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i+

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B

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∑

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∑

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sin(π

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mg

sin(π

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sin(π

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)−

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q=±

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cot(

π/3

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k

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2∼ =−

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1−

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mg

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x02 =⇒

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x

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dt2

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mx

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x x0

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0·x

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~ F1

=k

q2 L2·(

cos(

π 3)x−

sen(π 3

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q2 L2·(

x 2−√

3 2y)

~ F2

=k

q2 L2x

6*7,"/B*/"(,-#*%#""(

~ F=

~ F1+

~ F2

=3 2k

q2 L2x−√

3 2k

q2 L2y

=√

3·k

q2 L2·(√

3 2x−

y 2)

=√

3·k

q2 L2·(

cos(

π 6)x−

sen(π 6

)y)

R"-("%#$&'&"-*7,"/B*A1")'(0,"P

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π−

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r

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L

sen(2

π 3)

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1 √3·q

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R·(

cos(

θ)x

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n(θ

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√R

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2-(&)-:%-#

~ F=

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~ F2

=kqQ

(R2+

a2)3 2

·(R

cos(

θ)x

+R

sen(θ

)y−

az

+R

cos(

θ)x

+R

sen(θ

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az)

=2kqQ

R

(R2+

a2)3 2

·(co

s(θ)

x+

sen(θ

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.

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|~ F|=

2kqQ

R

(R2+

a2)3 2

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(R)

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dR

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2+

a2)3 2−

3R

2(R

2+

a2)1 2

(R2+

a2)3

=0

=⇒

(R2+

a2)1 2·(

(R2+

a2)−

3R

2))

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n(θ

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Tco

s(θ)

j

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∑

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θ)−

mg

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(2)

;'<=>7

T=

mg

cos(

θ)

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mg

cos(

θ)si

n(θ

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mg

tan(θ

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E

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tan(θ

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θ≪

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q(t)

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gθ(

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E

dθ dt

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mg

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;#&/%$E%&

+Q

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l3

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(xi−

d/2j)

l3

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q&'$N7

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l3i

(1)

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θ/0*/.-*%

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s(θ)

=⇒

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cos(

θ)

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d3

(2)

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m~a

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d2x

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−16

kqQ

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d3

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d2x

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x

md

3=

0

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w=

4√

kqQ

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√

md

3

kqQ

J

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Ey

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cos(

θ)>A,$%*(,%*#,.-$)%#,7.2+/.%=

E=#&)%+*#60/./408

E=

kq

d2+

r2

!

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θ)=

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Ey

=(

kq

d2+

r2·

r√

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/2

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~ Ey

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2qk(

1 r2−

r

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)

;'36,696#$%6'$'*.+:+.#$)6.%4$/0+-4,$53$*6%6E

~ E=

2qk(

1 r2−

1

r2(1

+d2

r2)3 2

)

=2q

k

r2(1−

(1+

d2 r2)−

3 2)

;58$5$.4,(,4)05*+>5

(1+

x)α

'$90$#$49.6F+%4.96.34A,6.(*045#6

x≈

04E

(1+

x)α

=1

+αx

B6%6

d≪

r('$3+$5$/0$

d r≈

0A96.,634536

(d r)2≈

01<6.,634536($5,4$F9.$'+>5#$,

*4%9696#$%6'G4*$.,4'+80+$53$49.6F+%4*+>5E

(1+

d2 r2)−

3 2≈

(1−

3 2

d2 r2)

;-4,045#6$'3$.$'0,34#6$5,4$F9.$'+>56:3$5+#494.4$,*4%96('$3+$5$/0$E

~ E=

3qk

d2

r4j

H

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

"#$ %&'()

I5#+96,6$,=*3.+*6$505*4%96$,=*3.+*605+)6.%$'$#$'9,4@4,+8$.4%$53$#$'096'+*+>5#$

$/0+,+:.+6(*6%6'$%0$'3.4$5,4J80.4(#65#$

θ$'9$/0$K61;,%6%$536#$+5$.*+4#$,#+96,6

$'

I1L+$,#+96,6'$,+:$.4#$'#$,496'+*+>5(#$%0$'3.$/0$'06.+$534*+>54580,4.9.$'$534

%6-+%+$5364.%>5+*6'+%9,$A$5*0$53.$,4).$*0$5*+4#$6'*+,4*+>51

*$%+,-./0

74')0$.@4'$,=*3.+*4'/0$4*3M45'6:.$,4'*4.84''65+804,$'$5%485+30#A$5#+.$**+>5(9$.6

$'3&5$5'$53+#6'690$'36'1;'34')0$.@4''658$5$.4#4'96.$,*4%96$,=*3.+*6

~ EA3+$5$5

%485+30#

F=

qE1;'36'$90$#$49.$*+4.$5,4'+80+$53$J80.4E

;,36./0$'6:.$$,#+96,6'$.&$5365*$',4'0%4#$36./0$'9.6#0*+#696.*4#4)0$.@4(*65N

'+#$.45#6$,$C$#$.634*+>5'6:.$$,90536

O1;5365*$''$3$5#.&/0$$,36./0$'6:.$$,#+96,6

$'E

τ=−

Fa

sin(θ

)+−

Fa

sin(θ

)=−

2F

asi

n(θ

)=−

2qE

sin(θ

)

;,'+8565$843+-6'$6:3+$5$494.3+.#$,4.$8,4#$,4%456#$.$*G4A'$90$#$*6%9.6:4.4,

G4*$.,469$.4*+65#$%6%$536'#$)6.%4-$*36.+4,1

L4:$%6'/0$$,36./0$'6:.$05'+'3$%4'$.&+804,4,%6%$536#$+5$.*+4

I96.,44*$,$.4*+>5

4580,4.

αO4%:6'.$'9$*364,$C$'6:.$

OD($'#$*+.

τ=

Iα

−2qE

asi

n(θ

)=

Iα

B6%6'$9.6#0C6'6,6059$/0$K6#$'9,4@4%+$536#$,#+96,6'$*0%9,4/0$

θ≪

1A96.,6

3453696#$%6'G4*$.,449.6F+%4*+>5

sin(θ

)≈

θ1"#$%&'5634%6'/0$

α=

d2θ

dt21?$$%9,4@45#6

$'36'-4,6.$'$5,4$*04*+>5453$.+6.'$3+$5$/0$E

!

−2qE

aθ

=Id

2θ

dt2

"#$%&'(%')*+,)-&.&+')%//%01,&21#&#'.)3).'1214

d2θ

dt2

+2qE

aθ

I=

0

51+

/1'(%/6&21#7(&/%1.)&+$%')*+

%+8(/%.,&/,)01/1#&8().9(+

216)2)&+$1%.2*+)'1

#)20/&'1+-.&'(&+')%%+8(/%.4

w=

√

2qE

a

I

:01./1$%+$1/%-.&'(%+')%#&.94

f=

w 2π

=1 2π

√

2qE

a

I

;

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

!"#$%&'(

!"#$%&'#()#

*+,'+-

.

!"#$%&'(

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

12λ

22'!3(,(%&'

0$(%#'"($/#(%45()/0)!)(10!,6(70!'!!8!,/!$(*+&'()(*+,!'4

)"$*+,-./9&,)()!:%!5&0)&*+2'(+!*&'70!)(10!,6(!);/",#/(70!!<3!,#*!$"(0$(/(,-(

7!$)(3&'#/#=$

~r%!+#%&(0$(%#'",#+0/#=$%!/(,-(

Ω!'

~ F=

q·∫

Ω

dq(

~r−

~ r‘)

4πǫ 0|~r−

~ r‘|3

9&$-(*&'(*+&'()(*+,!'!$!)!8!<2%!"()1&,*(70!0$&/0+,(%!

x=

0(

x=

L:!)

&",&%!

x=

L+

d(

x=

2L+

d4 !($

~r 1=

x1x2~r 2

=x

2x2/&$

0≤

x1≤

L:

L+

d≤

x2≤

2L+

d2

)&'>!/"&,!'3&'#/#=$%!/(%(()(*+,!4

?!$!*&'70!)(10!,6(70!!8!,/!@'&+,!0$!)!*!$"&%<%!A!'

