3- 3(-!/&1!- 3-( q '2 '$ 12,'(1-( !' $% %(.%!/(% 'C,'(1-("7 9'(- $%0 !'201!%!'0 !' $%0 0/3'(@&1'0 12,'(2%0 2- 0/A('2 &%.E >1-7 G1 05$- &-201!'(%.-0 %(.%!/(%0+ $% &%()% !' $% %(.%!/(% 'C,'(2% 0'(F %,(%D!% 3-( $% &%()% q > 0+ : $% !'201!%! 0/3'(@&1%$ ,'2!(F /2 .FC1.- ?!' &%()% 2')%,14%" '2 $% 3%(,' !' $% '0A'(% .F0 &'(&%2% % q 7 A" B% %(.%!/(% 'C,'(2% 0' 3-2!(F % /2 3-,'2&1%$ !10,12,- !' &'(- ?01 01)/' &-201!E Rb ~ · dˆ ~ 6= 0 : 2- I%: '(F2!-0' V (+∞) = 0"7 H2 'A'&,-+ V (b) = − +∞ E r 6= 0+ 3/'0 E 01.',(D% */' 3/!1'(% %2/$%( $% 12,')(%$7 B% !1A'('2&1% !' 3-,'2&1%$ '2,(' %(.%!/(%0 0')/1(F 01'2!- $% .10.% ?'0,- 0' !'03('2!' !'$ I'&I- */' $% )'-.',(D% !' $%0 %(E .%!/(%0 2- I% &%.>1%!-+ 3-( $- */' 0/ &%3%&1,%2&1% 2- I% &%.>1%!-+ : ,%.3-&I% &%.>1%!- $% &%()% '2 '$$%0"7

!

!"#$%&' ()

"#$%&'()( *$+ ,+)&--+ '(-.+'+ '( '($%&'+' -&$(+- *$&/#)0( λ 1 -+).# L2 3$4*($5)( %* 6#5($4&+(-745)&4# ($ 5#'# (- (%6+4&# 8*( -+ )#'(+2

*"$+,-./0

9+:(0#% 8*( (- 6#5($4&+- (%5; '+'# 6#)

V (~r) =

Z

Ω

dq 4πǫ0 |~r − r~1 |

3$ (%5( 4+%#< %& ~r = xˆ x + y yˆ + z zˆ 1 r~1 = x1 x ˆ< 4#$ 0 ≤ x1 ≤ L< %#$ -+% 6#%&4&#$(% '(- 6*$5# 1 '( -+ ,+)&--+ )(%6(45&,+0($5(< 5($(0#% 8*(

V (~r) =

Z

L

0

3$4#$5)(0#%

R

√ dx 2 x2 +1

9& x = tan(θ)< ($5#$4(%

sec2 (θ)dθ p = tan2 (θ) + 1 R dx1 3=5($'&($'# (%5# + √ 2 2 Z

√

dx = x2 + 1

Z

Z

(x1 −x) +y +z 2

Z

p sec(θ)dθ = ln(sec(θ)+tan(θ)) = ln(x+ x2 + 1) = arccosh(x) )(%*-5+ >,()?@8*(-#A

dx1

p

λdx1 p 4πǫ0 (x1 − x)2 + y 2 + z 2

(x1 − x)2 + y 2 + z 2

= ln(x1 − x +

B%?< (- 6#5($4&+- (%

V (x, y, z) = = = =

!"#$%&' 12

p

((x1 − x)2 + y 2 + z 2 )

L

λdx1 p 2 2 2 0 4πǫ0 (x1 − x) + y + z Z L λ dx1 p 4πǫ0 0 (x1 − x)2 + y 2 + z 2 L p λ · ln(x1 − x + (x1 − x)2 + y 2 + z 2 ) 4πǫ0 0 " ! p 2 2 2 L − x + (L − x) + y + z λ p · ln 4πǫ0 −x + x2 + y 2 + z 2

Z

"#$%&'()( (- '&6#-# (-745)&4# '( -+ @.*)+2 3$4*($5)( 6+)+ (- '&6#-# (-745)&4# 0&4)#%4C6&4# >(% '(4&)< 6+)+ '&%5+$4&+ 0*4D# 0;% .)+$'(% 8*( 2d< -+ %(6+)+4&C$ ($5)( 4+).+%AE +A 3- 6#5($4&+- (-745)&4# ($ 5#'# (- (%6+4&# > !"# E F%( 4##)'($+'+% 6#-+)(%< 1 *%( -+ -(1 '( -#% 4#%($#% 6+)+ (=6)(%+) -+% '&%5+$4&+% )(-(,+$5(% ($ 5+-(% 4##)'($+'+%A2 :A 3- 4+06# (-745)&4# ($ 5#'# (- (%6+4

!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

!

"# $% &'&()*% +,-&'".%/ 0&/ 0.+,/, 1&2-,3 +%(% 4' 0.+,/, &' )&'&(%/#5

!"#$%&'( %# 6,( &/ +(.'".+., 0& 24+&(+,2.".7'3 &/ +,-&'".%/ &/8"-(.", &' &/ +4'-, 6 &2-9 0%0, +,( q 1 1 V = − 4πǫ0 r+ r− 6,( /% /&: 0&/ ",2&',3 -&'&;,2 <4& 2 r+ = r2 + d2 − 2rd · cos

=⇒

1 r+

π

−θ

2 = r2 + d2 − 2rd · sen(θ) 1 = p 2 2 r + d − 2rd · sen (θ) 1 1 ·q = r 1 + ( d )2 − 2 d · sen (θ) r

r

6&(, &2-%;,2 "%/"4/%'0, &/ +,-&'".%/ +%(% r >> d3 +,( /, <4&

1 r+

=

∼ = =

= 24 >&?3 -&'&;,2 +%(% r−

d r

<< 1 :

1 1 ·q r 1 + ( dr )2 − 2 dr · sen (θ) ! "" ! d 1 d 2 1 − 2 · sen(θ) · 1− r 2 r r ! " 2 d 1 1 d + · sen(θ) · 1− r 2 r r

!

2 r− = r2 + d2 − 2rd · cos

π 2

+ θ = r2 + d2 + 2rd · sen (θ)

" #$ %&'() )*+,&-) ) ,& )*.$'/&'

1 ∼1 = · r− r

!

1 1− 2

" 2 d d − · sen(θ) r r

0&' ,& 12$ $, 3&.$*4/), 3)') r >> d '$52,.)

V (r, θ) = ∼ = = = = V (r, θ) =

1 1 − r + r− !! " ! "" q 1 d 2 d 1 d 2 d + · sen(θ) − 1 − − · sen(θ) · 1− 4πǫ0 r 2 r r 2 r r 2d q · · sen(θ) 4πǫ0 r r 2qd sen(θ) · 4πǫ0 r2 p sen(θ) · 4πǫ0 r2 p~ · rˆ 1 · 4πǫ0 r2 q 4πǫ0

#&*#$ p~ $5 $, (&($*.& #/3&,)' $,64.'/4& #$, #/3&,&7 89 :)8$(&5 12$

~ = −∇V ˆ = − ∂V rˆ − 1 ∂V θˆ E ∂r r ∂θ ;5<= .$*$(&5 12$

∂V 1 psen(θ) =− ∂r 2πǫ0 r3 1 pcos(θ) 1 ∂V = r ∂θ 4πǫ0 r3 ~ θ) = =⇒ E(r,

p ˆ 2sen(θ)ˆ r − cos(θ)θ 4πǫ0 r3

49 >&*5/#$'$(&5 ,) 5/-2/$*.$ ?-2')7 @)#& 12$ ,) '$-/A* &423)#) 3&' $, #/3&,& $5 )4&.)#)= 3&#$(&5 .&()' V (+∞) = 0 4&(& 3&.$*4/), #$ '$%$'$*4/)7 ;, .')$' ,)5 4)'-)5 #$5#$ $, /*?*/.& " %&'()' $, #/3&,&= ,) $*$'-<) )5&4/)#) ), #/3&,& $5

!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

!

U

= qV (x + dx, y + dy, z + dz) − qV (x, y, z)

= q (V (x + dx, y + dy, z + dz) − V (x, y, z))

U

= qdV ∂V ∂V ∂V = q dx + dy + dz ∂x ∂y ∂z ∂V ∂V ∂V px + py + pz = ∂x ∂y ∂z ~ = p~ · ∇V ~ = −~ p·E

!

!"#$%&' ()

"#$%&'()( *$+ )(,&-$ (%./)&0+1 '( )+'&# b1 2*( 3&($( *$+ '&%3)&4*0&-$ '( 0+),+ *$&.#)5( ρ(r) = ρ0 6+)+ 7+ )(,&-$ '(3()5&$+'+ 6#) a < r < b 8 '($%&'+' $*7+ 6+)+ r < a9 :(3()5&$( (7 6#3($0&+7 (7(03)#%3;3&0# ($ 3#'# (7 (%6+0

*"$+,-./0 '1 <# 2*( =+)(5#% 6)&5()# %(); 0+70*7+) (7 0+56# (7/03)&0# 6)#'*0&'# 6#) (%3+ '&%3)&4*0&-$ '( 0+),+ ($ 3#'# (7 (%6+0&# 8 0#$ (%3( )(%*73+'# #43($')(5#% (7 6#3($0&+7 *3&7&>+$'#? Z p ~ · d~ℓ E Vp = − ∞

"#$%&'()+$'# (7 6#3($0&+7 &,*+7 + 0()# ($ (7 &$@$&3#9 ~ = E rˆ1 '#$'( rˆ (% (7 C(03#) A7 6)#47(5+ 3&($( %&5(3)B+ (%./)&0+ 8 6#) 7# 3+$3# %( 3($'); 2*( E *$&3+)&# ($ '&)(00&-$ )+'&+79 "#$%&'()+)(5#% D )(,&#$(%? EF r < a EEF a < r b

• 21 r < a0 "#$%&'()# 0#5# %*6()@0&( ,+*%%&+$+ *$ 0+%0+)#$ (%./)&0# '( )+'&# r < aG%*6()@0&( HF9 H( 3&($( 2*( (7 C(03#) $#)5+7 + (%3+ %*6()@0&( (% 6+)+7(7# + rˆ9 :( (%3+ 5+$()+ %( 3($'); 2*( (7 0+56# ~ (% 6+)+7(7# +7 C(03#) $#)5+7 n ~ ·n E ˆ '( 7+ %*6()@0&( H 6#) 7# 2*( E ˆ = E 8 6#) 7# 3+$3#? Z Z ~ dS = E4πr2 E·n ˆ dS = E φ= S

S

I#) #3)# 7+'# 7+ 0+),+ ($ (7 &$3()&#) '( 7+ %*6()@0&( (% $*7+ 8+ 2*( (%3+ 0#$3($&'+ ($ (7 0#$'*03#)? qint = 09 J67&0+$'# 7+ <(8 '( K+*%% %( 3($'); 2*(?

E4πr2 = 0 L 6#) 7# 3+$3# E = 01 %& )M+9 • 221 a < r < b0 "#$%&'()# 0#5# %*6()@0&( ,+*%%&+$+ *$+ 0+%0+)+ (%./)&0+ '( )+'&# )G%*6()@0&( HF 0#5# 5*(%3)+ 7+ @,*)+? ~ ·n J7 &,*+7 2*( ($ (7 0+%# +$3()&#) %( 3($'); 2*( E ˆ = E 8 6#) 7# 3+$3#?

!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

!

φ=

Z

S

~ ·n E ˆ dS = E4πr2

"# $#%&# '()*%'+% qint ,* -.*/* $#0$.0#% 12$'03*()* 4# 5.* 0# /*(,'/#/ /* $#%&# *, $+(,)#()*6 7* )*(/%2 5.* qint = ρ0 V 6 V *, *0 8+0.3*( *()%* 0# *,1*%# /* %#/'+ r 4 0# *,1*%# /* %#/'+ a $+( 0+ 5.* %*,.0)#9 4 qint = ρ0 π(r3 − a3 ) 3 :-0'$#(/+ 0# 0*4 /* &#.,,9

E4πr2 = ρ0 ;+( 0+ 5.* ,* +<)'*(*9

E=

4 π(r3 − a3 ) 3ǫ0

ρ0 (r3 − a3 ) 3r2 ǫ0

;+3+ $+3*()#3+, #()*%'+%3*()* *,)* $#3-+ *,)# *( /'%*$$'=( %#/'#0 rˆ> -+% 0+ )#()+9 3 3 ~ = ρ0 (r − a ) rˆ E 3ǫ0 r2

• ! r > b" ?%+$*/'*(/+ /* '&.#0 3#(*%# 5.* *( 0# %*&'=( @@A 8*%*3+, 5.* 0# $#%&# #03#$*(#/# *( 0# /',)%'<.$'=( 5.* %+/*# #0 $+(/.$)+% ,*%29 4 qint = πρ0 (b3 − a3 ) 3 B* *,)# 3#(*%# #-0'$#(/+ 0# 0*4 /* &#.,, )*(/%*3+, 5.*9

E4πr2 =

E=

ρ0 (b3 − a3 ) 3rǫ0

ρ0 (b3 − a3 ) 3ǫ0 r2

C*$)+%'#03*()* 5.*/# *D-%*,#/+ $+3+9

ρ0 (b3 − a3 ) ~ rˆ E(r) = 3ǫ0 r2

! "#$%&'#()*+ #, -.&/* #,0-12'-* #( 1*)* #, #$/.-'*  0   ρ0 (r 3 −a3 ) ~ rˆ E= 3ǫ0 r 2   ρ0 (b3 −a3 ) rˆ 3ǫ0 r 2

2#$%,1.

rb

34*2. 5%# 1#(#&*$ #, -.&/* #,0-12'-* #( 1*)* #, #$/.-'* /*)#&*$ -.,-%,.2 #, /*1#(-'., #,0-12'-*6 7#8#&*$ #,#9'2 %(. 12.:#-1*2'. /.2. 1#(#2 %(. #;/2#$'<( /.2. #, =#-1*2 d~ℓ6 >,#9'&*$ %(. ,?(#. 2#-1. 5%# =.:. )#$)# #, '(@('1* 4.$1. %(. )'$1.(-'. r > b ~ · d~ℓ = E rˆ · d~ℓ = Edr6 7# #$1. &.(#2. #, /*1#(-'., #,0-12'-* $#2BC A*&* /*)#&*$ =#2 E Z r Z r Z r ρ0 (b3 − a3 ) ~ ~ Edr = − E · dℓ = − V (r) = − )dr 3ǫ0 r2 ∞ ∞ ∞ D(1#92.()* $# *81'#(# 5%#C

V (r) =

ρ0 (b3 − a3 ) , 3ǫ0 r

si r > b

34*2. 5%#2#&*$ -.,-%,.2 #, /*1#(-'., . %(. )'$1.(-'. r+ )*()# a < r < b6 E*2 )#@('-'<( 1#(#&*$ 5%# Z r ~ · d~ℓ E V (r) = − ∞

7#$-*&/*(#&*$ #$1. '(1#92., )# ,?(#. $#/.2.()* ,. 12.:#-1*2'. )#$)# #, '(@('1* 4.$1. r+ #( ,.$ 12.:#-1*2'.$ )#$)# '(@('1* 4.$1. b &.$ ,. 12.:#-1*2'. )#$)# b 4.$1. r6

V (r) = −

Z

b

∞

~ · d~ℓ + − E

Z

b

r

~ · d~ℓ E

F. /2'. '(1#92., 2#$%,1. )# #=.,%.2 #, /*1#(-'., .(1#2'*2 #( r = bC

V (b) = −

b

3 3 ~ · d~ℓ = ρ0 (b − a ) E 3ǫ0 b ∞

Z

E*2 *12* ,.)*+ /.2. ,. *12. '(1#92., 1#()2#&*$

−

Z

b

r

~ · d~ℓ = − E

Z

r

b

ρ0 = − 3ǫ0 ρ0 = − 3ǫ0 =

ρ0 (r3 − a3 ) dr 3ǫ0 r2 Z r Z b 1 3 rdr − a ·( dr) 2 b r ∞ r r2 1 r · (( ) − a3 ( − )) 2 b r b

b2 − r 2 (r − b) ρ0 ·( + a3 ) 3ǫ0 2 rb

!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

!

"# #$%& '&(#)&* #+ ,-%#(./&+ #( r )#$0+%& $#)

V (r) =

(r − b) ρ0 b3 − a3 b2 − r2 ( + + a3 ), 3ǫ0 b 2 rb

si a < x < b

1&)& +& )#2/3( r < a 4#$.-',-(#'-$ +& %)&5#.%-)/& 4#$4# #+ /(6(/%- 7&$%& ) #( 8 %)&5#.%-)/&$9 :#(4)#'-$ ;0# #+ ,-%#(./&+ #$

V (r) = −

Z

r

∞

~ · d~ℓ = − E

Z

b

∞

~ · d~ℓ + − E

Z

a

b

~ · d~r + − E

Z

a

r

~ · d~ℓ E

<+ $#) (0+- #+ .&',- #+=.%)/.- ,&)& r < a %#(4)#'-$ ;0# Z r ~ · d~ℓ = 0 E − a

> 4# #$%& '&(#)& #+ ,-%#(./&+ )#$0+%& $#)?

V (r) =

(a − b) ρ0 b3 − a3 b2 − a2 ( + + a3 ), 3ǫ0 b 2 ab

si r < a

!

!"#$%&' ()

"#$%&'()( (* %&%+(,- '-'# ($ *- ./0)-1 2( +&($($ 3 4&*&$')#% ,05 *-)/#%6 70(4#%6 4-'- 0$# '( )-'&# r 5 '($%&'-'(% '( 4-)/-% %08().4&-*(% 4#$%+-$+(% σ 5 −σ 9

'* :$40($+)( (* 4-,8# (*;4+)&4# %#<)( *- *=$(- AB 6 >0( (>0&'&%+- '( *#% 4&*&$')#% ($ 0$- '&%? +-$4&- &/0-* - %0 %(8-)-4&@$ d9 #* A(+(),&$( *- '&B()($4&- '( 8#+($4&-* ($+)( *#% 4($+)#% '( *#% 4&*&$')#%9

+"$,-./01 '* C-)- #<+($() (%+( )(%0*+-'# '(<(,#% 4-*40*-) (* 4-,8# (*;4+)&4# '( 4-'- 4&*&$')# 70(4# 0$- '&%+-$4&- r '( ;*D0+&*&E-)(,#% (%+( )(%0*+-'# 8#%+()&#),($+(F 5 (G-*0-) ($ r = b9 A-'# >0( *#% 4&*&$')#% %#$ ,05 *-)/#% 5 (%+H$ 4-)/-'#% 4#$ '($%&'-' %08().4&-* 0$&B#),( 8#'(,#% '(%8)(4&-) *-% 4#$'&4&#$(% '( <#)'( 5 0+&*&E-) *#% -)/0,($+#% '( %&,(+)=-9 :* 4-,8# (*;4+)&4# >0( 8)#'04( 4-'- 4&*&$')# (%+-)H ($ '&)(44&@$ )-'&-* 5 '( (%+- ,-$()- 0+&*&E-$'# 4#,# %08().4&( /-0%%&-$- 0$ 4&*&$')# '( *-)/# L 5 )-'&# r

φ=

qint ~ ·n E ˆ dS = ǫ0 S

Z

(1)

A(%-))#**-,#% *- &$+(/)-* 4#,# Z Z ~ dS = E2πrL E·n ˆ dS = E S

(2)

S

"#,# *- '($%&'-' '( 4-)/- %08().4&-* (% σ 4#$%+-$+(D%( 7-4( '( &/0-* ,-$()- 8-)- 8-)- (* 4&*&$')# F6 *- 4-)/- '(* 4&*&$')# %()H %&,8*(,($+(

qint = σV = σ2πaL

(3)

!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

!

