Ejercicios Resueltos de Electricidad y Magnetismo

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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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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'%($&"/")'( ,%* .''/&"%*&* 9 "% #'/%' -* ;'($.$?% &" "0,$-$
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( #$%
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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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mg tan(θ) B 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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k(−q)Q(xî − d/2ˆj) F~2 = l3 ") )$,& *&+)'&; 7& D0)'>& ')$07,&+,) $#?') 7& %&'(& −q $)'FA −2kqQx ˆ i F~ = F~1 + F~2 = l3

(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

!"#$%&' () '*+% $',-. !"#$% &'(!)*+!% !&*!" ,& &''"- .&/ -0 &1)&*+%*2 &- #3- 40&*)& 50& &/ *&6+%/&- !%/ #&/%* *",+% ,& !0*7")0*"8 9-"/,% '" *&'"!+:/ &/!%/)*"," &/ '" &!0"!+:/ ,& !%/-&*7"!+:/ ,& !"*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

!

!"#$%&' ()

"#$ %&$%&'&$ ($)*'+%&$ %#,-.%/#'&$ %#,%*,/'+%&$ -( '&-+#$ a b3 %#,%*,/'+%& %#, 1&$ &,/('+#'($6 =7.>,/# /'&<&?# ($ ,(%($&'+# '(&1+@&' 0&'& %#1#%&' -+%:& %&'8&A

+"$,-./01

"(;,+5#$ %#5# Q1 2 Q2 1&$ %&'8&$ -( 1& ($)('& %#, 0#/(,%+&1 V2 2 V2 3 '($0(%/+4&5(,/(6 7#5# 2& :(5#$ 4+$/# (, &2.-&,/+&$ &,/('+#'($3 (1 %&50# (1(%/'+%# 0'#-.%+-# 0#' %&-& %&$%&'& $( 0.(-( %&1%.1&' ./+1+@&,-# 1(2 -( 8&.$$ ./+1+@&,-# %#5# $.0(';(%+( 8&.$$+&,& .,& %&$%&'& ($)('+%& %#, '&-+# a < r < b6 B& %&'8& (,%(''&-& 0#' ($/& $.0(';%+( ($ 1& &0#'/&-& 0#' 1& %&$%&'& -( '&-+# a3 ($ -(%+'3 Q1 0#' 1# /&,/#C Z Q1 ~ ·n E ˆ dS = ǫ0 S R ~ ·n E ˆ dS = E4πr2 6 "( ($/& 5&,('& (1 %&50# (1(%/'+%#EF.( 2& :(5#$ 4+$/# F.( /+(,( D('# S

-+'(%%+#, '&-+&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

!

!"#$%&' ()

"#$% &'%&'#'% $%($#)&'% &*+,-&.*#'% &*+&$.#)&'% /-0 ,$12','% 3*%$$+ #',)*% a4 b 0 c4 #$%3$&.)5 6'/$+.$4 %)$+,* a < b < c7 8+)&)'1/$+.$ 1' &'%&'#' )+.$#)*# $%.' ,$%&'#2','4 1' ,$1 /$,)* 3*%$$ -+' &'#2' .*.'1 +$2'.)6' −Q 0 1' $9.$#+' ,$ &'#2' .*.'1 3*%).)6' +Q7

*'+ :+&-$+.#$ $1 3*.$+&)'1 $1$&.#)&* $+ &',' -+' ,$ 1'% &'%&'#'% &*+,-&.*#'%7 *#+ ;) 1'% &'%&'#'% )+.$#)*# 0 $9.$#)*# %*+ &*+$&.','% /$,)'+.$ -+ '1'/<#$ =-$ $%.' ')%1',* '1 3'%'# 3*# 1' &'%&'#' &$+.#'17 >?-'1 $% '@*#' $1 3*.$+&)'1 $1$&.#)&* ,$ &',' -+' ,$ 1'% &'%5 &'#'%A>?-'1 $% 1' &'%&'#' $+ &',' -+' ,$ 1'% &'%&'#'%A

,"$-./"0

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

!

!"#$%&' ()

"#$ %$&%$'$ (&)*'+%$ %,#-.%/,'$ -( '$-+, a 0 (&1(&,' δ ≫ a %,#/+(#( %$'2$ #(/$ Q3 4( -+&/'+5.0( .#$ %$'2$ q (# (6 7,6.8(# +#/('+,' -(6 %$&%$',# -( '$-+, a9.# $+&6$#/( (# 6$ 1$'/( +#/('+,' -(6 %$&%$',# +81+-( :.( (&/$ -(#&+-$- -( %$'2$ &( 1$&( $6 %,#-.%/,';3 <,& -+%(# :.( (6 %$81, (6*%/'+%, (# (6 +#/('+,' -(6 %$&%$',# (&/$ -$-, 1,'

~ = 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

3 qr3 2 πa6 ! 3&*)+$, 45) )/ )0 ./-)%.$% () 5/ 1&,1&%$/ 0& 1&%6& /)-& ), /50&2 ",. ,. 1$/,.()%&+$, 5/& ,57)%=1.) ),<)%.1& 1$/ %&(.$ r = a + αδ ' 1$/ 0 < α < 1' 8)%)+$, 45) 7&%& 45) ,) +&/-)/6& /50& 0& 1&%6&' )0 1$/(51-$% ./(51) 5/& 1&%6& −q )/ )0 ./-)%.$% ()0 1$/(51-$%2 :) ),-& +&/)%&' 0& ()/,.(&( () 1&%6& )/ 0& ,57)%=1.) ./-)%.$% ()0 1$/(51-$% ),9 ρ(r) =

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

9'0'(*1 '$ )*,'0%.#$ )-*/&%./* )*- $# /.1,-.2&%.30 /' %#-4# ρ(r) : ;'(*1 # $# ;'< 5&' '1,' )*,'0%.#$ '1,# '0 =&0%.30 /' r7 )*- $* 5&' '$ )*,'0%.#$ ,.'0' 1.(',-># '1=+-.%#8 ?@*-#7 &,.$.<#0/* '$ @'%@* /' 5&' ~ = −∇V E A*/'(*1 '0%*0,-#- =B%.$('0,' '$ %#()* '$+%,-.%*C

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

!

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

Ejercicios Resueltos de Electricidad y Magnetismo

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