~dF

21

=dq 2·∫

Ω1

dq 1

(~r 2−

~r 1)

4πǫ 0|~r 2−

~r 1|3

=dq 2·∫

L

0

λ1dx

1(x

2−

x1)x

4πǫ 0|x

2−

x1|3

=dq 2·

λ1

4πǫ 0·x·∫

L

0

dx

1

(x2−

x1)2

=dq 2·

λ1

4πǫ 0·x·

1

x2−

x1

∣ ∣ ∣ ∣

L 0

=λ

1x

4πǫ 0·d

q 2·(

1

x2−

L−

1 x2)

=λ

1λ

2x

4πǫ 0·d

x2·(

1

x2−

L−

1 x2)

@B

AC

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

=⇒

~ F21

=λ

1λ

2x

4πǫ 0·∫

2L

+d

L+

d

(

1

x2−

L−

1 x2

)

dx

2

=λ

1λ

2x

4πǫ 0·(

ln(x

2−

L)−

ln(x

2))|2L

+d

L+

d

=λ

1λ

2x

4πǫ 0·l

n(x

2−

L

x2

)∣ ∣ ∣ ∣

2L

+d

L+

d

=⇒

~ F21

=λ

1λ

2

4πǫ 0·l

n

(

(L+

d)2

d(2

L+

d))

·x

D!'"(!')(10!,6(!);/",#/(70!'!!8!,/!$(*+&'()(*+,!'230!''(+!*&'70!)()!:%!5&0)&*+

'("#'1(/!)(E,()!:%!F!G"=$%!1&,*(10!,"!4

!

!"#$%&'()

"#$#$%&'()(#$*+,'$(-.$/0'%$)010',2$1$'.(-(%343'4,#$'

BC

D5)('$)30

R6%75810'

)0-'(4.3#92($-)(#02+3.,)(-

AB

=2R

6%78

DE

=R

6%7:;0-$'40-

BC

8C

D(-.<2,23=

/0'%(%(2.(4$'+$)0-4024$'+$-

q>?8−

q5'(-1(4.3@$%(2.(5%3(2.'$-A,(-0&'(

AB

.$%&3B2

-()3-.'3&,8(,23/0'%(%(2.(,2$4$'+$

q:C,(4$2.3)$))(4$'+$)(&('(1$'.3'-(402)(2-3)$)

402-.$2.(-0&'((#.'$D0

DE51$'$A,((#4$%10(#B4.'340-($2,#0(2(#4(2.'0

O:

*"$+,-"./

;0A,(E$'(%0--('<)3@3)3'(#$#$#%&'(40%1#(.0(2F1$'.(-.$#40%0%,(-.'$#$*+,'$G

H$#4,#$'(%0-(#4$%10(#(4.'3401'0),43)010'#$-F'(+302(-(2

O8#,(+0#0--,1('102)'(%0-

(3%102)'(%0-#$402)34302)(A,((#4$%10(#(4.'340(2(-.$'(+302-($2,#0:

0!"1"23/

"#$#%$&'((-'(4.3#32(08)(#$'+0

2R

8.3(2((2(#,2$4$'+$

q,23/0'%(%(2.()3-.'3&,3)$:I(

(-.$%$2('$10)(%0-)(*23'-,)(2-3)$))(4$'+$40%0

λ1

=q 2R:I((-.$%$2('$5-3.0%$%0-

,2$1(A,(J$4$'+$

dq)(#$#$%&'(-$&(%0-A,(-(4,%1#(#$'(#$4302

dq

=λ

1dx5)02)(

dx(-

,21(A,(J0.'0D0)(#$#$%$&'(

"#4$%10(#(4.'340A,(1'0),4(4$)$(#(%(2.0

dq(-.$'$(2)3'(44302

i 510'#0.$2.0(#4$%10

(#(4.'340A,(1'0),4(-('<G

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

d~ E

1=

(kdq

x2

) i=

(kλ

1dx

x2

) i

K2.(+'$2)0)(-)(

x=−

3R

E$-.$

x=−

R

~ E1

=k(λ

1

∫

1 x2dx)i

=(k

λ1(−

1 x2

∣ ∣ ∣ ∣

−R

−3R

))i

;0A,('(-,#.$G

~ E1

=2kλ

1

3Ri

L$'$(#4$-0)(#0-$'40-.(2)'(%0-A,($#-,1('102('2#0-(#4$%10(#(4.'340'(-,#.$2.((-.$'$

(2)3'(44302

hati:"-.0A,()$32%()3$2.$%(2.4#$'0$@('#$-3+,3(2.(*+,'$G

H0%0@(%0-5(#$'40

BC

.3(2(4$'+$10-3.3@$(2(#810'#0.$2.04$)$(#(%(2.0)(4$'+$

+(2('$,24$%10-$#3(2.(5%3(-2.'$-A,((#$'40

CD

.3(2(4$'+$2(+$.3@$84$)$(#(%(2.0

)(4$'+$+(2('$,24$%10(2.'$2.(:I((-.$%$2('$5#0A,()(&(%0-E$4('(--3%1#(%(2.(

4$#4,#$'#$-40%102(2.(-(2

x)(4$)$4$%108)(-1,(--,%$'#$-:

;$-)(2-3)$)(-)(4$'+$1$'$4$)$$'40-(0&.3(2(2/$43#%(2.($#-$&('A,(.3(2()(10-3.$)$

,2$4$'+$A8=A8.3(2(2,2#$'+0

πR 2G

λ2

=2q πR

λ3

=−

2q

πR

2!,"34/H02-3)('$%0-,2$1(A,(J$4$'+$

dq)(#$'405(#%0),#0)(#4$%10(#(4.'301'0=

),43)010'(-.$4$'+$-('<-3%1#(%(2.(G

dE

=kdq

R2

H02$8,)$)(#$)(2-3)$))(4$'+$

λ210)(%0-(-4'3&3'

dq

=λ

2ds5

)02)(

ds(-,21(A,(J0

.'0D0)(#$'40:"-.(.'0D0)($'40#010)(%0-(-4'3&3'(2/,24302)(#$2+,#0A,(#0'(40''(G

ds

=R

dθ

!

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

dE

2=

kλ

2R

dθ

R2

=kλ

2dθ

R

12%3#4./*#&0#(%()%%(

x$%.*/&0#

dE

2'%+/

dE

x2

=dE

2co

s(θ)

=kλ

2co

s(θ)

dθ

R

5()%3+/($#$%'$%

θ=

06/')/

θ=

π 27

Ex2

=kλ

2

R

∫π 2

0co

s(θ)

dθ

=kλ

2

R

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

~ Ex2

=kλ

2

Ri

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

x-!("')0$0%$-1"&-$0$%!,#!'%"$,!%2&31'(

'('*#!%&$%4-!!,#')'*!%'5

~ Ex3

=kλ

2

Ri

6,&,!#!*-%'71!

~ E2+

~ E3

=2kλ2

Ri

'!#(#%)&

8!'

q′('"'%3'71!#&!*!!,#!#%$9$-!'('):%!#'(71!'*1('!("')0$!(!"#%&"$0-%$-1"&-$

0$%#$-$!('('):%!!*

O;<$)$#&!*!('%3$

R4('-!*,&-'--!"'%3',!%2

λ4

=q′ R

=1!3$40%$"!-&!*-$-!&31'(>$%)'71!0'%'!(#%$9$

AB/!)$,71!

~ E4

=(k

λ4

∫

2R

R

1 x2dx)i

=(−

kλ

4

2R)i

81)'*-$#$-$,($,"')0$,"'("1('-$,!&31'('*-$'"!%$5

~ E1+

~ E2+

~ E3+

~ E4

=2kλ

1

3R

i+

2kλ

2

Ri+

(−kλ

4

2R

)i=

0

?@

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

A!,0!B'*-$

λ45

λ4

=2 3(2

λ1+

6λ

2)

C!!)0('9'*-$($,/'($%!,-!

λ14λ

2D

λ4-!.*&-$,'*#!%&$%)!*#!4!*"$*#%')$,!(/'($%-!('

"'%3':1,"'-'5

q′=

2 3q(

1+

12 π)

8&"$*,&-!%')$,

π≈

35

q′=

10 3q

!