"##$%&'(')*+ ,-. / ,0. #) ,1.2 +34#)#$+5 67#

E=

σa rǫ0

(4)

89'&7')*+ #) r = d 4#)#$+5 67#

E=

σa dǫ0

:;+<' 57%#<%+)=#)*+ #& >'$%+ #&?>4<=>+ %<+*7>=*+ %+< >'*' >=&=)*<+ 5+3<# #& %7)4+ %#*=*+ 5# *#3# >+)5=*#<'< &' 5=@7=#)4# >+)A@7<'>=B)C

D+$+ /' $#)>=+)'$+5 #& >'$%+ #&?>4<=>+ #5 <'*='&2 / %+< &+ 4')4+ #54'>=B) =)*=>'*' #) &' A@7<'F D+$+ &+5 $B*7&+5 *# &+5 >'$%+5 %<+*7>=*+5 %+< '$3+5 >=&=)*<+5 5+) =@7' #) #& %7)4+ %#*=*+2 &'5 >+$%+)#)4#5 #) y 5# ')7&'<') / %+< &+ 4')4+ #& >'$%+ #&?>4<=>+ #54'>=B) î / 4#)*=<

E=

σa dǫ0

! G'<' >'&>7&'< &' *=H#<#)>=' *# %+4#)>='& #)4<# &+5 *+5 >=&=)*<+5 *#3#$+5 >'&>7&'< #& >'$%+

#&?>4<=>+ 5+3<# &' &I)#' 67# 7)# ' '$3+5 >=&=)*<+5F D+)5=*#<'<# #& +<=@#) #) #& >#)4<+ *#& >=&=)*<+ *# &' =(67=#<*'2 >+$+ $7#54<' &' A@7<'C

G'<' 0 < x < D #& >'$%+ #&?>4<=>+ 5=#$%<# #54'>=B) îF D+$+ ;#$+5 9=54+ ')4#<=+;+ *# 67# >'*' >=&=)*<+ 4#)@' >'<@' )7&' #) 57 =)4#<=+< =$%&=>''$%+ #&?>4<=>+ 67# %<+*7(>') #) 57 =)4#<=+< 5#' )7&+F :;+<'2 >'&>7&'<#$+5 #& >'$%+ #&?>4<=>+ <#57&J 4')4# 5+3<# #& #K# x 57%#<%+)=#)*+ &+5 >'$%+5 %<+*7>=*+5 %+< >'*' >=&=)*<+C

"#$%& '()*+,-*& %,&./*-.& %&, '( *-(-0.,& -12/-',.&

!"#" $% &' ()*"+ ', -%#." ',/-01)-" '2 ', )20'1)"1 (', -),)2(1" &'13 24,"+ ."1 ," 0%20"

~ 1 = 0, E

si 0 < x < a

54'1% (', -),)2(1" ', -%#." ',/-01)-" &' -%,-4,%1% (' )64%, #%2'1% -"#" ," 7)-)#"& '2 ,% .%10' %8+ (' '&0% #%2'1% 0'2'#"& 94':

~ 1 = σa î, E xǫ0

si x > a

!"#$ %&'()*+($ #*$,-(+,$ #$* %& (+&+.,*$ ,%*%(/$ ;2 ', )20'1)"1 (', -),)2(1" ', -%#." .1"(4-)(" ."1 /, &'13 24,":

~ 2 = 0, E

si d − a < x < d

<"1 "01" ,%("+ -"2&)('1%2(" ,% =641% #"&01%(% %20'1)"1#'20'+ >'1'#"& 94' % .%1% 42 -)'10" x 94' -4#.,% 0 < x < d − a+ ,% ()&0%2-)% (', -'201" (', -),)2(1" ('1'-7" % x &'13 d − x+ (' '&0% #%2'1% ', -%#." ',/-01)-" 1'&4,0% &'1:

~2 = E

σa ˆ i (d − x)ǫ0

?' '&0% #%2'1%+ %, &4.'1."2'1 ,"& -%#."& ',/-01)-"& "@0'2)("&+ >'#"& 94' ', -%#."& 1'&A,0%0' '&:

~ = E

σa ˆ (d−x)ǫ0 i σa σa ( xǫ + (d−x)ǫ )î 0 0 σa ˆ xǫ0 i

    

0
B7"1% .%1% -%,-4,%1 ,% ()C'1'2-)% (' ."0'2-)%, δV '201' ,"& -),)2(1"& ('@'#"& -%,-4,%1 ,% )20'61%, $ .%1% '&0" 40),)D%#"& ,% 01%$'-0"1)% #%& "@>)%: ', -%#)2" &"@1' ', '*' x:

∆V

= − = − = = = =

Z

d

0

Z

0

a

a

~ · d~ℓ E ~ · d~ℓ + − E

Z

a

d−a

~ · d~ℓ + − E

Z

d

d−a

~ · d~ℓ E

Z d−a Z d σa σa σa ˆ σa ( dx + + )dx + i xǫ0 (d − x)ǫ0 0 (d − x)ǫ0 a d−a xǫ0 σa σa σa (− ( ln(d − x)|a0 )) + ( ( ln(x) − ln(d − x)|d−a (( ln(x)|dd−a ))) a )) + ( ǫ0 ǫ0 ǫ0 −σa 2σa −σa d−a d−a d−a ( )) + ( )) + ( )) ln( ln( ln( ǫ0 d ǫ0 a ǫ0 d d−a d−a 2σa (ln( ) − ln( )) ǫ0 a d

Z

?' ("2(' &' "@0)'2'

!

!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, ' ∆V =

d 2σa ln( ) ǫ0 a

!

!"#$%&' (( "# $%&'( '%)'*#+) $, )+$%( R $, #+ -.*)+ ,&/0 '+).+$( '(1 *1+ $,1&%$+$ σ = σ0 (1 − σ0 > 02 &%,1$( r #+ $%&/+1'%+ $, *1 3*1/( '*+#4*%,)+ $,# $%&'( + &* ',1/)( O5 ') "1'*,1/), ,# 3(/,1'%+# ,1 ,# 3*1/( P &(6), &* ,7,8OP = R95

R r )2

'(1

#) :;+'%+ $(1$, &, <(=,)0 *1+ '+).+ 3*1/*+# q2 $,7+$+ ,1 ),3(&( ,1 P >

*"$+,-./0 ') ?( 4*, @+),<(& &,)0 '(1&%$,)+) ,# 3(/,1'%+# 3)($*'%$( 3() *1 +1%##( $, )+$%( a A '+).+

/(/+# q &(6), &* ,7, + *1+ $%&/+1'%+ d A #*,.(2 */%#%B+1$( ,&/, ),&*#/+$(2 $%=%$%),<(& ,# $%&'( ,1 3,4*,C(& +1%##(& A &*<+),<(& #(& 3(/,1'%+#,& 3)($*'%$(& 3() '+$+ *1(5 "# 3(/,1'%+# $, *1 +1%##( #( 3($,<(& '+#'*#+) */%#%B+1$( Z kdq V = r "# =+#() $, r #( (6/,1,<(& 3() 3%/+.()+&D

r=

p a2 + d2

"&/+ $%&/+1'%+ &, <+1/%,1, '(1&/+1/, A 3() #( /+1/( 3*,$, &+#%) $, #+ %1/,.)+# Z kq k dq = √ V =√ a2 + d2 a2 + d2 E@()+2 '(1&%$,)+), +1%##(& $,# $%&'( '(1 '+).+ dq 5 F,# ),&*#/+$( +1/,)%() A '(1&%$,)+1$( d = R A a = r &, /,1$)0 4*,

dV = √

kdq r 2 + R2

!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

!

! "!#$! dq %& '! ()*&+)% &%"#,-,# ")+) dq = σ(r) · dA

.)+) '! *&/%,*!* *& "!#$! *&(&/*& *& '! *,%0!/",! r 0&/*#&+)% 12& "!*! !/,'') *&' *,%") 12& ")/%,*&#&+)% 0&/*#3 ,$2!' *&/%,*!* 4! 12& "!*! &'&+&/0) *& 5' &%0! ! ,$2!' *,%0!/",! r *&' "&/0#) *&' *,%")6 7& &%0! +!/&#! ")/%,*&#!#& ")+) dA &' 3#&! *& 2/ !'!+-#& *& !/"8) dr9 dA = 2πrdr

7& &%0! +!/&#! )-0&/&+)% dq = 2πσ0 (r − R)dr

:' ()0&/",!' *& (#)*2",*) ()# &%0& !/,'') % &/0)/"&%9 dV =

;/0&$#!/*) *&%*& r = 0 8!%0! r = R9 V

<, ")/%,*&#!+)%

2kπσ0 (r − R)dr √ r 2 + R2

R

(r − R)dr √ r 2 + R2 0 Z R Z R r 1 √ √ = 2kπσ0 ( dr − R dr) 2 2 2 r +R r + R2 0 0 R R p p = 2kπσ0 (( r2 + R2 ) − ( ln( r2 + R2 + r) )) 0 0 √ √ = 2kπσ0 (R 2 − R − R ln(R 2 + R) + R ln(R)) √ √ = 2kπσ0 (R 2 − R − R ln(R( 2 + 1)) + R ln(R)) √ √ = 2kπσ0 (R 2 − R − R ln(R) − R ln( 2 + 1)) + R ln(R)) √ √ = 2kπσ0 (R 2 − R − R ln( 2 + 1)) √ √ = 2kπσ0 R( 2 − 1 − ln( 2 + 1)) √ √ 2 − 1 − ln( 2 + 1) ≈ 21 #&%2'0!9 = 2kπσ0

Z

V ≈ −kπσ0 R

=)*&+)% &%"#,-,# &%0& #&%2'0!*) &/ >2/",?/ *& '! "!#$! 0)0!' Q *&' *,%")6 :%0! "!#$! '! ()*&+)% "!'"2'!# ,/0&$#!/*) %)-#& &' *,%") *& '! &@(#&%,?/ dq = 2πσ0 (r − R)dr Q =

Z

0

7& &%0! +!/&#!

R

2πσ0 (r − R)dr

R r2 = 2πσ0 ( ( − Rr) ) 2 0 = −πσ0 R2

V ≈

Q − kπσ0 R −πσ0 R2

!

V ≈

kQ R

"#$# %&$#'( &) *#+&,-./) &,-#,+0/1# &' .23/) /) *#+&,-./) 43& *0#13-& 3,/ -/02/ *3,+3/) / 3,/ 1.'+/,-./ R 1& &))/5

! "/1/ &)&$&,+# 1& -/02/ dq &' ,&2/+.%/( )3&2# &) -/$*# &)6-+0.-# '&07 /+0/8&,+&5 9#0 '.$&+0:/( &) -/$*# &)6-+0.-# /*3,+/0/ &, 1.0&--.;, 1&) &<& 1&) 1.'-# 8 '&,+.1# =/-./ &) -&,+0# 8 -#$# )/ -/02/ 1& *03&>/ &' *#'.+.%/( &'+/ '& $#%&07 +/$>.6, =/-./ &) -&,+0# 1&) 1.'-#5

!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

!

!"#$%&' ()

"#$%&'()( '#% *+),+% -.$/.+0(% '( 1+%+ m 2 *+),+ q +/+'+% -#) .$+ *.()'+ &'(+0 '( 0+),# ℓ3 &$&*&+01($/( ($ )(-#%# ($ 0+ -#%&*&4$ I 56() 7,.)+89 :( %.(0/+$ 2 *+($ -#) 0+ +**&4$ '( 0+ ,)+6('+' + 0+ -#%&*&4$ II 3 +0*+$;+$'# .$ <$,.0# θ *#$#*&'#9 =>+)+ ?.( 6+0#) '( 0+ *+),+ (0 %&%/(1+ +0*+$;+ (%/( <$,.0# θ@9 AB-)(%( %. )(%.0/+'# ($ /C)1&$#% '( ℓ3 m 2 θ9

*"$+,-./0

A0(,&1#% (0 *()# '( 0+ ($(),D+ ,)+6&/+*&#$+0 ($ 0+ -#%&*&4$ ℓ 1+% +E+F# '( 0+ -#%&*&4$ &$&*&+0 56() 7,.)+89 G+ ($(),D+ -#/($*&+0 ($ 0+ -#%&*&4$ H %()<9

UI = 2mgℓ +

kq 2 2ℓ

G+ ($(),D+ -#/($*&+0 ($ 0+ -#%&*&4$ HH

UII = 2mgℓ(1 − cos(θ)) +

kq 2 2ℓ sin(θ)

"#1# 0+ ($(),D+ %( *#$%()6+3 -#'(1#% I+*() UI = UII 2 -#) 0# /+$/#

2mgℓ +

kq 2 kq 2 = 2mgℓ(1 − cos(θ)) + 2ℓ 2ℓ sin(θ)

J(%-(F+$'# ?K

s

q = ±2ℓ

mg sin(θ) k(sec θ − tan(θ))

!

!"#$%&' ()

"#$%& '() *)+) ,&-.&,/0%.*) 1& %)1.# R ,& 2.&(& '() 1.,2%.$'*.3( 1& *)%4) '(./#%-& σ 5 6)7*'7&8

'* 97 +#2&(*.)7 &7&*2%#,2:2.*# ) 7# 7)%4# 1&7 &;& Z < +)%) z > 05 #* 97 *)-+# &70*2%.*# ) 7# 7)%4# 1&7 &;& Z < +)%) z > 05 +* 97 *)-+# &70*2%.*# &( &7 +'(2# O5

,"$-+./01 '* 6#(,.1&%)-#, &7 #%.4&( &( O5 =)%) *)7*'7)% &7 +#2&(*.)7 >'& +%#1'*& 2#1) 7) ,'+&%?*.& ,#$%& '( +'(2# z kˆ 2#-)%&-#, '( &7&-&(2# 1& *)%4) dq 1& &77) @ *)7*'7)%&-#, &7 +#2&(*.)7 +%#1'*.1# *#-#8

dV =

kdq r

A#(1& r &, 7) 1.,2)(*.) 1&,1& dq B),2) z kˆ5 =#,2&%.#%-&(2&< +)%) *)7*'7)% &7 +#2&(*.)7 2#2)7< .(2&4%)-#, dV ,#$%& 2#1) 7) ,'+&%?*.&5 6#-# C.-#, &( 7) )@'1)(2.) D< 7) *)%4) dq 7) +#1&-#, &E+%&,)% *#-# &7 +%#1'*2# &(2%& 7) 1&(,.1)1 ,'+&%?*.)7 1& *)%4) σ +#% &7 )%&) >'& #*'+) &,) *)%4) >'& 77)-)-#, dA @ C.-#, >'&

dA = R2 sin(γ)dγdθ F) 1.,2)(*.) r &( &,2& *),# 7) #$2&(&-#, '2.7.G)(1# &7 2&#%&-) 1&7 *#,&(#8

r2 = R2 + z 2 − 2zR cos(γ)

H,I< &7 +#2&(*.)7 +%#1'*.1# +#% dq &,

kσR2 sin(γ)dγdθ dV = p R2 + z 2 − 2zR cos(γ)

F'&4#< .(2&4%)-#, &,2) &E+%&,.#( &7.4.&(1# 7#, 7.-.2&, 1& .(2&4%)*.#( 1& 2)7 /#%-) 1& %&*#%%&% 7) ,'+&%?*.&8

V =

Z

0

2π

Z

π π 2

9,2) .(2&4%)7 ,& +'&1& %&1'*.% )

V = 2kπσR2

Z

kσR2 sin(γ)dγdθ p π

π 2

R2 + z 2 − 2zR cos(γ)

sin(γ) p 2 R + z 2 − 2zR cos(γ)

!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

!

"#$# $%&'()%$ %&*'+ ,%-%.'& &#-%$ /#(/0(#$ (# 12*%3$#( ,% (# 4'$.#5 Z sin γ p dγ A + B cos(γ)

6#/%.'& %( /#.-1' ,% )#$1#-(% u2 = A + B cos(γ)7 8% %&*# .#2%$# dγ = Z Z 2 2 sin γ p 1du = − u dγ = − B B A + B cos(γ) p :'.' u = A + B cos(γ)5 Z sin γ 2p p dγ = − A + B cos(γ) B A + B cos(γ)

2udu −B sin(γ) 7

9 #&1

"#$# %( /#(/0(' ,% ;'*%2/1#( *%2%.'& <0% A = R2 + z 2 = B = −2zR7 >%%.;(?#2,' %2 (# 12*%3$#(5 Z sin γ 1 p 2 p R + z 2 − 2Rz cos(γ) dγ = zR R2 + z 2 − 2zR cos(γ)

@A'$#+ ;',%.'& /#(/0(#$ %( 12*%3$#( &12 ,1B/0(*#,%&5

V (z) =

!

π p 2kπσR2 p 2 2kπσR ( R + z 2 − 2Rz cos(γ) π ) = (z + R − R2 + z 2 ) zR z 2

@A'$# <0% *%2%.'& %( ;'*%2/1#( %(C/*$1/' %2 z kˆ+ %( /#.;' %(C/*$1/' &% ;0%,% /#(/0(#$ /'.'5

~ = −∇V E D%.'& <0% %( ;'*%2/1#( %&*# &'(' %2 402/1'2 ,% (# )#$1#-(% z = ;'$ (' *#2*' *%2,$%.'& <0% %( /#.;' %(%/*$1/' %&5

~ = − dV kˆ E dz 8%$1)#.'& V (z)5

dV dz

p d 1 ( (z + R − R2 + z 2 )) dz z p −1 1 1 1 = 2kπσ( 2 (z + R − R2 + z 2 ) + (1 + √ 2z)) 2 z z 2 R + z2 √ 1 −1 R R2 + z 2 1 − − + +√ = 2kπσ( ) 2 2 z z z z R + z2 √ R2 + z 2 − R 2 √ = −2kπσR ( ) z 2 R2 + z 2 = 2kπσR

8% %&*# .#2%$# %( /#.;' %(%/*$1/' %2 %( ;02*' z kˆ &%$E5

!