!"#$%&'((

"#$%&$''$%#()*'$+%,-.%*)-*/&%./*)*-&0)$#0*.010.,*/'/'02./#//'%#-3#&0.4*-('

σ5

)"$*+,-./

"*02./'-'(*2'$)$#0*'0%&'2,.60'$)$#0*78/'%**-/'0#/#2%#-,'2.#0#259#:'(*2;&''$

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

~ E(~r

)=

∫∫

Ω

dq(

~r−

~r 1)

4πǫ 0|~r−

~r 1|3

%*0

~r 1'$<'%,*-)*2.%.60/'$#/.2,-.:&%.60/'%#-3#

Ω;&'3'0'-#'$%#()*'$+%,-.%*'0

~r5

='0'(*2;&'

~r=

xx

+yy

+zz>

~r 1=

x1x

+y 1

y8|~r−

~r 1|2

=(x−

x1)2

+(y−

y 1)2

+z25

=⇒

~ E(~r

)=

∫

+∞

−∞

∫

+∞

−∞

σdx

1dy 1

((x−

x1)x

+(y−

y 1)y

+zz)

4πǫ 0

((x−

x1)2

+(y−

y 1)2

+z2)3 2

?#3#(*2%#(:.*/'<#-.#:$'25

w=

x−

x1>

v=

y−

y 1=⇒

x1

=x−

w,

y 1=

y−

v5@$

A#%*:.#0*/'$#,-#024*-(#%.60'2

J=

∣ ∣ ∣ ∣

(x1) w

(x1) v

(y1) w

(y1) v

∣ ∣ ∣ ∣

=

∣ ∣ ∣ ∣

−1

00

−1

∣ ∣ ∣ ∣

=1

=⇒

~ E(~r

)=

∫

+∞

−∞

∫

+∞

−∞

σdw

dv(w

x+

vy

+zz)

4πǫ 0

(w2+

v2+

z2)3 2

=σ

4πǫ 0

∫

+∞

−∞

∫

+∞

−∞

dw

dv·(

zz)

(w2+

v2+

z2)3 2

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

w=

rcos

(θ)

v=

rsen

(θ)

J=

∣ ∣ ∣ ∣

wr

wθ

v rv θ

∣ ∣ ∣ ∣

=

∣ ∣ ∣ ∣

cos(

θ)−

rsen

(θ)

sen(θ

)rc

os(θ

)

∣ ∣ ∣ ∣

=r

=⇒

~ E(~r

)=

σ

4πǫ 0

∫

2π

0

∫

+∞

0

rdrd

θ·(

zz)

(r2+

z2)3 2

=σ

4πǫ 0·2

πzz

∫

+∞

0

rdr

(r2+

z2)3 2

per

ou

=r2

+z2→

du

=2r

dr

=σ 4ǫ 0·z

z

∫

+∞

z2

du

(u)3 2

=σ 4ǫ 0·z

z(

2u−

1 2

)∣ ∣ ∣

z2

+∞

=σ 2ǫ 0·

z |z|z

=σ 2ǫ 0·s

ign(z

)z

D

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

=⇒

~ E(~r

)=

σ 2ǫ0·s

ign(z

)z

B*,';&''$2#$,*/'/.2%*0,.0&./#//'$(6/&$*/'$%#()*'$+%,-.%*E;&''20*-(#$#$)$#0*F

'2

+σ ǫ 05

!

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

σ40#$*$#)&.0&#

0&)0*,-)'()-'&#56

-7"-,0*,((,0-3+#(,80/)&0#($0*-,9*&()+*$/#'(,(:(9*(+-%-+#)(,0($/)#'(,#)&.0&#4

+()+($'&0*,-)3($/(-,+,-$#6

;7<

,#,-)1#'(,(:(%(0#,#0-*$

-,-3;)(=$#0#$'*0/#)7'(,-)1#-40#$

'($%&'-'

'(

0-)1-,&$(-,*$&2#)3(

λ>0*>-'&%/-$0&-3?%+)@A&3--,+,-$#(%'6B$0#$/)-),-2*()C-

(,80/)&0-9*((A+()&3($/-(,-,-3;)(6

*"$+,-./0

-7D%-$'#(,+)&$0&+&#'(%*+()+#%&0&@$4+#'(3#%($0#$/)-),#'(%(-'#0#$%&'()-$'#,-

%*3-E(0/#)&-,'(,0-3+#(,80/)&0#9*(1($()-*$+,-$#&$.$&/#'('($%&'-''(0-)1-

σ%#;)((,(:(>*$'&%0#'()-'&#5

>'($%&'-''(0-)1-−

σ6F(-(,+,-$#>C0##)'($-'#

(,+,-$#($0*(%/&@$>(,(:(A(,(:(($0*(%/&@$6

G(,#%&/(3%-$/()&#)(%4/($(3#%9*((,0-3+#(,80/)&0#1($()-'#+#)(,+,-$#(%

~ E1(~r

)=

σ 2ǫ 0·s

ign(x

)x

>(,0-3+#(,80/)&0#1($()-'#+#)(,'&%0#(%

~ E2(~r

)=−

σ

2ǫ 0·x(

sign(x

)−

x√

R2+

x2

)

6

=⇒

~ E=

~ E1+

~ E2

=σ 2ǫ 0·(

x√

R2+

x2

)

x

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

dx'(,-,-3;)('(;&'#-,

I

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

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

d~ F

=dq~ E6

=⇒

~ F=

∫

d+

a

ddq~ E

con

dq

=λdx

=

∫

d+

a

dλdx

σ 2ǫ 0·(

x√

R2+

x2

)

x

=λσ

2ǫ0·∫

d+

a

d

xdx

√R

2+

x2·x

=λσ

2ǫ0·(√

R2+

x2)∣ ∣ ∣

d+

a

d·x

=λσ

2ǫ0·(√

R2+

(d+

a)2−√

R2+

d2)

·x

!

!"#$%&'()

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

a)'%#%0#1&1$21).)13Q

0#'-.$+%+%#)%

/'*)$'40'/1%#*0*0(%$5&'%'#)%$'.$67#&0%#)$%%3&1+(.%38&)$'&.%#%3(0#).

O6

*"$+,-./0

7*)%($.43%+13./%4%+.*$%*.39%$0)'3':1#/.3131;/%&.03.+4(1$1/'*)$'40&'.#%**0(%$5<

&'13%*/%&1$2167*)13%;#.*/'&%(1$1%*)%&1*.=0%

~ E(~ 0

)=

∫

kdq(

~r−

~ r′)

‖~r−

~ r′‖3

>.+.;1,%+.*9'*).?%3%3%+%#)./%&1$213.(./%+.*%*&$'4'$&.+.

dq

=σdA?/.#/%

σ%*

31/%#*'/1//%&1$21*0(%$5&'13%#%3'#)%$'.$/%3$%&'('%#)%@&.#*)1#)%A6B%4%+.*%*&$'4'$%3

%3%+%#)./%C$%1

dA

/%)13+1#%$1=0%(./1+.*$%&.$$%$31*0(%$5&'%;(1$1%*).,1$%+.*3.

*'20'%#)%D>.#*'/%$1$%0#1&1*&1$1%*-8$'&1&.#31*'20'%#)%/'*(.*'&'E#(1$131*91$'143%*

θ;

γ F3,1&%$0#1(%=0%G191$'1&'E#/%3C#203.

θ?9%+.*=0%*%$%&.$$%0#(%=0%G.1$&./%3.#2')0/

ds 1

=R

sin(γ

)θ6H.$.)$.31/.13,1&%$0#1(%=0%G191$'1&'E#/%3C#203.

γ*%$%&.$$%0#1$&.