√ R2 + z 2 − R ˆ 2 ~ √ )k E(z) = 2kπσR ( z 2 R2 + z 2

! "#$%& '(# )* #+,-#&.%/ */0#-.%- #&0* ./1#2/.1* #/ z = 03 4./ #$5*-6%7 ,%1#$%& *8#-8*-/%& * O 0*/0% 8%$% '(#-*$%& 9 8*)8()*- #) :*)%- 1#) 8*$,% #);80-.8%3<-.$#-% :#$%& '(# )* #+,-#&.%/ 1#) 8*$,% #)#80-.8% &# ,(#1# -##&8-.5.- 8%$%= 2

2kπσR (

1−

√ R z 2 +R2 z2

)

>(#6%7 0%$*-#$%& #) ).$.0# 8(*/1% z → 0 1# )* ?(/8.%/ E(z)kˆ@= 2

l´ım E(z) = 2kπσR l´ım (

z→0

z→0

A%$% :#$%&7 #&0# ).$.0# #& 1# )* ?%-$*

2kπσR2 l´ım ( z→0

1−

1−

0 0

√ R z 2 +R2 z2

√ R z 2 +R2 z2

)

,%- )% '(# (0.).B*-#$%& >CDE,.0*) ,*-* 8*)8()*-)%=

) = 2kπσR2 l´ım ( z→0

(1 −

√ R )′ z 2 +R2 ) (z 2 )′ 3

R(z 2 + R2 )− 2 = 2kπσR l´ım ( ) z→0 2z 1 = 2kπσR2 · 2R2 = kπσ 2

~ F# #&0* $*/#-* E(0) = kπσ kˆ

!

!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

!"#$%&' (

!"#$%&!'() !"#$%&' ()

! "#!$!$ %&' !'()*+' ,&$%-,"&*+' %! *+%#&' r1 , r2 . ,+*/+' q1 , q2 '!0+*+%+' 0&* -$+ /*+$ %#'1 "+$,#+ d >> r1 , r2 2 # +34+' '! ,&$!,"+$ + "*+5)' %! -$ ,+46! ,&$%-,"&* 7%!'0*!,#+46!8 9-! '#*5! ':6& 0+*+ "*+$'0&*"+* ,+*/+ %! -$+ + &"*+;8 !$,-!$"*! 6+' %!$'#%+%!' %! ,+*/+ '-0!*<,#+6!' %! ,+%+ -$+ 7!$ (-$,#:$ %! 6+' 5+*#+46!' ,&$&,#%+'; -$+ 5!= 9-! !6 '#'"!3+ +6,+$=+ !6 !9-#6#4*#&2

*"$+,-./0

>+%& 9-! !6 '#'"!3+ !'"? !$ -$+ *!/#:$ +,&"+%+ %!6 !'0+,#&8 0&%!3&' "&3+* ,&3& 0-$"& %! *!(!*!$,#+ %!6 0&"!$,#+6 !6 #$<$#"& ! #/-+6+* !6 0&"!$,#+6 + ,!*& +66@8 !' %!,#*8 V (+∞) = 02 A'@8 !6 0&"!$,#+6 '&4*! 6+' '-0!*<,#!' %! 6+' !'(!*+' ,&$%-,"&*+' !' V1 = k

q1 q2 , V2 = k r1 r2

%&$%! B!3&' '-0-!'"& 9-! !6 0&"!$,#+6 %! -$+ !'(!*+ $& !' +(!,"+%& 0&* !6 %! 6+ &"*+ 7& 3?' 4#!$ !' +(!,"+%& %! (&*3+ %!'0*!,#+46!;8 %+%& 9-! !'"?$ 3-. 3-. '!0+*+%+8 !' %!,#*8 d >> r1 , r2 8 . "+30&,& !6 ,+30& %! -$+ !'(!*+ *!%#'"*#4-.! 6+ ,+*/+ %! 6+ &"*+2 C&* ,&$'!*5+,#:$ %! ,+*/+8 '# (q1 )f , (q2 )f '&$ 6+' ,+*/+' !$ ,+%+ !'(!*+ -$+ 5!= +6,+$=+%& !6 !9-#6#4*#&8 "!$!3&' 9-! q1 + q2 = (q1 )f + (q2 )f

A6 +6,+$=+* !6 !9-#6#4*#&8 !' %!,#*8 ,-+$%& %!D+ %! B+4!* "*+$'(!*!$,#+ %! ,+*/+' !$"*! 6+' !'(!*+'8 "!$!3&' 9-! 6+ %#(!*!$,#+ %! 0&"!$,#+6 !6),"*#,& !$"*! +34+' !' $-6&8 !' %!,#*8 V1 (q1 )f k r1 2 4π(r1 ) (σ1 )f r1 (σ2 )f =⇒ (σ1 )f

= V2 (q2 )f = k r2 4π(r2 )2 (σ2 )f = r2 r1 = r2

>! !'"& 5!3&' 9-! !$ /!$!*+68 6+' *!/#&$!' !$ 6+ '-0!*<,#! %! -$ ,&$%-,"&* ,&$ 3!$&* *+%#& %! ,-*5+"-*+ 70-$"+'; ,&$,!$"*+$ -$+ 3+.&* %!$'#%+% '-0!*<,#+6 %! ,+*/+8 0&* 6& ,-+6 !6 EF

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(q1 )f + (q2 )f 2

2

4π(r1 ) (σ1 )f + 4π(r2 ) (σ2 )f

= q1 + q2 = q1 + q2

4πr1 (σ1 )f (r1 + r2 ) = q1 + q2 1 q1 + q2 =⇒ (σ1 )f = 4πr1 r1 + r2 1 q1 + q2 (σ2 )f = 4πr2 r1 + r2

!

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+"$,-./01

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-+'(%%+#, '&-+&1G ($C

Q1 rˆ 4πǫ0 r2 9:#'&3 1& -+)('(,%+& -( 0#/(,%+&1 (,/'( &5<&$ $.0('%+($ ($C Z a Q1 1 1 Q1 dr = ( − ) V2 − V1 = − 2 4πǫ0 b a b 4πǫ0 r 7#, ($/#3 0#-(5#$ #
ab b−a 9:#'&3 %&1%.1&'(5#$ (1 %&50# (1(%/'+%# & .,& -+$/&,%+& r > b ./+1+@&,-# 1(2 -( H&.$$6 I1(8+5#$ .,& %#5# $.0(';%+( 8&.$$+&,& .,& %&$%&'& -( '&-+# r > b 2 0'#%(-(5#$ -( +8.&1 )#'5&6 B# F.( -(<(5#$ %#,$+-('&' (, ($/( %&$#3 ($ F.( 1& %&8& (,%(''&-& 0#' ($/& $.0(';%+( ($ Q1 + Q2 6 "( ($/& 5&,('& 4(5#$ F.(C Q1 = 4πǫ0 (V2 − V1 )

E4πr2 =

Q1 + Q2 ǫ0

!"#$%&' () '*+% $',-.

!

~ = Q1 + Q2 E 4πǫ0 r2 "#$ %&'#( )#*%+#& ,-,./-0 %/ )#'%$,1-/ %$ b2

V2 = −

Z

b

∞

Q1 + Q2 Q1 + Q2 dr = 4πǫ0 r2 4πǫ0 b

3% %&'- +-$%0-( /- ,-04- *% ,-&,-0- %5'%01#0 &%062

Q2 = 4πǫ0 b(V1 + (V2 − V1 )

ab ) a−b

! 7-0- ,-/,./-0 %/ '0-8-9# '#'-/( ,#$&1*%0-0%+#& :.% /-& ,-04-& :.% ,#+)#$%$ /- ,-&,-0- *% 0-*1# c > b /-& %&'-+#& '0-;%$*# *%&*% %/ 1$<$1'# =-&'- .$- *1&'-$1,- c> "/-0-+%$'%( %/ '0-8-9# $%,%&-01# )-0- /- ,-*- ,-04- dq ,#$'%$1*- %$ /- ,-&,-0- &%0- %/ +1&+#> ?&'% '0-8-9# &% ,-/,./,#+# dW = dqV (c) 3% %&'- +-$%0-( %/ '0-8-9# '#'-/ &%06 &1+)/%+%$'%

W = QV (c) > @# :.% '%$%+#& :.% =-,%0 %& ,-/,./-0 %/ )#'%$,1-/ %$ r = c :.% %& *10%,'-+%$'%

V (c) = A&1(

W =

Q1 + Q2 4πǫ0 c

Q(Q1 + Q2 ) 4πǫ0

!

!"#$%&' ()

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,"$-./"0

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!

!"#$%&' ()

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~ = K( r )4 rˆ E a =,#-( K (& .# %,#&/$#/( 1,' -(/('8+#$' 0 rˆ (& (6 7(%/,' .#+/$'+, '$-+$63 4( 1+-( (#%,#/'$'> '* ?$ -(#&+-$- -( %$'2$ ρ(r) (# (6 7,6.8(# +#/('+,' -(6 %$&%$',#3 #* ?$& -(#&+-$-(& -( %$'2$& &.1('@%+$6(& $6 +#/('+,' 0 $6 (A/('+,' -(6 %$&%$',#3 +* B6 1,/(#%+$6 (6(%/',&/C/+%, (# /,-, (6 (&1$%+,3

,"$-+./01 '* ?$ 6(0 -( 2$.&& &( 1.(-( (&%'+5+' (# &. ),'8$ 8$& %,8.# %,8,> φ=

qint ~ ·n E ˆ dS = ǫ0 S

Z

4+# (85$'2,D &( 1.(-( ./6+E$' /$85+(# 6$ ),'8$ -+)('(#%+$6 -( 6$ 6(0 -( F$.&& :.( (&/$56(%(>

~ ·E ~ = p(r) ∇ ǫ0 G,8, 7(8,&D (6 %$81, (6(%/'+%, (&/$ &,6, (# ).#%+,# -( rD -( (&/$ 8$#('$ /(#(8,&

~ ·E ~ = ∇ = = =

1 ∂ 2 (r E(r)) r2 ∂r 1 ∂ 2 r 4 (r K( ) ) r2 ∂r a 6 1 ∂ Kr ) ( r2 ∂r a4 6Kr3 a4

=( (&/$ 8$#('$D 6$ -(#&+-$- -( %$'2$ (# (6 +#/('+,' -(6 %$&%$',# &('C>

ρ(r) =

6ǫ0 Kr3 a4

! "#$%&' ()*)+$, ()-)%+./&% 0& 1$/,-&/-) K 2 3&*)+$, 45) 0& 1&%6& -$-&0 &0+&1)/&(& )/ )0 ./-)%.$% ()0 1&,1&%$/ ), q ' 7$% 0$ -&/-$ ,) 8& & 15+70.% 0& %)0&1.$/9

q =

Z

ρ(r)dV Z a 6ǫ0 Kr3 = 4π 4πr2 dr 4 a 0 a 24ǫ0 K r6 ) = a4 6 0

= 4πa2 ǫ0 K

:) ),-& +&/)%&

K=

q 4πǫ0 a2

",;' 7$()+$, ),1%.*.% 0& ()/,.(&( () 1&%6& )/ <5/1.$/ () 0& 1&%6& -$-&0 &0+&1)/&(&9

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σint = −

q 4πa2

>$/ ),-$ ? 5-.0.@&/($ 0& 1$/,)%8&1.$/ () 0& 1&%6& 8)+$, 45) )/ 0& ,57)%=1.) )A-)%.$% ,) -)/(%& 5/& 1&%6& Q + q 2 >$/ ),-) %),50-&($ 7$()+$, 1&0150&% 0& ()/,.(&( () 1&%6& ,57)%=1.&0 )A-)%.$%9

σext =

Q+q 4πa2

:$/() #)+$, (),7%)1.&($ δ 2

"! B&%& ,&*)% 7$-)/1.&0 )/ -$($ )0 ),7&1.$ ()*)+$, ,&*)% )0 1&+7$ )0C1-%.1$2 B&%& 0& %)6.$/ ./-)%.$% ?& -)/)+$, )0 1&+7$ )0)1-%.1$9

~ int = E

q r ( )4 rˆ, 4πǫ0 a2 a

para r < a

D/ )0 )A-)%.$% ()0 1$/(51-$% -)/)+$, -)/)+$, 45) )0 1&+7$ )0)1-%.1$ ), ,.+70)+)/-)9

~ ext = Q + q rˆ, E 4πǫ0 r2

si r > a

",. )0 7$-)/1.&0 7&%& 5/& %)6.$/ )/ 45) r > a ),9 Z Z Q+q r 1 Q+q ~ V (r) = − Eext d~r = − dr = 2 4πǫ0 ∞ r 4πǫ0 r "#$%&' 7&%& 0& %)6.$ ./-)%.$% Er < aF )0 7$-)/1.&0 0$ 1&0150&+$, 1$+$9

!"#$%&' () '*+% $',-.

!

V =−

Z

~ · d~r = − E

Z

∞

V (r) = − = − =

b

~ ext · d~r + − E

Z

Z

Z

b

r

~ int · d~r, E

si r > a

~ · d~r E b

∞

~ ext · d~r + − E

Z

r

b

~ int · d~r E

Q+q q − (r5 − a5 ) 4πǫ0 a 20πa6 ǫ0

"#$% #& '$&(& )*&

V (r) =

q Q+q (r5 − a5 ), − 4πǫ0 a 20πa6 ǫ0

si r < a

!

!"#$%&' ()

"#$%&$' '$ %#()* '$+%,-.%* )-*/&%./* )*- &0# /.1,-.2&%.30 /' %#-4# ρ(r) ,#$ 5&' '$ )*,'0%.#$ 5&' '1,# /.1,-.2&%.30 )-*/&%' '1,# /#/* )*-

V (r) = q

e−λr 4πr2 ǫ0

6&'4*7 '0%&'0,-' $# /.1,-.2&%.30 /' %#-4# ρ(r)8

*"$+,-./0

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q ~ = − ∂V rˆ = e−λr (1 + λr)ˆ r E ∂r 4πǫ0 r2 D1,' -'1&$,#/* 0*1 )'-(.,' '0%*0,-#- 1.0 /.E%&$,#/ $# /'01./#/ /' %#-4#8 D1,* 1' @#%' &,.$.<#0/* $# =*-(# /.='-'0%.#$ /' $# $': /' F#&11C

~ ·E ~ = ρ ∇ ǫ0 ~ = E rˆ7 /' '1,# (#0'-# 9'0'(*1 5&' '$ %#()* '1 -#/.#$ : )*- $* ,#0,* E 1 ∂ 2 (r E(r)) r2 ∂r 1 q = (−λe−λr (1 + λr) + λe−λr ) 4πǫ0 r2 1 q 2 −λr λ e = − 4πǫ0 r

~ ·E ~ = ∇

A*- $* ,#0,*

ρ(r) = −

1 q 2 −λr λ e 4π r

!

!"#$%&' () '*+% $',-.

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Z

a

b

~ · d~r E

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!

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S

;$ 7$,<$ -#(+,-., qint +* Q8 =64-7$#'. 4$ 4+% '+ <$&** /+0.* 1&+ E2πrℓ =

~ = E

Q ǫ0

% 6., 4. ($#(.

Q rˆ 2πǫ0 rℓ

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~ · d~r = E rˆ · d~r = Edr = E

Q dr 2πǫ0 rℓ

F+ +*($ 0$#+,$ -#(+<,$#'. +*($ +G6,+*-C# '+*'+ r = a D$*($ r = b:

Vb − Va = −

Z

a

b

Q Q dr = − 2πǫ0 rℓ 2πǫ0 ℓ

Vb − V a = −

b Q ln( ) 2πǫ0 ℓ a

Z

a

b

1 dr r

=D.,$? 7.#*-'+,$0.* ∆V 7.0. 4$ 0$<#-(&' '+ 4$ '-H+,+#7-$ '+ 6.(+#7-$4 % '+ +*($ H.,0$ '+ 4$ +G6,+*-C# 6$,$ 4$ 7$6$7-'$' /+0.* 1&+:

C=

Q = ∆V

Q Q ln( ab ) 2πǫ0 ℓ

=

2πℓǫ0 ln(b/a)

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Q (2πR)L 4

."( 1" ?)'

△V =

Q · ln ǫ0 πL

2d − R R

=⇒ |△V | =

Q · ln ǫ0 πL

;'("4 *'#'+"$ ?)' 1, 0,.,0%*,#0%, ."( )#%&,& &' 1"#6%*)& '$ C =

=⇒ C =

ǫ0 π ln

2d−R R

∼ =

ǫ0 π ln 2d R

2d − R R

Q |△V |L 4

!!

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!

!"#$%&' ()

"#$%&$' $# (&')*# '+,)' $#- .$#%#- /' &+ %0+/'+-#/0) /' .$#%#- .#)#$'$#- /' 1)'# A 2 -'.#)#%34+ d5 6#)# '-,07 #$'8' &+ .09&3,0 :3+;+3,'-3<#$<'+,'= $#- .$#%#- /'$ %0+/'+-#/0) 2 /3># 9&? .#-# %0+ $# (&')*# '+ $0- -3>&3'+,'- %#-0-@ #= A$ %0+/&%,0) '-,1 #3-$#/0 %0+ %#)># Q B= A$ %0+/'+-#/0) '-,1 %0+'%,#/0 # &+# B#,')C# /' /3(')'+%3# /' .0,'+%3#$ V0 5

*"$+,-./0 #= D'+'<0- 9&' $# %#.#%3,#+%3# 2 $# '+')>C# #%&<&$#/# /'$ %0+/'+-#/0) -0+

C(x) = ǫ0

A x

U=

1 Q2 1 Q2 x =⇒ U (x) = 2 C 2 Aǫ0

60) ,#+,07 $# (&')*# '-

1 Q2 ~ = − dU x ˆ=− x ˆ F~ = −∇U dx 2 Aǫ0 2 $#- .$#%#- -' #,)#'+ :+0,' 9&' $# (&')*# +0 /'.'+/' /' E=5 B= D'+'<0- 9&' $# %#.#%3,#+%3# 2 $# '+')>C# #%&<&$#/# /'$ %0+/'+-#/0) -0+

C(x) = ǫ0

A x

1A 1 C(x)(V0 )2 =⇒ U (x) = ǫ0 (V0 )2 2 2x

A+ '-,' %#-07

Aǫ0 (V0 )2 1 ~ = + dU x F~ = +∇U ˆ=− x ˆ dx 2 x2 2 $#- .$#%#- -' #,)#'+ :+0,' 9&' $# (&')*# /'.'+/' /' E=5 F3 G#%'<0- x = d7 V0 1 Q2 Aǫ0 (V0 )2 1 x ˆ = −CV x ˆ = − x ˆ F~ = − 0 2 d2 2d 2 Aǫ0

!"#$%&' () '*+,*-!+'.,-

!"

#$% &$ '() &* +()%,* )- &* ./-.* 01$.$ 2)3) -)%45 6) 2)7* *& &)18$% )91$98%*% )-8*- 2$- ):#%)-/$9)- 1$9-/2)%*92$ '() &* +()%,* #$% (9/2*2 2) ;%)* -$3%) &* -(#)%<1/) 2) (9 1$92(18$% )2

σ n ˆ F~ /A = − 2ǫ0

!"

!"#$%&' ()

#$%&'()*) +%, &+-)*./') /$%(+/0$*, )&12*'/, () *,('$ a /$%/2%0*'/, , $0*, () *,('$ b3 /$% a )?0)*'$* d3 /$%/2%0*'/$ , ,56,& &+-)*./')& > %)+0*$8 @' '%'/',:5)%0) :, &+-)*./') '%0)*'$* 0')%) /,*A, +Q > :, )?0)*'$* −Q3 ('&/+0, B+2 $/+**) /$% :, /,-,/'0,%/', (): &'&0)5, :+)A$ B+) &) -$%) ): ,*5,;<% > )%/+)%0*) :, /,-,/'0,%/', '%'/',: > .%,: (): &'&0)5,8

*"$+,-./0

C$()5$& )%/$%0*,* ): /,5-$ ):2/0*'/$ )%0*) :,& ($& &+-)*./')&3 ,%0)& B+) &) -$%A, ): ,*5,;<%3 +&,%($ :, :)> () D,+&&8 E)%)5$& B+) ): /,5-$ 0')%) &'5)0*F, )&12*'/, > G, )% ): &)%0'($ () ~ r) = E(r)ˆ rˆ3 )& ()/'*3 E(~ r8 E$5,%($ /$5$ &+-)*./') () '%0)A*,/'<% +%, &+-)*./') )&12*'/, /$%/2%0*'/, , :,& &+-)*./')&3 () *,('$ a < r < b3 0)%)5$&

Q ~ r) · n ~ r) = Q rˆ E(~ ˆ dS = 4πr2 E(r) = =⇒ E(~ ǫ 4πǫ0 r2 0 Γ

I

H&F3 :, ('1)*)%/', () -$0)%/',: )%0*) :,& &+-)*./')& )&

△V1 = −

Z

b

a

~ r) · d~r = − Q E(~ 4πǫ0

($%() ): /,5'%$ 0$5,($ 1+) *,(',:8 C$* 0,%0$3 :, /,-,/'0,%/', '%'/',: )&

Ci =

Z

b

a

1 1 dr Q 1 a Q = − = r2 4πǫ0 r b 4πǫ0 a b

Q = △V1

4πǫ0 − 1b

1 a

H: '%0*$(+/'* ): ,*5,;<%3 ): -$0)%/',: )%0*) :,& &+-)*./')& ('&5'%+>)3 -$* :$ /+,: )&-)*,5$& B+) :, /,-,/'0,%/', ,+5)%0)8 9% :, &+-)*./') '%0)*'$* (): ,*5,;<% &) ('&0*'6+>) +%'1$*5)5)%0) +%, /,*A, −Q8 9&0$ &) ()&-*)%() () :, :)> () D,+&&8 9% )1)/0$3 +&,%($ /$5$ &+-)*./') () '%0)A*,/'<% +%, &+-8 )&12*'/, /<%/)%0*'/, , :,& $0*,& > ()%0*$ (): ,*5,;<%3 /$5$ ): ,*5,;<% )& /$%(+/0$* > )&0= )% )B+':'6*'$3 ): /,5-$ ()%0*$ () 2: ()6) &)* %+:$3 -$* 0,%0$

Q + Qc ~ r) · n E(~ ˆ dS = 0 = =⇒ Qc = −Q ǫ0 Γ

I

> /$5$ ): ,*5,;<% )& %)+0*$3 , &+ G); &) ('&0*'6+>) +%'1$*5)5)%0) )% :, &+-)*./') )?0)*'$* (): ,*5,;<% +%, /,*A, Q8 9% ): '%0)*'$* () :, -*'5)*, &+-)*./')3 )& ()/'*3 -,*, a ≤ r3 ): /,5-$ )& %+:$ 4-+)& %$ I,> /,*A, ()%0*$ () 2:3 -$* :)> () D,+&&7 ,: 'A+,: B+) -,*, r > b8 C,*, )%/$%0*,* ): /,5-$ )% d < r a < r < c +&,5$& ): 5'&5$ -*$/)('5')%0$ B+) )% ): /,&$ ,%0)*'$*3 -$* 0,%0$3 $60)%)5$& B+) ): /,5-$ ):2/0*'/$ )&

~ r) = E(~

(

~0 Q rˆ 4πǫ0 r2

b < r, a > r c < r < b, a < r < c

!"#$%&' () '*+,*-!+'.,-

!"

#$%&'( %) *'+%,-./) %,+0% )/1 1$*%02-.%1 %3+%0,/ % .,+%0,/ %1

△V2 =

Z a Z b ~ r) · d~r = ~ r) · d~r = − E(~ E(~ b a Z b Z d Z c ~ r) · d~r ~ r) · d~r + ~ r) · d~r + E(~ E(~ E(~ = d c a Z b Z c ~ r) · d~r ~ E(~ E(~r) · d~r + = d a Z c Z b Q dr dr = + 2 2 4πǫ0 r d r a Q 1 1 1 1 = + − − 4πǫ0 a d c b

4'0 +/,+'( )/ -/*/-.+/,-./ 2,/) %1

Cf =

Q = △V2

1 a

4πǫ0 + − 1c − 1b 1 d

4'5%6'1 7%0 8$% Ci < Cf 5% )' 1.&$.%,+%9 :%,%6'1 8

%$c < d =⇒

1 1 1 1 1 1 4πǫ0 1 1 − < 0 =⇒ − + − < − =⇒ 1 1 = Ci < Cf = d c a b d c a b a − b

1 a

4πǫ0 − + d1 − 1 b

1 c

Cf *'50;/6'1 *.5/6%,+% ,'+/,5' 8$% /) .,+0'5$-.0 %) /06/?@, +%,%6'1 5'1 -',5%,1/5'0%1 %, 1%0.%9 4'0 +/,+'( -'6' -','-%6'1 )/ -/*/-.+/,-./ %,+0% 5'1 1$*%02-.%1 %1AB0.-/1 -',-B,+0.-/1( *'5%6'1 %,-',+0/0 )/ -/*/-.+/,-./ %8$.7/)%,+% 5%) 1.1+%6/9 C, %A%-+'( 1%/ C2 )/ -/*/-.+/,-./ 5% )/1 5'1 1$*%02-.%1 %3+%0.'0%1 D C1 )/ 5% )/1 .,+%0.'0%19 E% %1+/ A'06/ +%,%6'1 4πǫ0 4πǫ0 C2 = 1 1 , C1 = 1 1 d − b a − c 4'0 +/,+'( )/ -/*/-.+/,-./ %8$.7/)%,+% %1+> 5/5/ *'0 1 1 1 1 1 1 1 1 =⇒ Cf = = + = − + − Cf C1 C2 4πǫ0 a b d c 8$% %1 %) 6.16' 0%1$)+/5' '=+%,.5' /,+%0.'06%,+%9

1 a

4πǫ0 − + d1 − 1 b

1 c

!"

!"#$%&' ()

#$ %&'&%()*+ )(,$, '-&%&. %/&0+&0&.1 %&0& /$& 0, -&0* a1 2*+3&$0* /$ 4$5/-* θ %*3* 3/,.)+& -& 65/+&7 8,3/,.)+, 9/,1 '&+& θ ',9/,:*1 -& %&'&%()&$%(& ,.)& 0&0& '*+

C=

ǫ0 a2 aθ (1 − ) d 2d

*"$+,-./0

;* 9/, <&+,3*. ,. 0(=(0(+ -&. '-&%&. %/&0+&0&. ,$ -43($&. ',9/,:&. 0, &$%<* dx > -&+5* a )&%*3* 3/,.)+& -& 65/+&?

@*0,3*. %*$.(0,+&+ ,.)&. '-&%&. %*3* '&+&-,-&.7 @&+& /$ %*$0,$.&0*+ 0, '-&%&. '&+&-,-&. ., )(,$, 9/, -& %&'&%()&$%(& ,. ǫ0 A C= H 7 8*$0, H ,. -& 0(.)&$%(& ,$)+, -&. '-&%&.7 ;/,5*1 '&+& -&. ',9/,:&. '-&%&. ($0(%&0&. ),$0+,3*. 9/, ,- 4+,& ,. A = adx > -& 0(.)&$%(& H =&+(&+& &- =&+(&+ x? H = H(x)7 A*$.(0,+,3*. -& .(5/(,$), %*$65/+&%(B$?

A*3* =,3*. ., %/3'-, -& +,-&%(B$

h(x) a sin(θ) = x a

!"#$%&' () '*+,*-!+'.,-

!"

#$% &$ '()'$ h(x) = x sin(θ)* +,%$ -$.$ θ ,/ +,01,2$* +$3,.$/ -$)/43,%(% sin(θ) = θ -$) &$ -1(& 01,3( h(x) = xθ5 6, &( 781%( ()',%4$% 9,.$/ 01, H(x) = d + h(x) : %,,.+&(;()3$ -$) &( %,&(-4<) ()',%4$% %,/1&'( h(x) = d + θx =$) ,/'$* ',)3%,.$/ 01, &( -(+(-43(3 3, ,/', +,01,2$ -$)3,)/(3$% ,/

dC =

ǫ0 adx d + θx

>)',8%()3$ 3,/3, x = 0 ( x = a?

C=

Z

0

a

ǫ0 adx ǫ0 a d + θa = ln( ) d + θx θ d

=$.$ +$3,.$/ 9,%* (& /,% θ +,01,2$* /, -1.+&, 01, d+θa d ≈ 15 #$3,.$/ (@$%( (+%$A4.(% +$% d+θa '(:&$%B/,81)3$ $%3,)C &( ,A+%,/4<) ln( d )5 D& 3,/(%%$&&$ 3, '(:&$% +(%( ln(x) (&%,3,3$% 3, ,/ ?

1 ln(x) = (x − 1) − (x − 1)2 2 D9(&1()3$ ,) x =

d+θa d ?

aθ 1 (aθ)2 d + θa )= − d d 2 d2 E,,.+&(;()3$ ,/', 9(&$% ,) &( ,A+%,/4<) ,)-$)'%(3( +(%( &( -(+(-4'()-4( C %,/1&'(? ln(

C=

ǫ0 a aθ aθ ǫ0 a aθ 1 (aθ)2 )= ( − ( (1 − )) 2 θ d 2 d θ d 2d C=

F1, ,/ &$ 01, 01,%G(.$/ 3,.$/'%(%5

ǫ0 a2 aθ (1 − ) d 2d

!"

!"#$%&' ()

#$% &'(&)% *& )%*+, a '& -%).% % /,0&$-+%1 V0 2 '& %3'1%4 5,'0&)+,)6&$0& '& -,$&-0% % 0+&))% % 0)%78' *& 9$ -,$*&$'%*,) -92% -%/%-+*%* &' C : -,6, 69&'0)% 1% ;.9)%4<5,) *&;$+-+=$ 1% 0+&))% &'0% % /,0&$-+%1 -&), +$*&/&$*+&$0& *& 1% -%).% >9& %*>9+&)%? '* @%1-91& &1 /,0&$-+%1 ;$%1 *& 1% &'(&)% 2 1% -%).% ;$%1 &$ 1% &'(&)% 2 &$ &1 -,$*&$'%*,)4 #* A@9B$0% &$&).3% '& *+'+/, %1 C%-&) 1% -,$&D+=$ % 0+&))%E

+"$,-./01 '*F1 /,0&$-+%1 *& 9$% &'(&)% *& )%*+, a &$ '9 '9/&);-+& &' V

= kQ a *,$*& Q &' 1% -%).% %16%-&$%*% &$ 1% &'(&)%4 5%)% $9&'0), /),G1&6% '& 0+&$& >9& +$+-+%16&$0& 1% &'(&)% &'0% % 9$ /,0&$-+%1 V0 4 5,) 1, *+-C, %$0&)+,)6&$0& '& 0&$*)B >9& V0 =

kQ0 a

(1)

H,$*& Q0 &' 1% -%).% >9& 0+&$& 1% &'(&)% +$+-+%16&$0&4 H&'/98' *& -,$&-0%) 1% &'(&)% %1 -,$*&$'%*,): 1% &'(&)% >9&*% -,$ 9$% -%).% Qf 2 '9 /,0&$-+%1 '&)B &$0,$-&' kQf Vf = (2) a I &1 -,$*&$'%*,) >9&*% -,$ -%).% Qc 2 /,0&$-+%1 Vf : 2% >9& &'0% -,$&-0%*%, -,$ 1% &'(&)% % 0)%7&J *& 9$ -,$*9-0,)4 5,) *&;$+-+=$ C = QVc /,) 1, 0%$0,

Vf =

Qc C

(3)

K*&6B': /,) -,$'&)7%-+=$ *& -%).%

Q0 = Qf + Qc kCQ

H&
kQ0 a+kC

kQ0 a + kC Q0 QC = kC a + kC

VC = Vf =

(4)

!"#$%&' () '*+,*-!+'.,-

!"

! #$%&%'()*$+* (' *$*,-.' /* 0*1* ' (' 2,*/*$&%' 0*( &')23 *(4&+,%&3 *$ (' */5*,'6 7* +%*$* 89* 1 U = ǫ0 2

Z

E 2 dV

V

:/ &(',3 89* *( &')23 *(4&+,%&3 2,309&%03 23, (' */5*,' &',-'0' &3$ Q0 ' 9$' 0%/+'$&%' r > a */ kQ0 E= 2 r :/&,%1%)3/ *( *(*)*$+3 0* ;3(9)*$ &3)3< dV = 4πr2 dr =* */+' )'$*,' +*$*)3/ 89* (' *$*,-.' '/3&%'0' /*,>

1 kQ0 2πk 2 Q0 2 1 dr dUi = ǫ0 E 2 dV = ǫ0 2 4πr2 dr = ǫ0 2 2 r r2 #$+*-,'$03 ,*/9(+' 89* Z ∞ 2πk 2 Q0 2 kQ0 2 2πk 2 Q0 2 ǫ0 ǫ0 Ui = = dr = r2 a 2a a ?' *$*,-.' @$'( /* 0*1* '( &')23 *(4&+,%&3 0* (' */5*,' A '( &3$0*$/'03,6 ?' *$*,-%' '/3&%'0' kQ 2 '( $9*;3 &')23 *(4&+,%&3 /* &'(&9(' 0* %-9'( 53,)' A /*,> 2af 6 B3, 3+,3 ('03C (' &',-' '/3&%'0' ' 9$ &3$0*$/'03, */ UC =

Q2C 2C 6

=* */+' )'$*,' (' *$*,-.' @$'( /*,><

kQf 2 QC 2 + 2a 2C D**)2('E'$03 (3/ ;'(3,*/ 0* */+' &',-' *$ 59$&%3$ 0* Q0 < Uf =

Uf =

kQ0 2 2(a + kC)

=* */+' )'$*,'C (' *$*,-.' 0%/%2'0' /*,>

∆U = Uf − Ui =

kQ0 2 1 1 ( − ) 2 a + kC a

!"

!"#$%&' ()

#$%&'()*) )+ ,'*,-'.$ /-) &) 0-)&.*1 )% +1 23-*14 5*'0)*$ &) ,1*31 )+ ,161,'.$* C1 7 ,)**1%($ )+ '%.)**-6.$* S1 4 8-)3$ &) ,1*31 )+ ,$%()%&1($* C2 ,)**1%($ S2 4 #1+,-+) +1 ,1*31 2%1+ () ,1(1 ,$%()%&1($* 9()&6-:& () ,)**1* S2 ; < )+ ,10='$ () )%)*3>1 )% +1& &'3-')%.)& &'.-1,'$%)&? '* @+ '%.)**-6.$* S1 6)*01%),'A ,)**1($4 #* @+ '%.)**-6.$* S1 &) 1=*'A 1%.)& () ,)**1* S24

+"$,-./0 '* B+ ,)**1* S1 &) .)%(*C /-) +1 ('D)*)%,'1 () 6$.)%,'1+ 1 +1 ,-1+ )&.1 &$0).'($ )+ ,$%()%&1($*

C1 &)*C ∆V 4 B&' +1 ,1*31 /-) 1(/-')*) )&) &'06+)0)%.) Q1 = C1 ∆V

5$* $.*1 61*.) )+ ,$%()%&1($* C2 %$ &) ,1*317 6$* +$ .1%.$ Q2 = 0 E) )&.1 01%)*17 +1 )%)*3>1 '%','1+ ()+ &'&.)01 &)*C 1 Ei = C1 ∆V 2 2 @&,*'.1 )% D-%,'A% () Q1 ? 1 Q21 Ei = 2 C1 BF$*17 ,)**10$& )+ '%.)**-6.$* S2 01%.)%')%($ S1 ,)**1($4 81 ('D)*)%,'1 () 6$.)%,'1+ 61*1 10=$& ,$%()%&1($*)& &)*C +1 0'&01? ∆V 4 G 1&>7 &'06+)0)%.) +1& ,1*31& &)*C%?

Q′1 = ∆V C1 Q′2 = ∆V C2 81 )%)*3>1 ()+ &'&.)01 &)*C +1 16$*.1(1 6$* C1 < C2 ? Ef = 21 C1 ∆V 2 + 12 C2 ∆V 2 B&>7 )+ ,10='$ () )%)*3>1 &)*C? 1 Ef − Ei = C2 ∆V 2 2

#*

H' 1=*'0$& S1 1%.)& () ,)**1* S2 .)%(*)0$& /-) +1 ('D)*)%,'1 () 6$.)%,'1+ ∆V <1 %$ '%I-<)4 #$0$ C2 )&.1 ()&,1*31($ &) .)%(*C /-) +1 ,1*31 Q1 &) *)('&.*'=-'*C () .1+ 01%)*1 /-) )+ &'&.)01 /-)(1 )% )/-'+'=*'$4 8+101*)0$& Q′1 < Q′2 +1& ,1*31 /-) 1(/-')*)% +$& ,$%()%&1($*)& C1 < C2 7 *)&6),.'J10)%.)4 @% )&.) ,1&$ .)%)0$& -%1 ,$0='%1,'A% )% 61*1+)+$ () ,$%()%&1($*)& < &1=)0$& /-) )% )&.1 &'.-1,'A% +1 ('D)*)%,'1 () 6$.)%,'1+ )% 10=$& ,$%()%&1($*)& )& +1 0'&017 6$* +$ .1%.$ &) .)%(*C +1 &'3-')%.) ),-1,'A%?