/%3.#2')0/

ds 2

=R

dγ6B%%*)1+1#%$1%3%3%+%#)./%C$%1/%31&'$&0#-%$%#&'13.(./%+.*

%*&$'4'$&.+.

dA

=ds 1·d

s 2=

R2si

n(γ

)dγdθ

B%%*)1+1#%$1(./%+.*%*&$'4'$%3%3%+%#)./%&1$21&.+.

dq

=σR

2si

n(γ

)dγdθ

IJ

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

H.$.)$.31/.?/%4%+.*/%)%$+'#1$3.*9%&).$%*

~ r′;

~r6>31$1+%#)%

~ r′=

~ 0;

(.$.)$.31<

/.@+'$1#/.31520$1A9%$%+.*=0%

~r=

Rsi

n(γ

)co

s(θ)

i+

Rsi

n(γ

)si

n(θ

)j+

Rco

s(γ)k

K.%*/'-L&'39%$=0%‖~r−

~ r′‖

=R

B%%*)1+1#%$1%*(%&'5&1#/.3.*$%&.$$'/.*/%31*91$'143%*(1$1=0%$%&.$$1#%3$%&'('%#)%*%

)'%#%=0%

~ E(~ 0

)=

∫

2π

0

∫3π 2

π 2

kσ(−

Rsi

n(γ

)co

s(θ)

i−

Rsi

n(γ

)si

n(θ

)j−

Rco

s(γ)k

)

R3

R2si

n(γ

)dγdθ

H.$*'+%)$L1*14%+.*=0%%3&1+(.%38&)$'&.$%*03)1#)%%*)1$C%#/'$%&&'E#

k;(.$3.)1#).?

/%31'#)%2$131#)%$'.$?#.&.#*'/%$1+.*31*&.+(.#%#)%*$%*)1#)%*D

~ E(~ 0

)=

∫

2π

0

∫

π

π 2

kσ(−

cos(

γ)k

)si

n(γ

)dγdθ

=kσ

∫

2π

0dθ

∫

π

π 2

(−co

s(γ)k

)si

n(γ

)dγ

=πkσ

∫

π

π 2

(−si

n(2

θ)k))

dγ

B%%*)1+1#%$1 ~ E(~ 0

)=

πkσ(1 2

cos(

2θ)

∣ ∣ ∣ ∣

π π 2

)k=

πkσ

1 2(c

os(

2π)−

cos(

π))

k=

(πkσ)k

M'#13+%#)%?(1$1%*&$'4'$%*)%$%*03)1/.%#-0#&'E#/%

Q?/%4%+.*/%)%$+'#1$%3913.$/%

σ6

7*).%*-C&'3*14'%#/.=0%31&1$21%*)10#'-.$+%+%#)%/'*)$'40'/1%#31*0(%$5&'%'#)%$'.$?

)%#/$%+.*=0%

σ=

Q

2πR

2

B%%*)1+1#%$1

~ E(~ 0

)=

kQ

2R

2k

!

!"#$%&'()

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

AB

./+#+$0#'/.(1

2L

2+34+)4+#1/5($%)%$%$%,-)+

CD*1+$0#'/.(1

L67%1%(#01++$$03./+#+$%,/3,%5%)'%

Q*1/3.)/-(/1%(#/80),+,+#.+67%$5($+$%,%'#/.(1

21/)+55/9#1+$%8(+):%;(+(#01++$$03+<+)5+30-)++$0.)06

*"$+,-./0

"3.+4)0-$+,%3+)+3(+$=+1+/'(%$80),%;(++$4)0-$+,%%#.+)/0)6>04)/,+)0;(+?%)+,03

+35%$5($%)+$5%,40+$@5.)/50;(+4)01(5++$%$%,-)+=+)./5%$30-)+(#4(#.01+3(+<+1+

3/,+.)A%670,0$%5%)'%

Q+3.%(#/80),+,+#.+1/3.)/-(/1%*3+.+#1)B;(+30-)++$+<++$5%,40

+$@5.)/50+3.%)B+#1/)+55/9#

j

"$5%,40+$@5.)/504)01(5/1040)(#%5%)'%

dq3+)B

dE

=kdq

r26C0#1+

r+3$%1/3.%#5/%1+31+

$%5%)'%

dq(-/5%1%+#

xi?%3.%+$4(#.01+%4$/5%5/9#

djD

r=√

x2+

d26E$/'(%$50,0$0

?+,03?+5?0%#.+3*$%5%)'%

dq$%401+,03+35)/-/)50,0

dq

=λ

1dx10#1+

λ1+3$%1+#3/1%1

$/#+%$1+5%)'%2$%1+&#/,0350,0

λ1

=Q 2L6E3/*+$5%,40;(+1%+F4)+3%1050,0

dE

=kλ

1dx

x2+

d2

G%1/</,03;(++$5%,40+3.%)B+#1/)+55/9#

j2%;(+$%350,40#+#.+3+#

x3+%#($%)%#6>%3

50,40#+#.+3+#

y$%3401+,03+35)/-/)50,0

dE

y=

dE

cos(

θ)6"38%5/$=+);(+

cos(

θ)=

d√

(x2+

d2)

H

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

C++3.%,%#+)%

dE

y=

kλ

1ddx

(x2+

d2)3 2

I#.+')%#101+31+

x=−

L?%3.%

x=

LD

Ey

=kλ

1d

∫

L −L

1

(x2+

d2)3 2

dx

=kλ

1d(

x

(d2√

x2+

d2

∣ ∣ ∣ ∣

L −L

)=

2kλ

1L

d√

L2+

d2

C++3.%,%#+)%$%8(+):%;(++<+)5++$%$%,-)+?0)/:0#.%$30-)+(#%4+;(+J%5%)'%

dq1+$

%$%,-)+=+)./5%$;(++3.%%(#%%$.()%

y3+)B

dF

=dqE

=dq

2kλ

1L

d√

L2+

y2

"3.+%$%,-)+./+#+1/3./#.%1+#3/1%11+5%)'%2%;(+3/-/+#./+#+$%,/3,%5%)'%.0.%$%$,%K

5+#%1%*./+#+$%,/.%11+$%)'0;(++$%$%,-)+?0)/:0#.%$6C++3.%,%#+)%1+&#/,03

λ2

=Q L

E3/*dq

=λ

2dy2$%8(+):%;(+1%+F4)+3%1%50,0

dF

=λ

2dy

2kλ

1L

y√

L2+

y2

I#.+')%#101+31+

y=

L?%3.%

y=

2L

F=

2kλ

1λ

2L

∫

L −L

1

y√

L2+

y2dy

L+./+#+;(+

∫

1

y√

L2+

y2dy

=1 L

ln(√

L2+

y2−

L

y)

M0)$0.%#.0

F=

2kλ

1λ

2L

(1 L

ln(√

L2+

y2−

L

y)∣ ∣ ∣ ∣ ∣

2L

L

)

C+3%))0$$%#10

F=

2kλ

1λ

2ln

(

√5−

1

2(√

2−

1))

N++,4$%:%#10$03=%$0)+31+

λ12

λ2+#8(#5/9#1+

QD

F=

kQ

2

L2

ln(

√5−

1

2(√

2−

1))

!"#$%&'()

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

b64&-+/4"44&$"(3"%-/'5(2&

σ6%-"#"27(&/3%"#2&-1&#"(35$5-4&-+/4"44&$"(3"%-/'5(2&

λ8%&+15&-&#2/+258#"-5*%&#".%/-$."9"%-"4/+1"-$/"

d:

*"$+,-./0

!525#"'%&()"4&!5%#527$%28#&#" ("#&64&;&<1=-9&#&3/(&25+#"'5(2"2>++/28#&4&

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51(5+57(&&#51(5@:A5-3"25+&#"#"27(&&-&#&,&B96+&"&#8#"-545-4&&+1>-"275+57,&15+

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H&"-

~r 2=

~ 09

~r 1=

xx

+yy9$5-−∞≤

x≤

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b+

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+57(&1"#&#&2&-15&+

~dF

21

=dq 2

∫∫

Ω

dq 1·−

~r 14πǫ 0|~r 1|3

,co

ndq 1

=σdxdy

=−

dq 2

∫

+∞

−∞

∫

b+d

d

σdxdy·x

x

4πǫ 0

(x2+

y2)3 2

−dq 2

∫

+∞

−∞

∫

b+d

d

σdxdy·y

y

4πǫ 0

(x2+

y2)3 2

=−

dq 2

σy

4πǫ 0

∫

+∞

−∞

∫

b+d

d

dxdy·y

(x2+

y2)3 2

=dq 2

σy

4πǫ 0

∫

+∞

−∞

(

1√

x2+

y2

)∣ ∣ ∣ ∣ ∣

b+d

d

·dx

=dq 2

σy

4πǫ 0

∫

+∞

−∞

(

1√

x2+

(b+

d)2−

1√

x2+

d2

)

·dx,

per

odq 2

=λdx

2

=⇒

~dF

21

dx

2=

λσy

4πǫ 0

∫

+∞

−∞

(

1√

x2+

(b+

d)2−

1√

x2+

d2

)

·dx

=λσy

2πǫ 0·l

n

(

x+√

x2+

(b+

d)2

x+√

x2+

d2

)∣ ∣ ∣ ∣ ∣

+∞

0

=λσy

2πǫ 0·

lım

x→

+∞

ln

1+

√

1+

(b+d

x)2

1+

√

1+

(d x)2

−ln

(

b+

d

d

)

=λσy

2πǫ 0·l

n

(

d

b+

d

)

=⇒

~dF

21

dx

2=

λσy

2πǫ 0·l

n

(

d

b+

d

)

J

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dq%7/$"4"&-&#8%-15

~ r′=

xi4&#"#"27(&+57(&&#8%-154&

~r=

dj+&(>N

d~ E

=kdq(

~r−

~ r′)

|~r−

~ r′|3

=kdq(−

xi+

dj)

(x2+

d2)3 2

I.5("9+&1/&-&*%&

dq

=λdx685(#51"-15N

d~ E

=kλdx(−

xi+

dj)

(x2+

d2)3 2

P-1&3("-454&+4&−∞

"+∞

N

~ E=

∫

+∞

−∞

kλdx(−

xi+

dj)

(x2+

d2)3 2

=kλ((−∫

+∞

−∞

x

(x2+

d2)3 2

)i+

(∫

+∞

−∞

d

(x2+

d2)3 2

)j)

!

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f(x

)=

x

(x2+

d2)3 2

*+(,-#./-0.102#&201#(&2*3.#14**+2#$%&'()&*&%&(&2*.5#10

-#.20,#5#10.6780.02.01#4091#$%&'()&

g(x

)=

d

(x2+

d2)3 2

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4*+4*−∞

#+∞

+*.:*140;1*4*1#(&2*3.#14*3<=>4*+4*6#

+∞

?

∫

+∞

−∞

g(x

)dx

=2

∫

+∞

0g(x

)dx

80.102#&2091#*=-.*+()&-#.#*1'#,-0+*.*4%'*#

~ E=

2kdλ

∫

+∞

0

1

(x2+

d2)3 2

j

@*2(*&*A%*

∫

1

(x2+

d2)3 2

=x

d2(x

2+

d2)1 2

80.102#&20

~ E=

2kdλ(

x

d2(x

2+

d2)3 2

∣ ∣ ∣ ∣ ∣

∞ 0

)j=

2kλ d

j

BC0.#A%*+#;*,0+'0,0*+*1'#,-0*1D'2.('03*&*.#40-0.%&#1#,;.*(&E&(20-04*,0+'#1F

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2.0G0+4*'#;1*'0&'#.3#

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=dqE

9

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dq7

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dq*&$%&'()&4*1:&3%10

θA%*4#

*=-.*+#4#'0,0

d(θ

)=

R+

Rsi

n(θ

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dq*+-0.4*E&('()&(3%#1

#λds9

40&4*

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4*1:&3%10

θ'0,0

ds

=R

dθ7

L**+2#,#&*.#1#$%*.G#+0;.*%&#'#.3#

dq+*.:?

dF

=dqE

=λR

dθ·

2kλ

R+

Rsi

n(θ

)=

2kλ

2dθ

1+

sin(θ

)

M&2*3.#&404*+4*

θ=

0#

θ=

π0;2*&*,0+1#$%*.G#202#1+0;.**1#1#,;.*+*,('(.'%1#.78#.#

*+20%2(1(G#,0+*1C*'C04*A%*?

N

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

∫

1

1+

sin(θ

)=

sin(θ

)−

1

cos(

θ)

J0&*+202*&*,0+A%*1#$%*.G#.*+%12#&2**+

F=

4kλ

2

!"#$%&'(

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+'&*$.#/**+@

φ=

∫

S

~ E·d

~ S=

q ǫ 0(1

)

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#2!/#,C>'*))"1"1,+

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e+

φc

=⇒

φc=

φ+

φe

(2)

G**+("1"!*$"4*!#,!($"!-,*)8"),$-*

φe$*+,)8*1,+*)&$,7)*1"/!1*-/"("1*!(*9

φe+*

#")#')"#,1,@

φe

=

∫

Se

~ E·d

~ S

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q*+

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q R2r9G,!-*

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d~ S

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n*+

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r#,1,1'*+($")".%'$"@

HI

HJ

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~ E·d

~ S=

1

4πǫ 0

q R2r·n

dS

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4πǫ 0

q R2dS

K!(*%$"!-,+*,7(/*!**);'<,&"$")"+*1/*+0*$"@

φe

=

∫

Se

~ E·d

~ S=

∫

Se

1

4πǫ 0

q R2dS

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4πǫ 0

q R2

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Se

dS

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29G**+("

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φe

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4πǫ 0

q R2·2

πR

2=

q 2ǫ0

(3)

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φc=

q 2ǫ 0

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!

∮

Ω

~ E·n

dS

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int

ǫ 0

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Qin

t%&'()(*+(%$)%**("(,#*'(&-,%*.)/%

Ω01&2(3#*4(/$2%+*('"%'(5%6"%

7(-&&&%,-%"%%8,*%&(*%$3#*4("/3%*%$)/('-&($"#%'2%#*%4("%'("/9%*+%$)/(:'#)-('&%

"%;((''%)2#*<=#>2%$/%$"#'(%8,*%&/?$%@-/9('%$2%

~ ∇·~ E

=ρ ǫ 0

1&,#&/>'%-&(*'(5%6"%7(-&&,(*()(')-'(*%')(4,#%'A)2*/)#>(;#('2(&)#$"/)/#$%&"%

&/4%2*B(='#)-('&%)-4,'%%$%&2%)(&#0C#$+(4#&)#4#%;%D%'%;%"%&/4%2*B("%')/'/$"*#0

C#*'(&/4%2*B()/'/$"*/)(=%')(4,#%'A)2*/)#$#"%,%$"%"%

θ6,#*&%*/$.$/2#$#"%,%$"%"%

z0E&B(,*/#*/=&(>%4#&@-%

~ E(~r

)=

E(r

)r="#$"%

r&%4/"%"%&"%%'%;%(',-$2#%$)-%&2/?$

6r%&%'9%)2#*-$/2(*/#@-%9("%&"%%'%;%(',-$2#0

F%>%4#&%'%+/*-$(&-,%*.)/%"%/$2%+*()/?$"#$"%|~ E|$

#"%,%$"("%'(&9(*/(>'%&@-%"%G

&)*/>%$'(&-,%*.)/%:%&"%)/*=|~ E|"

%>%&%*)#$&2($2%&#>*%'(&-,%*.)/%<=,(*(,#"%*&()(*

|~ E|3-%*("%'(/$2%+*('0C#*%''#=)#$&/"%*%4#&-$(&-,%*.)/%)/'B$"*/)("%*("/#*)#$%;%"%

&/4%2*B(%'%;%D:6%'%;%"%')/'/$"*#)(*+("#<0C#"%4#&"/&2/$+-/*"#&)(&#&H

(<I/r

<R0

1$%&2%)(&#2%$%4#&@-%$#J(6)(*+(%$)%**("(,#*%')/'/$"*#=,#*'#)-('

∮

Ω

~ E·n

dS

=Q

int

ǫ 0=

0

E"%4K&=

∮

Ω

~ E·n

dS

=

∫

manto

E(r

)r·n

dS

+

∫

tapasE

(r)r·n

dS

=E

(r)

∫

manto

dS

=E

(r)(

2πr)

h

=⇒

E(r

)(2π

r)h

=0

=⇒

E(r

)=

0=⇒

~ E=

~ 0

1&/$2%*%&($2%%'J%)J#@-%%')(4,#%'A)2*/)#&%($-'#"%$2*#"%')/'/$"*#0

><I/R≤

r01$%&2%)(&#2%$%4#&@-%

∮

Ω

~ E·n

dS

=E

(r)(

2πr)

h=

(2πR

)hσ

ǫ 0=⇒

E(r

)=

σ ǫ 0

(

R r

)