V1 = V2

!"#$%&' () '*+,*-!+'.,-

!

Q′ Q′1 = 2 C1 C2 "# $%&# '()#(*+, -)' ,'('.*%#/$. .' $0%*',' 1$& '2 1&*,(*1*$ 3' ($,.'&4#(*+, 3' (#&5#6 '2 ($,3',.#3$& C1 -)' '.%# *,*(*#2/',%' (#&5#3$ ($, Q1 1$& 2$ %#,%$ 2# .)/# 3' 2#. ,)'4#. (#&5#. %',3&7, -)' .'& *5)#2 # '.%#8

Q′1 + Q′2 = Q1 9' '.%# /#,'&# ($/0*,#,3$ #/0#. '()#(*$,'. %','/$. -)'

Q′1 =

C1 Q1 C1 + C2

Q′2 =

C2 Q1 C1 + C2

9' '.%# /#,'&# 1$3'/$. '.(&*0*& 2# ','&5:# 3'2 .*.%'/# -)' .'&7 2# #1$&%#3# 1$& (#3# ($,; 3',.#3$&8

Ef

1 Q′2 1 Q′2 1 2 + 2 C1 2 C2 1 C1 Q2 1 C2 Q2 = ( )2 1 + ( )2 1 2 C1 + C2 C1 2 C1 + C2 C2 1 C C2 1 1 = Q21 ( + ) 2 2 (C1 + C2 ) 2 (C1 + C2 )2 =

<2 (#/0*$ 3' ','&5:# .'&7 ',%$,('.8

Ef − Ei = =

C2 1 Q21 C1 Q21 + ) − ( 2 (C1 + C2 )2 (C1 + C2 )2 2 C1 2 Q1 C1 C2 1 ( + − ) 2 2 2 (C1 + C2 ) (C1 + C2 ) C1

!"#$%&' ()

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

-"$.+/012 '* !$ "5 $%#"&&/,#)& S "'#+ +0$"&#)- ')5) '" (+&3+&+ "5 ()%*"%'+*)& C38 C/"3)- ()6) 5+ *$4"&"%($+

*" ,)#"%($+5 "' V0 . '+0"6)' '/ (+,+($*+*- 5+ (+&3+ +56+("%+*+ '"&>D

Q3 = V0 C3 9)6) C3 = 4C0 - )0#"%"6)'

Q3 = 4C0 V0

#* :5 ("&&+& "5 $%#"&&/,#)& S '" (+&3+&+% #)*)' 5)' ()%*"%'+*)&"' ()6) 6/"'#&+ 5+ 73/&+

A5 ()%*"%'+*)& C3 '" 6+%#"%*&> ()% $3/+5 (+&3+ .+ 2/" 5+ *$4"&"%($+ *" ,)#"%($+5 "' 5+ 6$'6+D Q3 = 4C0 V0 8 E+&+ (+5(/5+& 5+' (+&3+' Q1 . Q2 /#$5$F+&"6)' "5 ()6,)&#+6$"%#) *" ()%*"%'+*)&"' "% '"&$" 5) (/+5 %)' "%#&"3+&+ *)' "(/+($)%"' ,+&+ "'#+' *)' ?+&$+05"'8 C+ *$4"&"%($+ *" ,)#"%($+5 2/"

!"#$%&' () '*+,*-!+'.,-

!

"#$%&" "'&(" ) ! "% V0 * +,"-. /.0. C1 ) C2 "%&1' "' %"($"2 34 %,04 5" 3.% 6.3&47"% 5" /454 /.'5"'%45.( %"(1 V0 8

V1 + V 2 = V 0 9"(.2 :.( 5";'$/$<' &"'"0.% =," V1 =

Q1 C1

) V2 =

Q2 C2 *

>" "%&4 04'"(4

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C.' "%&4% ! "/,4/$.'"% &"'5("0.% =,"

Q2 − Q1 = 0

Q1 = Q2 =

C1 C2 V0 C1 + C2

9"(.2 '.% 5$/"' =," C1 = C0 ) C2 = 2C0 * ?%$2 34% /4(-4% %.'

2 Q1 = Q2 = C0 V0 3

! ?3 %4/4( 34 B4&"(D4 34% /4(-4% %"-,$(1' %$"'5. 34% 0$%04% =," /43/,340.% 4'&"($.(0"'&" )4 =," %" /,0:3" 34 /.'%"(64/$<' 5" 34 /4(-4 ) "' "3 /$(/,$&. /"((45. 34 %,04 5" 5$E"("'/$4% 5" :.&"'/$43 "% ',34* "! +4 %$&,4/$<' %" :,"5" 4:("/$4( "' 34 %$-,$"'&" ;-,(48

+4 /4(-4 %" /.'%"(64(42 5" "%&4 04'"(4 &"'"0.% =," %" /,0:3" 34 ("34/$<'

Q′1 + Q′2 = Q1 + Q2

(1)

Q3 − Q1 = Q′3 − Q′1

(2)

+4 %,04 5" &.5.% 3.% :.&"'/$43"% 5"3 /$(/,$&.F"% /"((45.G "% ',348

V1′ + V2′ + V3′ = 0 Q′1 Q′3 Q′2 + + =0 C1 C3 C2

(3)

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6& 1, &*+,*-7) 8:9 '& (-&)&

;(-1-<,)3# &'(#' %,1#/&' &) 8!9

Q′3 = Q3 − Q1 + Q′1

Q′1 Q3 − Q1 + Q′1 Q1 + Q2 − Q′1 + − =0 C1 C3 C2 Q′1 (

1 1 Q3 − Q1 Q1 + Q2 1 + + )+ − =0 C2 C2 C3 C3 C2

=) 1,' 2/&4+)(,' ,)(&/-#/&' ., #0(+%-$#' 1#' %,1#/&' -)-*-,1&' 3& 1,' *,/4,'>

2 Q1 = Q2 = C0 V0 3 Q3 = C3 V0 = 4C0 V0 ;(-1-<,)3# &'(#' %,1#/&'? (&)&$#'

Q′1

7 5 2 − V0 − V0 = 0 4C0 6 3 Q′1 =

2 C0 V0 21

"#) &'(# #0(&)&$#' 1,' 3&$@' *,/4,'>

10 C0 V0 7 24 Q′3 = C0 V0 7

Q′2 =

!

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!"#$%&' (

!"#"$%&!$'( !"#$%&' ()

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@@A

!"#$%&' () *+,&, $-+ '.

!

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C1 =

A/2 3A ǫ1 = ǫ0 2/3d 2d

C2 =

6A A/2 ǫ2 = ǫ0 1/3d 2d

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1 1 1 + = = ′ Ceq C1 C1

1 3A 2d ǫ0

+

1 6A d ǫ0

=

5 d 6 Aǫ0

′ = 6 Aǫ 8 =% *# :+% '% #61.%/% :+% Ceq 5 d 0 ′ >?#,)5 Ceq !"# $ %#&#' '( )($ C3 * %(& '( "#$"( '# )#%#)+,#, -.+/#' $" ! &# '# !.0# ,

#01#! )#%#)+,#, !2

6A A 17 A ǫ0 + ǫ0 = ǫ0 5d 2d 10 d ! 3# %& ! $)+# , ' ,+ ' )"&+)( 4#&5 -. '# )#&6# Q * −Q , %(!+"#,# $ '#! %'#)#! ! & ,7 +"&+1.*#$ , .$# , " &0+$#,# 0#$ &# # , 8$+&9 : 8$+0(! q3 '# )#&6# -. #,-.+ & ' )($, $7 !#,(& C3 * q '# )#&6# $ '(! )($, !#,(& ! C1 * C2 ; ! '# 0+!0# $ #01(! )($, !#,(& ! %(& ′ Ceq = Ceq + C3 =

!"#& $ ! &+ < )(0( 0. !"&# '# 86.

=(0( '# )#&6# ! )($! &/# ! " $,&5 '# ).#)+($2

q1 + q2 = Q

(1)

>(& ("&( '#,(? ' )+&).+"( ! ) &&#,( * %(& '( "#$"(2

∆V3 − ∆V1 − ∆V2 = 0 q3 q q − − =0 C1 C2 C3

(2)

: ;@<2 q3 = Q − q A 0%'#B#$,( !" /#'(& $ '# ).#)+($ ;C<2

1 1 1 (Q − q) − q( + )=0 C3 C1 C2

!

−q(

1 1 1 Q + + )+ =0 C3 C1 C2 C3

"##$%&'(')*+ &+, -'&+.#, *# &', /'%'/01'/0', #)/+)1.'*', #) &' %'.1# '23 .#,4&1' 54#6

12 Q 17 5 q3 = Q 17 q=

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!

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~ r) = ~0 E(~ 6D a < r %4 $%'?% 1,6')3 2)#).-& :8)

I

Ω

~ r) · n ~ r) = ~0 D(~ ˆ dS = 4πr2 D(r) = Qlibre = 0 =⇒ D(~

)# 2-0- )1 )&9%$,-3 9-' 2%#2-

~ r) = − 1 P~ (~r) E(~ ǫ0 @# )&2% ')?,(#3 )1 ;)$2-' 9-1%',/%$,(# )&

k P~ (~r) = rˆ r 9-' 1- $8%1

k ~ rˆ E(r) =− ǫ0 r

@&2- 980- $%1$81%'&) 2%.6,+# 0) -2'% *-'.% G%8#:8) .H& 1%'?%D< I)#).-& 9-' 1)4 0) E%8&&

Qencerrada ~ r) · n E(~ ˆ dS = ǫ0 Ω

I

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%$ !

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σp (a) = P~ (~a) · n ˆ k = rˆ · −ˆ r a k = − a

σp (b) = P~ (~b) · n ˆ k = rˆ · rˆ b k = b

~ · P~ (~r) ρp (r) = −∇ ∂(sen(φ)Pφ ) 1 1 ∂(Pθ ) 1 ∂(r2 Pr ) − − = − 2 r ∂r rsen(φ) ∂φ rsen(φ) ∂θ 2 1 ∂(r Pr ) = − 2 r ∂r 1 ∂rk = − 2 r ∂r k = − 2 r 7# #.,% 8'$-%5 ,#*#-'. 23#

2

Qencerrada = σp (a)4πa + = −4πka + 4π

Z

ρp (r)d3 x

Z Ωr

= −4πka − 4πk

a

Z

ρp r2 dr r

dr

a

= −4πka − 4πk(r − a) = −4πkr

9'$ ,%*,'5

Qencerrada −4πkr k ~ r) · n ~ E(~ ˆ dS = 4πr2 E(r) = = =⇒ E(r) =− rˆ ǫ0 ǫ0 ǫ0 r Ω

I

!"#$%&' () *+,&, $-+ '.

!"

#$ r > b %& '()* +',-.& '/ 0'#)1+ 21/*+-3*#-.& '( &4/15 '( 6'#-+5 P~ (~r) = ~05 21+ /1 74'

~ r) = − 1 P~ (~r) = ~0 E(~ ǫ0 8*9:-;& 4(*&61 /* /'< 6' =*4((5 )'&'91( 74'

Qencerrada ~ r) · n E(~ ˆ dS = ǫ0 Ω

I

< '& '()' #*(1 /* #*+,* '&#'++*6* '(

Z

ρp (r)d3 x Qencerrada = σp (b)4πb + σp (a)4πa + Ω Z b ρp r2 dr = 4πkb − 4πka + 4π a Z b dr = 4πk(b − a) − 4πk 2

2

a

= 4πk(b − a) − 4πk(b − a) = 0

21+ /1 74' >&*/9'&)'

I

Ω

~ r) · n ~ r) = ~0 E(~ ˆ dS = 4πr2 E(r) = 0 =⇒ E(~

?1+ )*&)15 +'(49-'&61 '/ #*921 '/;#)+-#1 6' 21/*+-3*#-.& '(

~ E(r) =

 

~0

r b

!

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Q △V = 2πǫ0 L

Z

b

a

dr Q 2πǫ0 L = ln(b/a) =⇒ C = r 2πǫ0 L ln(a/b)

$, /- )-8-)&%-')&- +$/ )*'+$',-+*. )&/0'+.&)*; I/ 56-. 9$.%&)-/A$'%$ $/ )*'+$',-+*.3 $/ >/*?($ +&$/@)%.&)* +$,)&$'+$ 8*. $/ 8$,* ?($+-'+* ('8-.%$ +$/ )*'+$',-+*. ,&' +&$/@)%.&)*3 8*. /* ?($ /- )-8-)&%-')&- +$/ ,&,%$A- )-A>&-; #$- x /+&,%-')&- ?($ $/ >/*?($ +&$/@)%.&)* +$,)&$'+$3 A$+&+- +$,+$ /- 8-.%$ ,(8$.&*. +$/ )*'+$',-+*. D-,%- /- 8-.%$ ,(8$.&*. +$/ +&$/@)%.&)*3 1 x ˆ $/ 9$)%*. ('&%-.&* $' $/ ,$'%&+* A$')&*'-+*; I/ +$,)$'+$. x $/ +&$/@)%.&)*3 ,$ E*.A-' +*, )*'+$',-+*.$, $' 8-.-/$/* +$ )-8-)&%-')&-,

C1 =

2πǫ0 x ln(b/a)

C2 =

2πǫ(L − x) ln(b/a)

8*. /* ?($ /- )-8-)&%-')&- +$/ )*'+$',-+*. )*A8/$%* $,

C(x) = C1 + C2 =

2πǫ(L − x) 2π 2πǫ0 x + = (ǫL − x(ǫ − ǫ0 )) ln(b/a) ln(b/a) ln(b/a)

8-.- 0 < x < L J- $'$.40- 8*%$')&-/ -/A-)$'-+- $' $/ )*'+$',-+*. $,

U (x) =

1 Q2 2 C(x)

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dU x ˆ + mgˆ x = ~0 F~ = − dx

!" #$%&'( )*+,&)&-+ ./0 10 ,020 )/'3$&4 0+ 0$ 0./&$&24&*5 6*4 %(+%*7

mg = = = =⇒ (ǫL − x(ǫ − ǫ0 ))2 = |ǫL − x(ǫ − ǫ0 )| = (ǫL − x(ǫ − ǫ0 )) = =⇒ x =

dU dx 1 Q 2 dC − 2 C dx 2 Q ln(b/a)(ǫ − ǫ0 ) 4π (ǫL − x(ǫ − ǫ0 ))2 Q2 ln(b/a)(ǫ − ǫ0 ) 4πmg s ln(b/a)(ǫ − ǫ0 ) Q 4πmg s ln(b/a)(ǫ − ǫ0 ) ±Q 4πmg s Q ln(b/a)(ǫ − ǫ0 ) ǫL ∓ (ǫ − ǫ0 ) (ǫ − ǫ0 ) 4πmg

8020'*1 (3$&)(4 $( 401%4&))&-+ 0 < x < L ( ('2(1 1*$/)&*+015 90+0'*1 34&'04('0+%0 ./0

ǫL Q x+ = + (ǫ − ǫ0 ) (ǫ − ǫ0 ) Q =⇒ (ǫ − ǫ0 )

s

ln(b/a)(ǫ − ǫ0 ) 4πmg

s


$* )/($ 01 /+( )*+%4(,&))&-+ 3/01

Q (ǫ−ǫ0 )

q

< L

ǫ < L 1− (ǫ − ǫ0 )

ln(b/a)(ǫ−ǫ0 ) 4πmg

=−

Lǫ0 <0 (ǫ − ǫ0 )

>0

6*4 %(+%*7 $( #+&)( 3*1&)&-+ ,0 0./&$&24&* 3*4 $* 34*+%* 3*1&2$0 01 s ǫL Q ln(b/a)(ǫ − ǫ0 ) x0 = − (ǫ − ǫ0 ) (ǫ − ǫ0 ) 4πmg :3$&)(+,* $(1 401%4&))&*+017 %0+0'*1 ./0

x0 = =⇒ L >

s Q ln(b/a)(ǫ − ǫ0 ) ǫL − >0 (ǫ − ǫ0 ) (ǫ − ǫ0 ) 4πmg s Q ln(b/a)(ǫ − ǫ0 ) ǫ 4πmg

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!"#$%&' () *+,&, $-+ '.

!"

x0 = =⇒ L <

s ǫL Q ln(b/a)(ǫ − ǫ0 ) −
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Q ǫ

s

ln(b/a)(ǫ − ǫ0 ) 4πmg Q =⇒ ǫ

Q ǫ0 Q ǫ0

s


,' -7%, $( :$3/%/$3'4 5'3 *%.*'; ,% 1'()-)0. /$ $67),)23)' $(

x0 =

Q ǫL − (ǫ − ǫ0 ) (ǫ − ǫ0 )

s


!"

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• r < a1 B) $*+$,) /$ ,)/&. ) $* -.'/(-%.,) 0 2., 6. %)'%. '. F)0 -):2. $6$-%&,-. $' *( &'%$,&.,> ~ = 0, E

si r < a

D$ $*%) :)'$,) $6 3$-%., /$*26)@):&$'%. *$,5>

~ = ǫ0 E ~ = 0, D

r
• a < r < 2a1 ?' $*%$ -)*. (%&6&@):.* 6$0 /$ 1)(**> Z qenc ~ ·n E ˆ dS = ǫ0 S

G*H4 /)/) 6) *&:$%,H) $*+7,&-) 0 I($ 6) -),1) $' 6) $*+$,) -.'/(-%.,) $* q .;%$'$:.* $6 ,$*(6%)/. /$ *&$:2,$>

~ = E

q rˆ, si a < r < 2a 4πǫ0 r2

J),) $6 3$-%., /$*26)@):&$'%.>

q rˆ, si a < r < 2a 4πr2 • 2a < r < 2, 5a1 G26&-):.* 6$0 /$ 1)(** 2),) /&$67-%,&-.*> Z ~ ·n D ˆ dS = qenc ~ = ǫ0 E ~ = D

S

!"#$%&' () *+,&, $-+ '.

!"

#$ %&'()* +&,-$./.01&2() &, -.*.$&$) .$ '.0-) &$3'(*1') 456& &, *.+1.$78 9& &,(. 0.2&*.

~ = D

q rˆ 4πr2

~ = ǫE ~ < +)2+& ǫ &, $. -&*01(1%1+.+ +&$ +1&$3'(*1') 56& %.*1. &2 >62'1?2 :;)*.< ,.=&0), 56& D +& r8 9& &,(. 0.2&*.< &$ '.0-) &$3'(*1') ,&*@A ~ = E

q rˆ, si 2a < r < 2, 5a 4πǫ0 r2 (1 + ar )

• r > 2, 5a B(1$1/.2+) $&C +& D.6,, )=(&2&0), +1*&'(.0&2(& 56&A ~ = E

q rˆ 4πǫ0 r2

~ = D

q rˆ 4πr2

#$ %&'()* -)$.*1/.'1?2 &E1,(1*@ &%1+&2(&0&2(& ,)$) &2 $. *&D1?2 +)2+& ,& &2'6&2(*. &$ +1&$3'F ~ + P~ C D = ǫE ~ 8 9& &,(. 0.2&*. (*1')8 G10), &2 &$ *&,60&2 56& D = ǫ0 E !"

~ P~ = (ǫ − ǫ0 )E H)2 ǫ = ǫ0 (1 + ar ) C &$ %.$)* +&$ '.0-) &$3'(*1') &2 &$ +1&$3'(*1') &2')2(*.+) &2 .7A

r q rˆ P~ = (ǫ0 (1 + ) − ǫ0 ) a 4πǫ0 (1 + ar )r2 I10-$1J'.2+)A

P~ =

q rˆ 4πar(1 + ar )

H)2)'1&2+) &$ %&'()* -)$.*1/.'1?2 -)+&0), '.$'6$.* +1*&'(.0&2(& $., +&2,1+.+&, +& '.*D. ,6-&*J'1.$ C %)$603(*1'. +&$ +1&$3'(*1')A $%&'()*) '+,%-.#(*/ σp G10), 56& $. +&2,1+.+ +& '.*D. ,6-&*J'1.$ ,& '.$'6$. -)*A #"

σp = P~ · (ˆ n) K&2+*&0), 62. +&2,1+.+ ,6-&*J'1.$ &2 &$ 12(&*1)* +&$ +1&$3'(*1') C &2 &$ &E(&*1)* +&$ +1&$3'(*1')8 L.*. &$ 12(&*1)* +&$ +1&$3'(*1') 4r = 2a7

σp int = P~ · −r =

q 24πa2

#2 &$ &E(&*1)* +&$ +1&$3'(*1') 4r = 2, 5a7A

σp ext = P~ · r = $%&'()*) 01/+234-(#*

q 35πa2

ρp M. +&2,1+.+ +& '.*D. %)$603(*1'. ,& '.$'6$. ')0) ~ · P~ ρp = −∇

!" #$%$ &'%$( )* +, -,./, ,01 )+ &)2/$. -$+,.'3,2'4* )( .,5',+ 6-,.,+)+$ ,+ 2,%-$ )+72/.'2$0 8 -$. +$ /,*/$ +, 5'&).9)*2', 5) )(/) &)2/$. () 2,+2:+, 2$%$;

~ · P~ = − ρp = −∇

1 ∂ 2 (r P ) r2 ∂r

<)(,..$++,*5$ )(/, )=-.)('4*1 $>/)*)%$(;

ρp = −

qa + r)2

4πr2 (a

!"#$%&' () *+,&, $-+ '.

!"

!"#$%&' ()

#$ %&$'($)*'&+ '( ,-*%*) ,*+*-(-*). )(,*+*'*) /$* '0)1*$%0* d. 10($( /$ '0(-2%1+0%& ($ )/ 0$1(+0&+. */)($1( '( %*+3*) -04+(). %/5* ,(+601070'*' (-2%1+0%* () ǫ 5 '(,($'( '( -* '0)1*$%0* * /$* '( -*) ,-*%*)8 9*-%/-*+ -* %*,*%0'*' '(- %&$'($)*'&+ )0 ǫ(x) 70($( '*'& ,&+:

ǫ(x) = ǫ0 (

*"$+,-".

1 1−

y2 3d2

)

;( ,0'( %*-%/-*+ -* %*,*%0'*' '(- )0)1(6* 5 )*4(6&)
C=

Q V

9*-%/-*6&) ($&$%() (- ,&1($%0*- * ,*+10+ '(- %*6,& (-(%1+0%& 5 ,*+* ()1&) )/,&$(6&) ?&+* ,&'(6&) *,-0%*+ -* -(5 '( 3*/)) ,*+* '0(-(%1+0%&)8 @*+* ()1& %&$)0'(+*6&) %&6& )/,(+A%0( 3*/))0*$* /$ %0-0$'+& %&$ 1*,*) '( *+(* S (- %/*- 10($( -* 1*,* )/,(+0&+ ($ (- 0$1(+0&+ '(- /$* '( -*) ,-*%*) %&$'/%1&+*) 5 -* &1+* 1*,* ($ (- '0(-(%1+0%&8 B(%&+'*6&) '(6*). -* %*+3* -04+( ($%(++*'* )(+D qenc = σL S Z ~ · dS ~=D ~1 ·S ~1 + D ~2 ·S ~2 = qenc = σL S D S

E( -* A3/+* 7(6&)
~1 ·S ~1 + D ~2 ·S ~2 = −D1 s + D2 S = σL S D −D1 + D2 = σL

F$ (- 0$1(+0&+ '(- %&$'/%1&+ (- %*6,& (-2%1+0%& () $/-& 5 ,&+ -&. ,*+* (- 7(%1&+ '(),-*C*60$+1& (-(%1+0%& ($ (- %&$'/%1&+ )( 10($ D1 = 0 @&+ -& 1*$1& 1($'+(6&) -* )03/0($1( (%/*%0&$:

D2 = ǫE2 = σL F- %*6,& (-(%10+%& )(+D ($1&$%() E2 = σǫL 8 >?&+*. (7*-/*$'& -&) 7*-&+() '( σL 5 ǫG'*'& ($ (- ($/$%0*'&H8

E2 =

Q Sǫ0 (

1 2 1− y 2

)

3d

>?&+* ()1* (I,+()0&$ -* ,&'(6&) +(()%+040+ %&6&:

E2 =

Q (3d2 − y 2 ) 3Sǫ0 d2

! "#$% &' ($)*+*,-$' (* ./0*,-$'& #* -'&-1&'+' -/2/

V =

Z

d

Es (y)dy =

Z

0

0

3* *#0/ #* /40$*,* 51*

V = 6' -'.'-$('( *# *,0/,-*#

C=

d

Q (3d2 − y 2 ) 3Sǫ0 d2

8 Qd 9 Sǫ0

Q 9 Sǫ0 = V 8 d

!"#$%&' () *+,&, $-+ '.

!"

!"#$%&' ()

#$%&'( )$ *% +(%)$%,-)(' )$ ./-+-, .-'-/$/-,0 )$ ,$++12% A 3 $,.$,(' d0 1%&'()*+14(, *% )1$/5+&'1+( )$ .$'41&161)-) 6-'1-7/$0 ,1$%)( y /- )1'$++12% .$'.$%)1+*/-' - /-, ./-+-,8 #$,.'$9 +1-%)( /(, $:$+&(, )$ 7(')$ 3 $% +-,( )$ %( $;1,&1' +-'<-, /17'$, -/ 1%&$'1(' )$/ )1$/5+&'1+(0 +-/+*/-'= '* >/ +-4.( $/5+&'1+(0 $/ )$,./-?-41$%&( $/5+&'1+( 3 $/ 6$+&(' )$ .(/-'1?-+12%0 +*-%)( -./19 +-4(, *%- )1:$'$%+1- )$ .(&$%+1-/ V0 $%&'$ /-, ./-+-,8 #* @-, )$%,1)-)$, )$ +-'<- )$ .(/-'1?-+12%8 +* @- +-.-+1)-) )$/ +(%)$%,-)('8

y ǫ = ǫ0 (1 + ) d

,"$-+."/ '* >% $/ 1%&$'1(' )$/ )1$/+&'1+( %( &$%$4(, +-'<- /17'$ 3 .(' /( &-%&( /- )$%,1)-) )$ +-'<- /17'$

$, %*/-= ρL = 08 A(4( B$4(, 61,&( )*'-%&$ $/ +*',(0 )$,.'$+1-%)( /-, +(%)1+1(%$, )$ 7(')$0 $/ +-4.( $/$+&'1+( ,$'- .$'.$%)1+*/-' - /-, ./-+-, 3 .(' /( &-%&(0 )$ /- $+*-+1(% C"D0 $/ 6$+&(' )$,./-?-41$%&( $/5+&'1+( &-471$% /( ,$'-8 #$ $,&- 4-%$'-0 )$ -/ :('4- )1:$'$%+1-/ )$ /- /$3 )$ E-*,, &$%)'$4(,=

~ ·D ~ = ∂ D = ρP = 0 ∇ ∂y #$ $,&(0 '$,*/&- F*$ $/ 6$+&(' )$,./-?-41$%&( $/$+&'1+( $, +(%,&-%&$8 G$%$4(, )-)( *% .(&$%+1-/ V0 F*$ +*4./1'- /- ,1<*1$%&$ $+*-+1(%= Z ~ · d~l V0 = E

~ = H$'( )$ /- $+*-+1(% C"D &$%$4(, F*$ E

~ D ǫ 8#(%)$

ǫ = ǫ(y)8 H(' /( &-%&(=

Z Z ~ 1 D ~ · dl = D dy V0 = ǫ ǫ

#(%)$ D .*$)$ ,-/1' )$ /- 1%&$<'-/ -/ ,$' +(%,&-%&$ 3 +(%,1)$'-4(, dy = dl8 I$$4./-?-%,( $/ 6-/(' )$ ǫ )-)( $% $/ $%*%+1-)( $ 1%&$<'-%)( &$%)'$4(, F*$=

V0 =

Dd ln(2) ǫ0

#$,.$J-%)( D=

D=

V0 ǫ0 d ln(2)

KB('- F*$ &$%$4(, $/ 6$+&(' )$,./-?-41$%&(0 .()$4(, +-/+*/-' $/ 6$+&(' +-4.,( $/$+&'1+( )$ /- $+*-+1(% C"D=

E=

D V0 = ǫ d ln(2)(1 + yd )

#$ /- $+*-+1(% C D .()$4(, ,-7$' +*-/ $, $/ 6$+&(' )$ .(/-'1?-+1(%=

!!

y ~ − ǫ0 E ~ = V0 ǫ0 P~ = D d ln(2) y + d

! "#$# %&'&$#( &) *&+%#, -#).,/0.+/#'1 #2%&'&$#( ).( +.,3.( 4& -#).,/0.+/#' 4/,&+%.$&'%& 4& &+5.+/#' 6789 :. +.-.+/4.4 ). +.)+5).$#( +#$# (/&$-,&; C = VQ0 9 < 4/=&,&'+/. 4&) &>&,+/+/# .'%&,/#,1 ?. %&'&$#( ). 4/=&,&'+/. 4& -#%&'+/.)9 @#, )# %.%'# %&'&$#( A5& *&, .)35'. $.'&,. 4& +.)+5)., ). +.,3. Q &' =5'+/#' 4& V0 9 @.,. &(%#1 .-)/+.,&$#( ). :&? 4& B.5(( .) /35.) A5& &' &) -,#2)&$. ; Z ~ · dS ~=D ~1 ·S ~1 + D ~2 ·S ~2 = qenc = σL S D S

C& ). D35,. *&$#( A5& )#( -,#45+%#( &'%,& )#( *&+%#,&( 4&(-).0.$/&'%# ? )#( *&+%#,&( 4& .,&. +5$-)& +#';

~1 ·S ~1 + D ~2 ·S ~2 = −D1 s + D2 S = σL S D −D1 + D2 = σL

E' &) /'%&,/#, 4&) +#'45+%#, &) +.$-# &)F+%,/+# &( '5)# ? -#, )#1 -.,. &) *&+%#, 4&(-).0.$/',%# &)&+%,/+# &' &) +#'45+%#, (& %/&' D1 = 0 @#, )# %.'%# %&'4,&$#( ). (/35/&'%& &+5.+/#';

D2 = σL :. +.,3. 4&) +#'4&'(.4#, (&,G ). 4&'(/4.4 4& +.,3. -#, ). (5-&,D+/& 4& ).( -).+.(6H<89 @,# +#$# ). 4&'(/4.4 4& +.,3. )/2,& &( /35.) .) *&+%#, 4&(-).0.$/&'%# &)F+%,/+#;

Q = SσL = SD2 = S

V0 ǫ0 d ln(2)

C& &(%. $.'&,. ). +.-.+/4.4 (&,G;

C=

Q ǫ0 =S V0 d ln(2)

!"

!"#$%&' ()

!"#$%&' () *+,&, $-+ '.

#$ %&$'$' ! $()$*+( ,-'./,%-*+( ,-',0'%*&,+( .$ *+.&-( a1 b 2 c3 45 $(6+,&- $'%*$ 5+( .-( 6*&7$*+( $(%+ 55$'+ .$ 7+%$*&+5 .&$50,%*&,- .$ ,-'(%+'%$ κ3 8'&,&+57$'%$ 5+ $()$*+ .$ *+.&- a $(%+ .$(,+*9 :+.+ 2 5+( $()$*+ .$ *+.&-( b 2 c %&$'$' ,+*:+( &'&,&+5$( q1 2 q2 1 *$(6$,%&;+7$'%$3 <+ $()$*+ &'%$*&-* .$ *+.&- + ($ ,-'$,%+ ,-' 5+ $()$*+ .$ *+.&- c ,-' /' ,+=5$ +&(5+.- .$5:+.-3 '* >+5,/5+* 5+ ,+*:+ 5&=*$ $' ,+.+ $()$*+3 #* >+5,/5+* 5+ ,+*:+ .$ 6-5+*&?+,&@' $' 5+ (/6$*A,&$ $B%$*'+ 2 $' 5+ (/6$*A,&$ &'%$*'+ .$5 .&$50,%*&,-3

!"

!"#$%&' ()

#$ %&$'$' ()* +)'($'*,()-$* ($ ./,+,* .,-,/$/,* +)' &01,/$* 2-$,* S 3 $ &01,/$* (&*%,'+&,* $'%-$ /,* ./,+,* d4 5/ .-&6$- +)'($'*,()- %&$'$ ,&-$ $'%-$ *1* ./,+,*7κ = 18 9 +,.,+&(,( C0 4 5/ *$01'() +)'($'*,()- %&$'$ 1', +,.,+&(,( X 4 '* #& $/ .-&6$- +)'($'*,()- %&$'$ &'&+&,/6$'%$ 1', +,-0, Q3 +,/+1/$ $/ +,6:&) ($ $'$-0;, $/$+%-)*%2%&+,3 $' <1'+&=' ($ X 3 *& ,6:)* +)'($'*,()-$* *$ +)'$+%,' $' .,-,/$/)4 #* #& $/ (&$/>+%-&+) ($/ *$01'() +)'($'*,()- %&$'$ 1', +)'*%,'%$ (&$/>+%-&+, ?,-&,:/$ $' $/ $*.,+&) ($ ?,/)- κ = κ0 (1 + xd ) +,/+1/$ /, +,.,+&(,( ($ $*%$ +)'($'*,()-4

!"

!"#$%&' () *+,&, $-+ '.

!"#$%&' ()

#$ %&'&$()* %*$(+%,*) -+.%* /&$ ./0./*) (. )1(&* r1 2 '1)3* ℓ4 ''.51 +$1 %1)31 +q +$&6*)7.8 7.$,. (&/,)&9+&(1: ;').(.(*) (. .'4 *,)* %&'&$()* %*$(+%,*) -+.%* /&$ ./0./*)4 %*1<&1' %*$ .' 1$,.)&*)4 (. )1(&* r2 2 7&/7* '1)3* ℓ 4 ,1' %*7* &$(&%1 '1 =3+)14 ''.51 +$1 %1)31 −q 4 ,179&>$ +$&6*)7.7.$,. (&/,)&9+&(1: ?. ''.$1 '1 ).3&@$ .$,). '*/ (*/ %&'&$()*/ %*$ +$ 71,.)&1' %+21 %*$8 /,1$,. (&.'>%,)&%1 ./ 6+$%&@$ (. '1 (&/,1$%&1 1' .A. (. '*/ %&'&$()*/4 κ = κ(r): B*$/&(.). ℓ 7+%-* 712*) C+. r1 2 r2 :

'* D$%+.$,). +$1 .<0)./&@$ 01)1 κ(r) C+. ''.51 1 +$ %170* .'.%,)*/,E,&%* )1(&1' &$(.0.$(&.$,. (. '1 (&/,1$%&1 1' .A.: #* B1'%+'. '1 %101%&(1( (.' %*$(.$/1(*) %&'F$()&%* %*))./0*$(&.$,.: +* B1'%+'. '1 (.$/&(1( (. %1)31 (. 0*'1)&G1%&@$ /+0.)=%&1' (.' 71,.)&1' (&.'>%,)&%*4 1 r = r1 2 r = r2 4 01)1 .' 51'*) (. κ(r) %1'%+'1(* .$ .' &,.7 a): ,* B*$/&(.). C+. .$ %**)(.$1(1/ %&'F$()&%1/ ∇~ · A~ = 1r ∂r∂ (rAr ) !"#!$#"% #"& '()&*'"'(& +,#$-./%*!"& '( !"%0"& '( 1,#"%*2"!*3) () (# '*(#.!/%*!,4

!

5"#!$#"% #" !")/*'"' /,/"# '( !"%0"& '( 1,#"%*2"!*3)

*)!#$6.)',&( !"%0"& () #" &$1(%7!*( 6

() (# +,#$-() '(# '*(#.!/%*!,4

"!

5"#!$#( #" ()(%08" 1,/()!*"# (#(!/%,&/9/*!" "#-"!()"'" () (# !,)'()&"',%4

!"#$%&' (

!""#$%&$ '($)&"#)* !"#$%&' ()

!"#$ %&' (&!%)("&#$' $'*+#,(&' (&!(+!"#,(&' %$ #-%,&' a . b/ '$ 00$!- (&! )! 1-"$#,-0 (&!%)("&# %$ (&!%)(",2,%-% 2-#,-30$ g(r) = gr0 b 4 5-0()0$ 0- #$','"$!(,- $!"#$ 0-' %&' 60-(-' (&!%)("&#-'4

*"$+,-./

70 8-3$# )!- %,*$#$!(,- %$ 6&"$!(,-0 $!"#$ -13&' (&!%)("&#$' $'*+#,(&'/ '$ 9$!$#- )!- (&##,$!"$ J~ :)$ $' #-%,-0; J~ = J(r)ˆ r

<&%$1&' $'(#,3,# 0- (&##,$!"$ $! *)!(,=! %$ J~ 1$%,-!"$ 0- #$0-(,=!; I=

Z

S

~=J J~ · dS

Z

>?@

S

dS = J4πr2

!"#$%&' () '**+,-$, ,&, $*+ !

!"