C#*2($2#=

~ E(r

)=

~ 0r

<R

σ ǫ 0

(

R r

)

·ρR≤

r

L#2%@-%%'&('2#"%"/&)#$2/$-/"(""%'4?"-'#"%')(4,#%'A)2*/)#:@-%%&$#*4('('(&-,%*G

.)/%<%&

+σ ǫ 00

MN

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

!"#$%&'()

I%2/%$%-$("/&2*/>-)/?$%&3A*/)("%)(*+(

Θ"%*("/#O

6"%$&/"("9(*/(>'%

ρ=

αr,(*(

r<

R0P(')-'%'(%$%*+B(,#2%$)/('()-4-'("(,#*'("/&2*/>-)/?$0

*"$+,-./0

I(>%4#&@-%'(%$%*+B(,#2%$)/('(&#)/("((-$("/&2*/>-)/?$"%)(*+(:#(')(4,#%'A)2*/)#

@-%A&2(,*#"-)%<%&2K"("(,#*

U=

ǫ 0 2

∫

R3

~ E2d

3x

"#$"%'(/$2%+*('%&&#>*%2#"#%'%&,()/#0C#*2($2#=)#$#)/%$"#%')(4,#%$2#"#%'%&,()/#=

,#"%4#&)(')-'(*'(%$%*+B(,#2%$)/('@-%@-%*%4#&0

F("#@-%'("/&2*/>-)/?$"%)(*+(%&&/4A2*/)(%&3A*/)(4%$2%62(4>/A$'("%$&/"(""%)(*+(

2/%$%&/4%2*B(%&3A*/)(=(,*/#*/&(>%4#&@-%%')(4,#%'A)2*/)#2/%$%&/4%2*B(%&3A*/)(=%&"%)/*=

~ E(~r

)=

E(r

)r="#$"%

r%&'("/&2($)/("%')%$2*#"%'(%&3%*((',-$2#%$)-%&2/?$6

r%&%'

9%)2#*-$/2(*/#"/*/+/"#"%')%$2*#"%'(%&3%*((',-$2#0

C#"%4#&"/9/"/*%',*#>'%4(%$"#&*%+/#$%&

(<

r<

RP#$&/"%*%4#&)#4#&-,%*.)/%"%/$2%+*()/?$-$()K&)(*(%&3A*/)("%*("/#*)#$

4/&4#)%$2*#@-%'("/&2*/>-)/?$"%)(*+(

Θ0C#*'('%6"%7(-&&2%$%4#&

∮

Ω

~ E·n

dS

=Q

int

ǫ 0∮

ΩE

(r)r·n

dS

=1 ǫ 0

∫

Θρd

3x

E(r

)4πr2

=1 ǫ 0

∫

2π

0

∫

π

0

∫

r

0αr3

sen(ϕ

)drd

ϕdθ

E(r

)4πr2

=4π ǫ 0

∫

r

0αr3

dr

E(r

)4πr2

=παr4

ǫ 0

=⇒

E(r

)=

α 4ǫ 0

r2

!

R≤

r"#$%&'()(*#%$+(,-*($.(/#*#%+0()1/&('(&$.(2)-/&3$+$-/4%/-)-(%56)&/-'(

)-'&#)/#$*&%*#/($.)#7+(8-'&%.)& +/&3$'(/-)2-

Θ9:#)8-8(;'(<-+%%.($(*#%

∮

Ω

~ E·n

dS

=Q

int

ǫ 0

E(r

)4πr2

=4π ǫ 0

∫

R

0αr3

dr

E(r

)4πr2

=παR

4

ǫ 0

=⇒

E(r

)=

α 4ǫ0

(

R r

)

2

R2

!

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

~ E(r

)=

α 4ǫ 0

r2r

r<

Rα 4ǫ 0

(

R r

)

2R

2r

R≤

r

./#$&,-)0#'#0)*#+)10&*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=

ǫ 0 2

∫

R3

~ E2d

3x

=ǫ 0 2

∫

2π

0

∫

π

0

∫

+∞

0E

(r)2

r2se

n(ϕ

)drd

ϕdθ

=4πǫ 0 2

∫

+∞

0E

(r)2

r2dr

=4πǫ 0 2

(∫

R

0

α2

16ǫ2 0

r6dr

+

∫

+∞

R

α2

16ǫ2 0

R8

r2dr)

=πα

2

8ǫ 0·(

r7 7

∣ ∣ ∣ ∣

R 0

−R

81 r

∣ ∣ ∣ ∣

+∞

R

)

=πα

2

8ǫ 0·(

R7 7

+R

7

)

=πα

2

7ǫ 0

R7

;

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

!"#$%&'()

<#'+45)$)*#+5#+)+=)$&+'#0#'03'%$40&+5)$&54#

R(0#'5)'+45&5)+5)0&$6&>#1-*3%$4?

0&+

ρ@−

ρ-'4=#$*)+:A#+0)'%$#+5)&*8&+)+=)$&+)+%B'&-'&54+%&'04&*)'#$,-)

2R:C)&

~ d)1>)0%#$,-)>&5)10)'%$#5)1&)+=)$&2#+4%4>&&10)'%$#5)1&)+=)$&')6&%4>&:

"$-)8),-))10&*2#)130%$40#)'1&4'%)$+)0049'5)1&+)+=)$&+)+0#'+%&'%)@)'0-)'%$)

+->&1#$:

*"$+,-./0

D)12$#81)*&;E(+&8)*#+,-))10&*2#)130%$40#)')14'%)$4#$5)-'&)+=)$&*&04F&5)

5)'+45&55)0&$6&-'4=#$*)

ρ)+

~ E(r

)=

ρr

3ǫ 0·r:

D)1&+4*)%$7&5)1&54+%$48-049'(&2$4#$4+&8)*#+,-)&5)*B+2&$&

R≤

r)10&*2#

0-*21)

~ E(~r

)=

E(r

)r:C484)''#)+')0)+&$4#2&$&1&$)+#1-049'5)12$#81)*&0#'#0)$)1

0&*2#)130%$40#=-)$&5)1&)+=)$&()'0#'%$3*#+1#-+&'5#1&1)@5)G&-++:

<#'+45)$)*#+0#*#+-2)$H04)5)4'%)6$&049'-'&0B+0&$&)+=3$40&5)$&54#

r>

R:I)'?

)*#+,-)

∮

Ω

~ E·n

dS

=Q

int

ǫ 0

E(r

)4πr2

=4πR

3ρ

3ǫ 0

=⇒

~ E(~r

)=

R3ρ

3ǫ 0

r2r

"#$%&'%#(%)')*#+,-))10&*2#)130%$40#6)')$&5#2#$1&)+=)$&*&04F&)+

~ E(r

)=

ρr

3ǫ 0·r

r<

RR

3ρ

3ǫ 0

r2r

R≤

r

"#'6&*#+)1#$46)'5)0##$5)'&5&+)')10)'%$#5)1&)+=)$&5)5)'+45&55)0&$6&

ρ:.+7(

%)')*#+,-)1#+0&*2#+)130%$40#+)'1#+4'%)$4#$)+5)0&5&)+=)$&)+%B'5&5#+2#$

~ E1(r

)=

ρ 3ǫ 0·~r

para

|~r|<

R

~ E2(r

)=

−ρ 3ǫ 0·(

~r−

~ d)

para

|~r−

~ d|<

R

A#+>)0%#$)+)'1&4'%)$+)0049'5)&*8&+)+=)$&++&%4+=&0)'+4*-1%B')&*)'%)

|~r−

~ d|<

R@|~r|<

R:.+7(2#$)12$4'0424#5)+-2)$2#+4049'()10&*2#)130%$40#)'1&

4'%)$+)0049'5)&*8&+)+=)$&+)+

~ E(~r

)=

~ E1(~r

)+

~ E2(~r

)=

ρ 3ǫ 0·~r−

ρ 3ǫ 0·(

~r−

~ d)

=ρ 3ǫ 0·~ d

)10-&1)+0#'+%&'%)(%&10#*#+),-)$7&2$#8&$:

!