#$%&' ($%)*&'+,-$) ($-$ S .%, )./'+0(*' ')12+*(,3 #' ')4, -,%'+, '5 6'(4$+ &'%)*&,& &' ($++*'%4' )' /.'&' ')(+*7*+8

I rˆ 4πr2 9, 5': &' ;<- %$) /'+-*4' +'5,(*$%,+ '5 (,-/$ '52(4+*($ ($% '5 6'(4$+ &'%)*&,& &' ($++*'%4'3 =% ')4' (,)$ ($-$ 5, ($%&.(4*6*&,& ') g(r) = gr0 b )' 4'%&+>8 J~ =

~ J~ = g(r)E #')/'?,%&$ '5 (,-/$ '52(4+*($ : .4*5*@,%&$ '5 6,5$+ '%($%4+,&$ /,+, J~

~ = E

I rˆ 4πbg0 r

A<$+,B )' /.'&' (,5(.5,+ 5, &*1'+'%(*, &' /$4'%(*,5 '%4+' ,-7$) ($%&.(4$+') :, C.' 4'%'-$) .%, 'D/+')*E% /,+, '5 (,-/$ '52(4+*($38

V0 =

Z

a

b

~ · d~r = E

I 4πbg0

Z

a

b

1 b I ln( ) dr = r 4πbg0 a

F,7*'%&$ 5, &*1'+'%(*, &' /$4'%(*,5 : 5, *%4'%)*&,& &' ($++*'%4' I /$&'-$) (,5(.5,+ 5, +')*)4'%(*, -'&*,%4' 5, +'5,(*E% R = VI0 3 G$+ 5$ 4,%4$B /,+, '5 )*)4'-, -$)4+,&$ 5, +')*)4'%(*, ')8

ln( ab ) V0 R= = I 4πbg0

!"

!"#$%&' ()

#$ %$&'()*+ ,-+ .+/0+ Q0 $- ,-+ &1+.+ .'-%,.*'/+ .,+%/+%+ %$ 1+%' a2 3,$ $(*4 ($&+/+%+ ,-+ %)(*+-.)+ d %$ '*/+ &1+.+ )0,+12 %$(.+/0+%+ 5 .'-$.*+%+ + *)$//+ 6.'-()%$/$ 3,$ a >> d &+/+ %$(&/$.)+/ $7$.*'( %$ 8'/%$9: ;-*/$ $11+( $<)(*$ ,- =+*$/)+1 %$ &$/=)*)>)%+% ǫ 5 .'-%,.*)>)%+% µ2 +=8+( .'-(*+-*$(: +9 ?$*$/=)-$ 1+ .'//)$-*$ 3,$ .)/.,1+ $- 7,-.)@- %$1 *)$=&': 89 ;-.,$-*/$ 1+ /$()(*$-.)+ %$1 ()(*$=+ 5 >$/)A3,$ 3,$ RC = µǫ : .9 B+1.,1$ 1+ $-$/0C+ %)()&+%+ $- $1 &/'.$(' .'=&1$*'2 $( %$.)/2 +1 &+(+/ *'%+ 1+ .+/0+ %$ ,-+ &1+.+ + '*/+:

*"$+,-./0 +9 #+8$='( %$ 1+( +5,%+-*C+( +-*$/)'/$( 3,$ $1 .+=&' $1D.*/).' 0$-$/+%' &'/ ,- &1+-' %$ %$-()%+% %$ .+/0+ (,&$/A.)+1 .'-(*+-*$ σ $(

~ = σ x E ˆ 2ǫ0 .'- x ˆ -'/=+1 +1 &1+-': ;- $(*$ .+('2 %$(&/$.)+-%' $7$.*'( %$ 8'/%$ %$8)%' + 3,$ a >> d2 5 &'/ 1+ &/$($-.)+ %$1 =+*$/)+1 %$ &$/=)*)>)%+% ǫ $-*/$ &1+.+(2 $1 .+=&' $1D.*/).' $-*/$ &1+.+( $(

Q ~ = σx x ˆ E ˆ= 2ǫ 2ǫa2 ~ 2 &'/ 1' .,+1 #+8$='( 3,$ J~ = µE µQ x ˆ = Jx ˆ J~ = 2ǫa2

!"#$%&' () '**+,-$, ,&, $*+ !

!"

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

i =

I

Ω

J~ · n ˆ dS

= Ja2 µQ = 2ǫ dQ = − dt ,(),* *& /-2)( 0*)(/ */ ,*>-,( $ 53* &$ %&$'$ /* */+= ,*/'$.2$),( 8 &$ /3%*.?'-* ,* -)+*2.$'-@) 63* 3) '3$,.$,( ,* &$,( a %$.$&*&( $ &$/ %&$'$/< A* &( $)+*.-(.4 +*)*0(/ &$ *'3$'-@) ,-6*.*)'-$&

−

µQ 2ǫ

=

ssi

0 =

ssi

0 =

ssi

0 =

=⇒ Q(t) =

dQ dt dQ µQ + dt 2ǫ dQ µQ µt µt + exp exp dt 2ǫ 2ǫ 2ǫ d µt Q · exp dt 2ǫ µ Q0 · exp − t 2ǫ

%3*/ Q(0) = Q0 < B(. +$)+(4 &$ '(..-*)+* *&1'+.-'$ *) 63)'-@) ,*& +-*0%( */

i(t) = − >: A*& -+10 $)+*.-(.4 /$>*0(/ 53*

µ dQ µQ0 = · exp − t dt 2ǫ 2ǫ

~ = Q(t) x ˆ = E(t)ˆ x E 2ǫa2 #$ .*/-/+*)'-$ ,* 3) (>C*+( ,*%*),* ,* &$ 2*(0*+.D$ ,*& 0-/0(4 ,* &$ 6(.0$ *) 53* /* $%&-53* &$ ,-6*.*)'-$ ,* %(+*)'-$& 8 ,*& 0$+*.-$& ,*& 53* */+= 9*'9( 7&( 53* /* +.$,3'* *) &$ .*/-/+-;-,$, ( '(),3'+-;-,$, ,*& (>C*+(:< E) */+* '$/( +*)*0(/ 53* &$ '(),3'+-;-,$, µ

! "# $%&#'(&'") *%+ ,% -." #" $./*," ,( ,"0 1" 23/) *%+ ,% $.(, ,( +"#4#'"&$4( "# $%&#'(&'" 0 &% 1"*"&1" 1", 5%,'(6" &4 1" ,( $%++4"&'" -." *(#( *%+ ", %76"'%8 9( ,"0 1" 23/ &%# 14$" -."

V =i·R

*%+ ,% -." "&$%&'+(&1% V (t)) i(t) "& .& 4&#'(&'" 1(1% *%1"/%# 3(,,(+ R8 :&$%&'+"/%# ,( 14;"+"&$4( 1" *%'"&$4(,

V (t) =

Z

d

0

<"&"/%# (1"/=# -."

i(t) =

E(t)ˆ x·x ˆdx = E(t)d =

Q(t)d 2ǫa2

µ 2ǫ Q(t) =⇒ Q(t) = i(t) 2ǫ µ

>%+ '(&'%)

V (t) =

Q(t)d 2ǫd 1 d 1 d = i(t) = i(t) = R · i(t) =⇒ R = 2 2 2 2ǫa 2µǫa µa µ a2

?(7"/%# 1" (0.1(&'@(# (&'"+4%+"# -." ,( $(*($4'(&$4( 1" .& $%&1"&#(1%+ 1" *,($(# *(+A (,",(# 1" =+"(# A) #"*(+($4B& d 0 #"*(+(1(# *%+ .& /"14% 14",C$'+4$% 1" *"+/4'4541(1 ǫ "#

C=ǫ

a2 d

>%+ '(&'%)

RC =

1 d a2 ǫ · ǫ = µ a2 d µ

$D 9( "&"+E@( 14#4*(1( "& ", *+%$"#% "#

U

=

Z

i(t)V (t)dt Z +∞ µd Q2 (t)dt 4a2 ǫ2 0 Z µ µQ0 2 d +∞ exp − t dt 4a2 ǫ2 0 ǫ 2 Q0 d µ 0 · exp − t 2 4a ǫ ǫ +∞ 2 Q0 d 4a2 ǫ 0

= = = =

+∞

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!"

!"#$%&' ()

#$ %&$'$' ()* *+,$-./&$* /)'(+/%)-0* $*12-&/0* 3 /)'/2'%-&/0*4 ($ -0(&)* 0 3 5 6/)' a < b74 80'%$'&()* 0 +'0 (&1$-$'/&0 ($ ,)%$'/&09 V0 6/)' $9 803)- ,)%$'/&09 $' 90 *+,$-./&$ &'%$-&)-7: ;9 $*,0/&) $'%-$ 90* *+,$-./&$* *$ 99$'0 /)' +' 80%$-&09 /)'(+/%)- <)8)=2'$) ($ /)'(+/%&>&(0( /)'*%0'%$ µ: ?$%$-8&'$ 90 /)--&$'%$ $92/%-&/04 $9 /08,) $92/%-&/) $'%-$ *+,$-./&$* 3 90 -$*&*%$'@ /&0 ($9 *&*%$804 3 >$-&.A+$ '+$>08$'%$ A+$ RC = µǫ :

*"$+,-./0

~ r) = E(r)ˆ ?$ 90 *&8$%-B0 ($9 ,-)59$804 %$'$8)* A+$ E(~ r4 ()'($ r $* 90 /))-($'0(0 /)--$*,)'@ ~ ~ 4 $>&($'%$8$'%$ J~ = J(r)ˆ (&$'%$ ($ /))-($'0(0* $*12-&/0*: C)8) J = µE r: ?$ $*%) >$8)* A+$ 90 /0-=0 *$ %-0'*,)-%0 -0(&098$'%$4 $* ($/&-4 90 /)--&$'%$ $* -0(&09: D($8E*4 (0() A+$ µ $* /)'*%0'%$4 *$ /+8,9$ 90 9$3 ($ )<8 3 V0 = iR4 /)' R /)'*%0'%$4 ,)- 9) A+$ 90 /)--&$'%$ i $* /)'*%0'%$: ;'/)'%-$8)* 90 /)--&$'%$: Z Z i rˆ ~ ~ = J ·n ˆ dS = J(r) dS = J(r)4πr2 =⇒ J(r) i= 4π r2 Ω Ω ()'($ +*08)* ($ *+,$-./&$ ($ &'%$=-0/&F' +'0 *+,$-./&$ $*12-&/0 ($ -0(&) - /)'/2'%-&/0 0 90* *+,$-./&$* /)'(+/%)-0*: ?$ $*%) %$'$8)* A+$

i rˆ 1~ ~ = E(r) = J(r) µ 4πµ r2 D<)-04 ,0-0 $'/)'%-0- 90 -$*&*%$'/&0 <099$8)* 90 (&1$-$'/&0 ($ ,)%$'/&09 $'%-$ *+,$-./&$*:

= = = = =⇒ R =

Z

~ · dˆ E r Z b i dr 4πµ a r2 i 1 a 4πµ r b 1 1 i − 4πµ a b iR 1 1 a − b 4πµ

V0 = −

Γ

D($8E*4 %$'$8)* A+$ 90 /)--&$'%$ $*

i= 3 $9 /08,) $92/%-&/)

~ E(r) =

4πµV0 1 1 a − b

i rˆ = 4πµ r2

1 a

V0 rˆ 1 r2 −b

!" #$ %&'(%)*+%, %)*$-./-$, ,%0$1/, 2'$ 3% 4%5%4.*%)4.% ($ (/, ,'5$-64.$, $,78-.4%, 4/)48)9 *-.4%, ($ -%(./, a & b 4/) a < b: 33$)%, $)*-$ ,+ ($ ') 1%*$-.%3 (.$384*-.4/ ($ 5$-1.*.;.(%( ǫ: $,

C=
RC =

4πǫ − 1b

1 a

− 1b · 4πµ

1 a

ǫ 4πǫ 1 = µ −b

1 a

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!!

!"#$%&' ((

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)"$*+,-./ "/,'%+'./) &,# 7#+'(*# -' -*8'(',%*# -' 9/+',%*#$ V0 # $#) %#(#) -'$ )'.*+/(/6 :/( $# )*.'+(;# -'$ +/(/2 '$ %#.9/ '$<%+(*%/ -'7' ')+#( -*(*=*-/ ', '$ )',+*-/ -' θˆ6 > 9(*.'(# 5*)+#2 ~ = E(r, θ)θˆ2 -/,-' r2 θ )/, $#) %/,/%*-#) %//(-',#-#) %*$;,-(*%#)6 @/)+('./) +','./) ?&' E ?&' )* $# %/((*',+' ')+3 ', (<=*.', ')+#%*/,#(*/2 ',+/,%') E = E(r)6 A, '8'%+/2 ', (<=*.', ')+#%*/,#(*/ B/ 9'(.#,',+'C +','./) ?&'

~ · J~ = 1 ∂ (rJr ) + 1 ∂Jθ + ∂Jz = 0 ∇ r ∂r r ∂θ ∂z :'(/

~ = µE(r, θ)θˆ = J θˆ =⇒ Jθ = J J~ = µE :/( +#,+/

~ · J~ = 1 ∂J = 0 =⇒ J = f (r) + C = J(r) ∇ r ∂θ ~ = E(r)θˆ6 9/( $/ ?&' 5'./) ?&' J~ = J(r)θˆ 1 E D# %/((*',+' ')

i = =

Z

J~ · n ˆ dS

Ω Z a Z b+a 0

= aµ

Z

J(r)drdz

b b+a

E(r)dr

b

-/,-' 4'./) +/.#-/ %/./ )&9'(E%*' -' *,+'=(#%*0, &,# %#(# -'$ +/(/6 D# -*8'(',%*# -' 9/+',%*#$ ',+(' $#) %#(#) 'F+('.#) -'$ +/(/ ') V0 6 >);2 +','./) ?&'

V0 = =

Z

~ · dˆ E r

ZΓπ 0

E(r)θˆ · rdθθˆ

= E(r)rπ V0 =⇒ E(r) = πr

!" #$%#& '&($) *$(+#$ ,% -+(.%$ )&(.-./-,0+/ #& /+#.$ /1 -$% dˆ r = rdθθˆ2 3& &)*+ 4$/(+1 *&%&($) 5,&

i = aµ

Z

b+a

E(r)dr

b

= = =⇒ R = 6,&) V0 = iR

Z aµV0 b+a dr π r b aµV0 a ln 1 + π b π aµln 1 + ab

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!"

!"#$%&' ()

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~ · J~ = ∂Jx + ∂Jy + ∂Jz = ∂Jz = 0 =⇒ J = Jz = C ∇ ∂x ∂y ∂z ∂z ()$(' H '* /$- .)$*%-$%'@ I)& %-$%)4 J~ = J zˆ '* /$ 7'.%)& .)$*%-$%'@

~ 4 ,)* .-6+)* '$ .-(- &'923$ *)$ F-() =/' J~ = µE

J E~1 = zˆ µ1 J>)&-4 ', +)%'$.2-, '*

J E~2 = zˆ µ2

√

A >>

!"

Z

~ · dˆ r V0 = − E Γ Z d2 Z = E2 dz + 0

d2 +d1

E1 dz

d2

Jd2 Jd1 + µ2 µ1 d1 d2 = J + µ2 µ1 V0 = d1 d2 + µ2 µ1 =

=⇒ J

#$%& '()(*+$ ,-(

i= .+/ 0+ 1-20

V V0 A = 0 J~ · n ˆ dS = JA = d1 d2 R Ω µ 1 + µ2

Z

R=

1 A

d1 d2 + µ1 µ2

34 5+)$67(/(*+$ -) 1606)7/+ /(1'+ 1+) (8( 7( $6*9'/62 () (0 $()'67+ 7( zˆ& 1+) 12/2$ 16/1-02/($ 7( :/(2 S & (0 1-20 ($ 1+/'27+ .+/ (0 6)'(/;2< ()'/( *2'(/620($= >02*(*+$ $- $-.(/?16( Ω= ~ = ǫE ~ & ()'+)1($ D ~ = D(z)ˆ 5+*+ D z = @+/ 02 0(A 7( B2-$$ C()(/206<272& '()(*+$

I

Ω

~ ·n D ˆ dS = Qlibre = σS

n ˆ = zˆ

(D1 − D2 )S = σS

=⇒ σ = D1 − D2

= ǫ1 E1 − ǫ2 E2 ǫ2 ǫ1 = J − µ1 µ2 ǫ1 V0 ǫ2 = − d d1 µ1 µ2 2 + µ2 µ1 ǫ1 µ2 − ǫ2 µ1 σ = V0 d1 µ2 + d2 µ1

!"#$%&' () '**+,-$, ,&, $*+ !

!"