!"#$%&'()

"#$%&'()&*+,&-#%.,$)/$012+34()&,$56$%.%.#'&%$%+#&71)3.

ρ>

01,+8$+#012+3.#

.'74)&,1%.)$%&19:

$;<$2,+2..2,$381.24,()&,1.#(1%1.2.'8$,&1=+'$#%12.>%.?$+'';:

*;@.,121,$+#$,$)/$8+#(+$2−

Q<

0.#.2,.#()1%.2$.'7.)$A>'.2$%.6$2&*).:BC+.%$

1#1.#

.D+&2&*)&12$,$)/$.#

.2,.#()1E@&.'$5)3$(&0$2$).'8+.'($AB.'.'($*2.1

&#.'($*2.E

@&2$,$)/$(&.#.+#$3$'$

m>'.%.'82$F$%.2,.#()1%.2$.'7.)$A8.)1'&.38).D+.%$#%1

%.#()1%..22$A,$2,+2.2$7+.)F$D+.2$,$)/$.G8.)&3.#($>8)+.*.D+.'+310&3&.#(1

.'+#310&3&.#(1$)3-#&,1'&382.:H#,+.#().'+8.)I1%1%.1',&2$,&-#:

,;@+81#/$31'D+.2$,$)/$'-218+.%.310.)'..#.2.6.G:@&$%.3J''.$82&,$+#,$381

.24,()&,1.G(.)#1

~ E0

=E

0xA

E0

>0A

,$2,+2.2$'#+.0$'81'&,&1#.'%..D+&2&*)&1%.2$

,$)/$>%.(.)3&#.,+J2.'%..22$''1#.'($*2.'1&#.'($*2.':K&',+($D+.,1#%&,&1#.'%.*.

,+382&)

E08$)$2$.G&'(.#,&$%.($2.'81'&,&1#.'%..D+&2&*)&1>'+#L3.)1:

*"$+,-./0

$;M1#/$31'.21)&/.#,11)%.#$%1.#.2,.#()1%.2$.'7.)$:K.2&(43

$#(.)&1)A$2+'$)2$2.>

%.?$+''A'$*.31'D+..2,$381.24,()&,1/.#.)$%181)2$.'7.)$.'

~ E(r

)=

ρr

3ǫ 0·r

r<

RR

3ρ

3ǫ 0

r2r

R≤

r

*;N.#.31'D+..2,$381.#.2,.#()1%.2$.'7.)$.'

~ E(~ 0

)=

~ 0A81)21,+$2.2,.#()1.'

+#8+#(1%..D+&2&*)&1:@&310.31'+#81D+&(12$,$)/$A.'($.G8.)&3.#($+#$7+.)F$

~ F=−

Qρr

3ǫ 0·rA2$,+$2$8+#($O$,&$.2,.#()1%.2$.'7.)$='&#&381)($).#D+4%&).,,&-#

'.O$>$310&%12$,$)/$;A81)21,+$2.'+#8+#(1%..D+&2&*)&1.'($*2.:

@&310.31'2$,$)/$D+.%$#%1'&.38).%.#()1%.2$.'7.)$A(.#.31'D+.'+.,+$,&-#%.

310&3&.#(1.'

m~a

=~ F

=−

Qρr

3ǫ 0·r

~a+

Qρr

3m

ǫ 0·r

=~ 0

=⇒

r+

Qρ

3m

ǫ 0·r

=0

r+

w2·r

=0

M1)($#(1A.2310&3&.#(1%.2$,$)/$.'+#310&3&.#(1$)3-#&,1'&382.>'+8.)I1%1%.

1',&2$,&-#.'

T=

2π w

=2π

√

3m

ǫ 0Q

ρ

,;M1).28)&#,&8&1%.'+8.)81'&,&-#A.2,$381.24,()&,1).'+2($#(..'

~ E(r

)=

ρr

3ǫ 0·r

+E

0x

r<

RR

3ρ

3ǫ 0

r2r

+E

0x

R≤

r

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

9..',)&*&.#%1.2,$381.24,()&,1$212$)/1%.2.6.GA).'+2($

~ E(r

)=

(ρx

3ǫ 0

+E

0)·x

−R

<x

<R

(R

3ρ

3ǫ 0

x2

+E

0)·x

R≤

x

(E0−

R3ρ

3ǫ 0

x2)·x

x≤−

R

H#,1#().31'2$'#+.0$'81'&,&1#.'%..D+&2&*)&1:

&;−

R<

x<

RC+.).31'D+. ρx

3ǫ 0

+E

0=

0=⇒

x1

=−

3ǫ 0

E0

ρ

M$)$D+.

x1.G&'(.A'.%.*.,+382&)D+.

x1

=−

3ǫ0E

0

ρ>−

R=⇒

E0

<R

ρ

3ǫ0

H'($,1#%&,&-#%.*.,+382&)

E08$)$D+.O$>$+#8+#(1%..D+&2&*)&1%.#()1%.2$

.'7.)$:

@$*.31'D+.'&.22$82$,&$#1%.2$.#.)/I$81(.#,&$2.24,()&,$∇

2U

.#

.28+#(1

%..D+&2&*)&1.'3$>1)D+.,.)1A.#(1#,.'($28+#(1.'.'($*2.P'&.'3.#1)D+.

,.)1A.#(1#,.'($28+#(1.'&#.'($*2.A>D+.

U=−

qVA%1#%.Q

.'2$%&7.).#,&$%.

81(.#,&$2.24,()&,1:R'IA(.#.31'D+.

V(x

)=

−∫

~ E(x

)·x

dx

+c

=−

∫(

ρx

3ǫ 0

+E

0

)

dx

+c

=−

E0x−

ρx

2

6ǫ 0

+c

U(x

)=

−Q

V(x

)

=Q

E0x

+Q

ρx

2

6ǫ0−

Qc

=⇒∇

2U

=d

2U

dx

2=

Qρ

3ǫ 0

>0

M1)21($#(1A.'8+#(1%..D+&2&*)&1.'($*2.:

&&;

R≤

xC+.).31'D+.

R3ρ

3ǫ 0

x2

+E

0=

0=⇒

x2

imagin

ari

o

81)21D+.#1O$>81'&,&-#%..D+&2&*)&18$)$

R≤

x:

!

"""#

x≤−

R$%&'&()*+%&

E0−

R3ρ

3ǫ 0

x2

=0

=⇒

x3

=−

R

√

ρR

3ǫ 0

E0

,-'-+%&

x1&."*/&0*&1&2&3%(45"'+%&

x3

=−

R

√

ρR

3ǫ0E

0<−

R=⇒

√

ρR

3ǫ 0

E0

>1

=⇒

E0

<R

ρ

3ǫ 0

+%&&*5-("*(-3)61"3"76+%&&63)6/'-()*-6/&'")'(&6/&4-'-+%&8%2"&'-&+%"9

5"2'"):;*/&4%6/)1&&+%"5"2'")&*"6&*/-25&0*&1&<--55&3/)'=&'">3-'5):

?-4'%&2-1&5)*"@%"&6/&*&1&<--55&3/)':A&()*="*/)+%&4-'-

E0

<R

ρ3ǫ 0

&."*/&61)*4%6/)*

1&&+%"5"2'")0%6)&*/-25&B&5)/')"6&*/-25&:,-'-

E0

=0&."*/&%6-*)5-4)*"3"761&&+%"5"2'")0

&6&53&6/')1&5-&*C&'-B&*&*/-25&:,-'-

E0

=R

ρ3ǫ 0

&."*/&%6-*)5-4)*"3"761&&+%"5"2'")0&*

&6

x=−

RB&*"6&*/-25&:D

4-'-

E0

>R

ρ3ǫ 0

6)8-B4)*"3")6&*1&&+%"5"2'"):E)1)&*/)*&4%&1&

=&'35-'-(&6/&8-3"&61)%6@'F>3)1&;G.#=&'*%*.G*&1&<--55&3/)'#:

H

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

!"#$%&'((

I)6*"1&'&%6

3"5"61')(%B5-'@)1&'-1")