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~ r) = dB(~ = ~ r) = =⇒ B(~ = = = = = =

~ × (~r − r~′ ) µ0 idl 4π |~r − r~′ |3

µ0 idz zˆ × (rˆ r − z zˆ) 4π (r2 + z 2 )3/2 Z L µ0 rdz θˆ ·i 2 2 3/2 4π −L (r + z ) Z L dz µ0 · irθˆ z = rtan(α) 2 + z 2 )3/2 4π (r −L Z arctan(L/r) µ0 rsec2 (α)dα ˆ · irθ 2π r3 sec3 (α) 0 Z µ0 iθˆ arctan(L/r) cos(α)dα · 2π r 0 µ0 iθˆ arctan(L/r) · sen(α)|0 2π r ! " µ0 iθˆ L/r p · 2π r 1 + (L/r)2 L µ0 iθˆ √ · 2π r r2 + L2

FG$*,3 4,*, )%5$%7*,* )- 5,.4$ .,0%:7'5$ () +% ,-,./*) '%=%'7$ %$& /,&7, 7$.,* L → ∞

!" #$% &$'(%)*#+ *,)$&-+&. /+, %+ /(*% +0)$,$1+'

~ r) = µ0 · i θˆ B(~ 2π r /*12+ 3($ )-$,$ '-1$)&4* /-%4,#&-/*. /+1+ $&* #$ $'2$&*&'$5

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!"#$%&' ()

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~ r) = µ0 · i θˆ B(~ 2π r G&H2 ,+ 93)*:+ <3) ), 4+-1$ ;)%)*+($ 1$* ), +,+-.*) 1 )=)*4) &$.*) 3% ),)-)%0$ ('9)*)%4'+, () 4$**')%0) (), +,+-.*) 2 )&0E (+($ 1$*

~2 × B~1 dF~21 = i2 dl µ0 i1 · x ˆ = i2 dz2 zˆ × 2π a µ0 = · i1 i2 yˆdz2 2πa µ0 dF~21 = · i1 i2 yˆ =⇒ dz2 2πa F) ,$ 43+, C)-$& <3) +-.$& +,+-.*)& &) +0*+)% 43+%($ ,+& 4$**')%0)& 0')%)% -'&-$ &)%0'($ B &) *)1),)% 43+%($ 0')%)% &)%0'($& ('9)*)%0)&7 .6 8% )&0) 4+&$ 0)%)-$& <3)

i=

λdz dz dq = =λ = λv dt dt dt

1$* ,$ <3) i = i1 = i2 = λv D$* 0+%0$2 (), '0@- +%0)*'$*2 ,+ 93)*:+ -+;%@0'4+ 1$* 3%'(+( () ,+*;$ &$.*) ), +,+-.*) 12 ().'($ + ,+ '%0)*+44'?% -+;%@0'4+2 )&

µ0 µ0 dF~m = · i2 yˆ = · (λv)2 yˆ dz 2πa 2πa

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~ E(r) =

λ rˆ 2πǫ0 r

B4 &2,& +#2/; )# $%&'(# &)*+,'-+# 2/@'& %4 &)&1&4,/ 0-$&'&4+-#) 0&) #)#1@'& &2

~ dF~e = dq E(r) ~ = λdz E(r) λ2 · yˆdz 2πǫ0 a λ2 = − · yˆ 2πǫ0 a

= − =⇒

dF~e dz

C/' ,#4,/; )# +/40-+-<4 D%& 0&2/2 &2 D%&

dF~e dF~m ~ + =0 dz dz )/ D%& &D%-=#)& # D%&

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~ r) = µ0 B(~ 4π

I

Γ

~ × (~r − r~′ ) idl |~r − r~′ |3

~ = dsθˆ = Rdθθˆ8 B$* 2,%2$@ 9% )&2) .,&$ ~r = z zˆ@ r~′ = Rˆ ρ@ dl

~ r) = B(~ = = = = =

µ0 4π

I

Γ

~ × (~r − r~′ ) idl |~r − r~′ |3

iRdθθˆ × (z zˆ − Rˆ ρ) 3/2 2 2 (R + z ) 0 Z 2π µ0 dθ(z ρˆ + Rˆ z) iR 3/2 2 2 4π (R + z ) 0 Z 2π Z 2π µ0 µ0 dθ(z ρˆ) dθ(Rˆ z) + iR iR 2 + z 2 )3/2 2 + z 2 )3/2 4π 4π (R (R 0 0 2πˆ z µ0 2 iR 4π (R2 + z 2 )3/2 iR2 zˆ µ0 2 (R2 + z 2 )3/2 µ0 4π

Z

2π

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λds rdθ dθ dq = =λ = λr = λrw dt dt dt dt ($%() i )& 1- ,$**')%4) <+) /-&- /$* ,-(- )&/'*- '%0%'4)&'3-1 @ r &+ *-('$6 A)1 /*$51)3- -%4)*'$*> 4)%)3$& <+) )1 ,-3/$ .)%)*-($ /$* ,-(- )&/'*- )& i=

~ r) = dB(~

µ0 ir2 zˆ µ0 λwr3 zˆ 1 r3 drˆ z µ σw = = 0 3/2 3/2 2 2 2 2 2 2 2 (r + z ) 2 (r + z ) 2 (r + z )3/2

K$* 4-%4$> )1 ,-3/$ 3-.%;4',$ .)%)*-($ /$* )1 ('&,$ -1 .'*-* ,$% *-/'()L -%.+1-* w )& R

r3 dr (r2 + z 2 )3/2

~ r) = B(~

1 µ0 σwˆ z 2

Z

=

1 µ0 σwˆ z 2

Z

=

1 µ0 σwˆ z 2

Z

1 µ0 σwˆ z 2

! p

= = =

0

√

R2 +z 2

|z| √

R2 +z 2

|z|

u2 = r2 + z 2 → udu = rdr

(u2 − z 2 )udu u3 1−

z2 u2

R2 + z 2 − |z| +

√R2 +z 2 " 2 z u |z|

p 1 z2 2 2 − 2|z| R +z + √ µ0 σwˆ z 2 R2 + z 2 2 1 2z + R2 µ0 σwˆ z √ − 2|z| 2 R2 + z 2

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#$%&'()*) +% ,-.%$ ,$* )- /+.- 0+1) /$**')%2)3 /$% ()%&'(.( () /$**')%2) 4 &+,)*5/'.-6 J~ = J0 yˆ7 8%/+)%2*) )- /.9,$ 9.:%;2'/$ )% 2$($ )- )&,./'$7

*"$+,-./0

~ ~ 7 @$9)9$& /$9$ ,+%2$ () $>A <$* -. &'9)2*=. ()- ,*$>-)9.3 2)%)9$& ?+) B(x, y, z) = B(z) ′ ~ &)*B./'C% ~r = z zˆ 1 r = xˆ x + y yˆ )- B)/2$* ?+) ()&/*'>. )- ,-.%$7 @)%)9$& ?+) -. ()%&'(.( () /$**')%2) &+,)*5/'.- )&2D (.(. ,$* J~ = J0 yˆ3 ,$* -$ ?+) -. /$**')%2) &) 2*.%&9'2) )% )- &)%2'($ () yˆ7 E&=3 2)%)9$& ?+) di = J~ · yˆdS = J0 dS 7 <$* 2.%2$3 )- /.9,$ 9.:%;2'/$ :)%)*.($ ,$* +% )-)9)%2*$ ('F)*)%/'.- () /$**')%2) )& ~ r) = dB(~

µ0 diˆ (xˆ z + zx ˆ) y × (z zˆ − xˆ x − y yˆ) µ0 J0 dS 2 = 4π (x2 + y 2 + z 2 )3/2 4π (x + y 2 + z 2 )3/2

/$% dS = dxdy 7 E&=3 )- /.9,$ 9.:%;2'/$ :)%)*.($ ,$* )- ,-.%$ )&

~ r) = B(~ = = = = =

Z ∞Z ∞ µ0 (xˆ z + zx ˆ)dxdy J0 2 2 2 3/2 4π −∞ (x + y + z ) Z Z−∞ Z ∞Z ∞ ∞ ∞ µ0 µ0 xˆ z dxdy zx ˆdxdy + J0 J0 3/2 2 2 2 2 2 2 3/2 4π 4π −∞ −∞ (x + y + z ) −∞ −∞ (x + y + z ) Z ∞Z ∞ µ0 dxdy J0 z x ˆ 2 2 2 3/2 4π −∞ −∞ (x + y + z ) Z ∞ Z 2π rdrdθ µ0 J0 z x ˆ 2 4π (r + z 2 )3/2 Z 0∞ 0 rdr µ0 J0 z x ˆ 2 + z 2 )3/2 2 (r 0 1 z µ0 J0 x ˆ 2 |z|

($%() -. ,*'9)*. '%2):*.- )& /)*$ ,+)& )- '%2):*.%($ )& '9,.* 4.- '%2):*.* )% G6 1 -$& -=9'2)& () '%2):*./'C% &$% &'9;2*'/$& *)&,)/2$ 07 H$2) -$ &'9'-.* ?+) )& )- 9C(+-$ /$% )- 9CA (+-$ ()- /.9,$ )-;/2*'/$ :)%)*.($ ,$* +% ,-.%$ () ()%&'(.( &+,)*5/'.- /$%&2.%2) σ 3 )- /+.- )& ~ r) = σ z zˆ7 E(~ 2ǫ0 |z| 8- I)/I$ ?+) )- /.9,$ 9.:%;2'/$ )&2+B')*. .,+%2.%($ I./'. x ˆ )*. () )&,)*.*&)7 <.*. B)* )&2$ /$%&'()*) )- ,-.%$ /$9$ +%. &+/)&'C% '%5%'2. () .-.9>*)& () /$**')%2) )% )- &)%2'($ yˆ7 J'&+.-A 'K.%($ )- /.9,$ 9.:%;2'/$ ?+) :)%)*. /.(. +%$ () )&2$& .-.9>*)& '%5%'2)&'9.-)& &+/)&'B$&3 1 .,-'/.%($ &+,)*,$&'/'C%3 )& /-.*$ ?+) )- /.9,$ 9.:%;2'/$ ()>) )&2.* )% )- &)%2'($ x ˆ7 LM$:*.& B)*-$N7

!!

!"#$%&' ()

!"#$ %&' ()*)!%#&' (&!%+("&#$' (&!(,!"#)(&'- .+/ *0#1&'- %$ #0%)&' a / c- '$ **$!0 %$ .0"$#)0* %$ (&!%+(")2)%0%$' g1 / g2 - / %$ 3$#.)")2)0%$' ǫ1 / ǫ2 - (&.& .+$'"#0 *0 41+#05 '* !(&!"#0# *0 #$')'"$!()0 3&# +!)%0% %$ *0#1& $!"#$ $* !6(*$& / $* (&!%+("&# $7"$#!&5 #* 8) '$ (&!$("0 +!0 %)9$#$!()0 %$ 3&"$!()0* V0 $!"#$ $* !6(*$& / $* (&!%+("&# $7"$#!&- (0*(+*$ *0 %$!')%0% %$ (&##)$!"$ J~ / *0 (&##)$!"$ I :+$ ()#(+*05 +* ;0*(+*$ *0 %$!')%0% %$ (0#10 *)<#$ :+$ '$ 0(+.+*0 $! *0 )!"$#90= %$ 0.<&' .0"$#)0*$' >)5$ $! r = b? $! (&!%)()&!$' 3$#.0!$!"$'- 0* $'"0# *0' 3*0(0' (&!$("0%0' 0 +!0 %)9$#$!()0 %$ 3&"$!()0* V0 5

,"$-+./0 '* @0#0 $!(&!"#0# *0 #$')'"$!()0 '$ %$<$ 3#&($%$# %$ )1+0* 9&#.0 :+$ $! $* 3#&<*$.0 A5 B$!$.&' :+$ *0 (&##)$!"$ '$ (0*(+*0 (&.&C I=

Z

S

J~ · n ˆ dS

~ * 2$("&# %$!')%0% %$ (&##)$!"$ $' #0%)0*CJ(r) = J(r)ˆ r5 D')C I = J2πrL

;&! $'"&- 3&%$.&' $'(#)<)# $* 2$("&# %$!')%0% %$ (&##)$!"$ $! 9+!()E! %$ *0 (&##)$!"$C J~ =

I rˆ 2πrL

;&! $'"$ #$'+*"0%& 3&%$.&' (0*(+*0# *&' (0.3& $*,("#)(&' $! 0.<&' .0"$#)0*$' +")*)=0!%& *0 (&!%+(")2)%0% %$ (0%0 +!&C ~1 = E

I 2πrLg1

~2 = E

I 2πrLg2

DF�- (0*(+*0.&' $* 3&"$!()0* $!"#$ r = a / r = cC ∆V =

Z

a

c

~ · d~r E

! $'"0 #$1)E! $* (0.3& $*,("#)(& 20#)0 $! 0.<&' (&!%+("&#$' / 3&# *& "0!"& '$30#0.&' $'"0 )!"$1#0* %$ *G!$0 %$ *0 ')1+)$!"$ .0!$#0C

!"#$%&' () '**+,-$, ,&, $*+ !

!"

∆V =

Z

a

#$%&%'()*+ r =

∆V I

b

~ 1 · d~r + E

+,$-)-.+/0

R=

Z

c

b

~ 2 · d~r = E

I I b c ln( ) + ln( ) 2πLg1 a 2πLg2 b

b 1 c 1 ln( ) + ln( ) 2πLg1 a 2πLg2 b

1+.+ 2+*-.+/ (23-4%(3 -) &( -523-/%6) *- 3-/%/$-)4%( /+&+ %)$-37%-)-) &+/ *($+/ 3-8-3-)$-/ ( &(/ 23+2%-*(*-/ *-& .($-3%(& 9 &( :-+.-$3;( *-& /%/$-.(<

! =% /- 4+)-4$( >)( *%8-3-)4%( *- 2+$-)4%(& δV = V0 ? /- $%-)- @>- &( 4+33%-)$- 4>.2&- 4+) I = RV0 < A--.2&('()*+ -) &( -523-/%6) *- 3-/%/$-)4%( -)4+)$3(*( ()$-3%+3.-)$-0 I=

V0 2πL ln( cb ) ln( ab ) + g1 g2

1+) -/$( 4+33%-)$- -/ *%3-4$+ -)4+)$3(3 -& 7(&+3 *- &( *-)/%*(* *- 4+33%-)$-< "! B(3( 4(&4>&(3 &( *-)/%*(* *- 4(3:( &%,3- @>- /- (4>.>&( -) &( %)$-38(' *- (.,+/ .($-3%(&-/ >$%&%'(.+/ &( &-9 *- :(>// 2(3( *%-&C4$3%4+/< B(3( -/$+ 4+)/%*-3(.+/ 4+.+ />2-3D4%- :(>//%()( >) .()$+ 4%&;)*3%4+ 4+) />/ 3-/2-4$%7(/ $(2(/ *- E3-( .>9 2-@>-F(

Z

S

~ ·n D ˆ dS = qenc

B+3 &( /%.-$3;( 2+*-.+/ *-/4+.2+)-3 -/$( %)$-:3(& -)0

~ 2 · (Aˆ ~ 1 · (−Aˆ D n) + D n) = σA ~2 −D ~ 1) · n (D ˆ=σ

1+.+ 7%.+/ -) -& 3-/>.-)? -& 7-4$+3 *-/2&('(.%-)$+ 2(3( >) *%-&C4$3%4+ /- 4(&4>&( /%.2&-.-)$4+.+ &( 2-3.%$%7%*(* 2+3 -& 4(.2+ -&C4$3%4+0

~ 1 = ǫ1 E ~1 D ~ 2 = ǫ2 E ~2 D 1+) &+/ 7(&+3-/ *- 4(.2+ -&C4$3%4+ -)4+)$3(*+/ -) &( 2(3$- (G $-)-.+/ @>- &+/ 7-4$+3-/ *-H /2&('(.%-)$+ -&C4$3%4+ /+)0

!"

~1 = D

ǫ1 I rˆ 2πrLg1

~2 = D

ǫ2 I rˆ 2πrLg2

#$ %&'()* *+,-+$ &. -/0+$ +$ %&'()* 1)*2+$ + $+ .03&*4'-&5 rˆ = n ˆ 6 7& &.(+ 2+1&*+ 3),&2). '+$'0$+* $+ ,&1.-,+, ,& '+*/+ $-8*&5

σ=

ǫ1 I ǫ2 ( − ) 2πbL g2 g1

9)1 &$ %+$)* ,& ')**-&1(& ,& $+ 3+*(& 8: *&.0$(+ 41+$2&1(& ;0& $+ ,&1.-,+, ,& '+*/+ $-8*& &.5

V0

σ= b(

ln( ab ) g1

+

ln( cb ) g2

ǫ2 ǫ1 − ) ) g2 g1 (

!"#$%&' () '**+,-$, ,&, $*+ !

!"

!"#$%&' ()

#$%&'($) *+, '&-(-%(.(/$/ 0,(1+'2& ρ -& 1+'2$ *+2+ 0,$ *03$ /& )$ 2$,&'$ (,/(*$/$ &, )$ 450'$6 #0&-%'& 70& )$ '&-(-%&,*($ &,%'& )$- *$'$- A 8 B /& &-%$ *03$ &-9

R=ρ

L y2 ln( ) w(y2 − y1 ) y1

*"$+,-./

:$ '&-(-%&,*($ -& /&4,& *+2+ R = ρℓ A 6 :+ 70& ;$'&2+- &- /(.(/(' )$ *03$ &, <&70&3+- =)+70&/& $,*;+ dx 8 $)%0'$ H(x) *+2+ 20&-%'$ )$ 450'$>.(-%$ )$%&'$)?9

@+/&2+- *+,-(/&'$' &-%+- =)+70&- *+2+ <&70&3+- <$'$)&)&
!" #$%$&'( )*$ ($ +*&,-./01

y2 − y1 h(x) = L x (y2 − y1 )x h(x) = L

1 )x 1 )x 2*$3'4 H(x) = h(x)+y1 = (y2 −y +y1 5 6'% $(7$ /$(*-789' :$&'( )*$ A(x) = w( (y2 −y +y1 ) L L ;$ $(78 &8%$/8 -8 ,$)*$<8 /$(.(7$%+.8 )*$ 8,'/78 ($/01

dR =

dx ρdx ρL ρdx = = −y )x (y A(x) w (y2 − y1 )x + y1 L w( 2 L 1 + y1 )

#$%$&'( )*$ $(78( /$(.(7$%+.8( $(70% $% ($/.$4 9$ $(78 &8%$/8 7$%$&'( )*$ (*&8/ -.%$8-&$%7$ 7'98( -8( /$(.(7$%+.8( 8,'/7898( ,'/ +898 =-')*$ > '=7$%$&'( -8 /$(.(7$%+.8 7'78- 9$ -8 +*<85 ?8/8 $(7'4 .%7$3/8&'( 9$(9$ x = 0 8 x = L1

ρL R= w

Z

0

L

dx (y2 − y1 )x + y1 L

@*-7.,-.+8&'( ,'/ *% *%' +'%:$%.$%7$ ,8/8 ,'9$/ .%7$3/8/ 9./$+78&$%7$1

(y2 − y1 ) ρL R= (y2 − y1 ) w

Z

0

L

dx ρL = (y2 − y1 )x + y1 L w(y2 − y1 )

Z

0

L

(y2 − y1 )dx ρL (y2 − y1 )L + y1 L = ln( ) (y2 − y1 )x + y1 L w(y2 − y1 ) (y2 − y1 ) · 0 + y1 L

A.&,-.B+8%9' $(78 $C,/$(.D% ($ '=7.$%$ B%8-&$%7$

R=ρ E*$ $( -' )*$ )*$/F8&'( 9$&'(7/8/5

L y2 ln( ) w(y2 − y1 ) y1

!"

!"#$%&' ()

!"#$%&' () '**+,-$, ,&, $*+ !

#$%&'() *$+ +, -&'.+'/).&0 .+ ,) 1($0) +/2) 3'3-3),4+'2+ .+/-)0().&5'& 6)7 -&003+'2+ -$)'.& +, 3'2+00$%2&0 S +/2) )83+02&9: ;'-$+'20+ ,)/ +-$)-3&'+/ *$+ 4&.+,)' ,) -)0() .+, -&'.+'/).&0 +' <$'-3&' .+, 23+4%& -$)'.& /+ -3+00) +, 3'2+00$%2&0: =>&4& /+ -&4%&02) ,) -&003+'2+ +' <$'-3&' .+, 23+4%&?:

!

!"#$%&' ()

"#$%&#'(& )* +&(,-,* ,& &#&(.-* +/( $*)/( ,& 0/%)& ,&) $-($%-'/ ,& )* 1.%(*2 3&4%&5'(& 6%& &5'& #%4&(/ #/ ,&+&#,& ,& )* (&5-5'&#$-* R2 7#-$-*)4&#'& &) $/#,*,/( ,& $*+*$-,*, C1 '-&#& %#* $*(.* Q0 8 &) 5&.%#,/ $/#,*,/( &5'* ,&5$*(.*,/2

Ejercicios Resueltos Electricidad y Magnetismo (Garrido - Narrias)

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