R+%&*&3-'@-&6

*%

"6/&'")'3)6

%6-1&6*"1-1

ρ=

ρ0(1−

r R)0

1)61&

ρ0&*%6-3)6*/-6/&4)*"/"=-0*"&61)

r5-1"*/-63"-(&1"1-1&*1&&5&<&

1&53"5"61'):;63%&6/'&-+%&1"*/-63"-1&5&<&&53-(4)&5J3/'"3)&*(F."()B3-53%5&&*/-

(-@6"/%1(F."(-:

)"$*+,-./

K-1)+%&&53"5"61')&*(%B5-'@)0&53-(4)&5J3/'"3)&*/-'F&61"'&33"76'-1"-5&63))'1&6-1-*

3"5L61'"3-*:I)6&*/)4)1&()*3)6*"1&'-'3)()*%4&'>3"&1&@-%**%63"5"61')1&'-1")

r<

R0

3)6"@%-5&<&+%&&53"5"61')1&'-1")

R:;*/)*&4%&1&-4'&3"-'&65-*"@%"&6/&>@%'-M

,)1&()*&6/)63&*&*3'"2"'&53-(4)&5J3/'"3)3)()

~ E=

Er0

1)61&

r&*&5=&3/)'%6"/-'")

'-1"-5&63))'1&6-1-*3"5L61'"3-*:,)'5-5&B1&@-%***&/&61'F+%&M

∫

S

~ E·n

dS

=

∫

tapas

~ E·n

dS

+

∫

manto

~ E·n

dS

=q i

nte

rio

r

ǫ 0(1

)

;5=&3/)'6)'(-5n1&5-*/-4-*&*4&'4&61"3%5-'-

r:I)6&*/)*&/&61'F+%&

r·n

=0B4)'

5)/-6/)

∫

tapas

~ E·n

dS

=

∫

tapasE

r·n

dS

=0

(2)

;5=&3/)'

n06)'(-5-5(-6/)0*&'F4-'-5&5)-

rB4)'5)/-6/)

r·n

=15)+%&"(45"3-+%&M

∫

manto

~ E·n

dS

=

∫

manto

Er·n

dS

=E·∫

manto

dS

?-"6/&@'-5

∫

manto

dS

3)''&*4)61&-5F'&-1&5(-6/)M

∫

manto

dS

=2π

rL:,)'5)/-6/)M

∫

manto

~ E·n

dS

=E·2

πrL

(3)

N&&(45-O-61)GP#BGQ#&6GR#'&*%5/-+%&M

E·2

πrL

=q i

nte

rio

r

ǫ 0(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′ )·d

V(5

)

A(,(1()1%)(,

dV

1#5-67&,(0#-&).#)%0&5&5*,&7#-16)657,#-7&)()(,2#"406-0#&B&$%&

&)16)657,#7&)+,#/)&0(C1#5,(76#-

r′4

r′+

dr′

1#0#0%&-*,()(92%,(@

"%&2#CdV

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

dV

=2π

r′dr′

L(6

)

D&&0+)(E(57#FGH&5FIH@

dq

=2π

r′·d

r′·L·p

(r′ )

dq

=2π

r′·d

r′·L·ρ

0(1−

r′ R)

=⇒

dq

=2π

L·ρ

0(r′ −

r′2

R)d

r′

J5*&2,(57#@

q=

2πL

ρ0

∫

r

0(r′ −

r′2

R)d

r′=

2πL

ρ0·(∫

r

0r′

dr′−

∫

r

0

r′2

Rdr′

)

=⇒

q=

2πL

ρ0·(

r2 2−

r3 3R

)=

2πL

ρ0·(

3R

r2−

2r3

6R

)

=⇒

q=

πL

ρ0

3R

(3R

r2−

2r3

)(7

)

K&F H4F!H@

E·2

πrL

=πL

ρ0

3R

ǫ 0(3

Rr2−

2r3)

K&-+&B(57#

E@

E=

ρ0

6R

ǫ 0(3

Rr−

2r2

)(8

)

L

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

M%&,&0#--(/&,+(,($%&.()#,7&

r&)1(0+#&)<1*,61#&-0(N60#:A(,(&-*#7&,6.(0#-

E,&-+&1*#(,@

dE dr

=ρ0

6R

ǫ 0(3

R−

4r)

J2%()(57#(O-&#/*6&5&$%&&).()#,7&

r/%-1(7#&-@

r=

3R 4

K&&-*(0(5&,(

|E| m

ax

=3ρ0R

16ǫ

0

!

!"#$%&'()

"#$%&'()*+,#()-#.#-$()*,%,+&+()/0&)&&#10&#$%,#,0#,'-)$,#1-,

2a&+0#('&+($%()&

$-&#&0#,'-)$%-201-3#4(5(67#&,'&1,%6,1(#'&#)-','

ρ89,#6&#$&,+*+,#('&+,'&%&14,

4,:0#1,)1,%3#1,%6,'(;'&%,'-(

R:'&#)-','

)0*&%.1-,+

σ<=&%.60%,>8?,+10+,%;1(#&+

(%-6&#&#&+*0#$(5&'-('&+,%&1$,/0&0#&+()*+,#();&+1,5*(&+&1$%-1(*,%,$('(

x>

08

*"$+,-./0

@,1(#.60%,1-3#'&+*%(2+&5,&)+,)-60-&#$&A

B,%,&#1(#$%,%&+1,5*(&+71$%-1()(2%&&+&C&

x<x

>0>

1,+10+,%&5()&+1,5*(&+71$%-1(*%(D

'01-'(*(%+,%&6-3#&#$%&+()*+,#()-#.#-$()'&'&#)-','

ρ:&+1,)1,%(#&)E7%-1('&'&#)-','

σ:*()$&%-(%5&#$&0$-+-F,%&5()&+*%-#1-*-('&)0*&%*()-1-3#8

1'&2"%$3,4!-,"2!"5+,-5"2"!$'!%6-./%/4!%$"72$'/"7-/8/-4"79E

ρ:0G,',+,

)-5&$%H,/0&*%&)&#$,&)$,%&6-3#%&)*&1$(,+&C&

y;&+1,5*(&+71$%-1(

Eρ*%&)&#$,%,$,52-7#

&)$,)-5&$%H,1(5(50&)$%,+,)-60-&#$&.60%,A

IJ

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

?,+10+,%&&+1,5*(&+71$%-1(&#'()%&6-(#&)8?0,#'(

0<

x<

a:

x>

aA

•0

<x

<a0

?(#)-'&%,%&1(5()0*&%.1-&6,0))-,#,0#,1,C,'&K%&,+,$&%,+A

:+,'(

2x1(5(50&)$%,+,

)-60-&#$&.60%,A

G,',+,'-%&11-3#/0&$-&#&&+1,5*(&+71$%-1()&$&#'%K/0&*(%+,)$,*,)E%(#$,+;*()$&%-(%;

-#E&%-(%:)0*&%-(%&+L0C()&%K1&%(;:,/0&&+=&1$(%#(%5,+,&)$,)1,%,)&)*&%*&#'-10+,%,+

1,5*(:'&&)$(

~ E·n

=0+(/0&-5*+-1,/0&

∫

~ E·n

dS

=0

M-#&52,%6(;)-4,2%KL0C(*(%+,)1,%,)+,$&%,+&):)&$&#'%K/0&&+1,5*(&+71$%-1(&)*,%,+&+(

,+()=&1$(%&)#(%5,+&)'&&)$,)1,%,)1(#+(10,+)&105*+-%K

~ (E

)·n

=E

+(/0&-5*+-1,/0&

∫

S

~ E·n

dS

=E

∫

SdS

G&&)$,5,#&%,;,*+-1,#'(+,+&:'&6,0));)&$&#'%K/0&&+L0C($($,+)(2%&+,1,C,)&%K&+

L0C(,$%,=7)'&+,)1,%,)+,$&%,+&)A

φ=

E

∫

S1

dS

+E

∫

S2

dS

=E

A+

EA

=2E

A=

q int

ǫ 0(1

)

G(#'&

S1:

S2)(#+,))0*&%.1-&)+,$&%,+&)-#'-1,',),#$&%-(%5&#$&8

B('&5()1,+10+,%+,1,%6,-#$&%-(%

q int:,/0&),2&5()+,'-5&#)-(#&)'&+,1,C,:+,'&#)-','

=(+057$%-1,'&1,%6,

ρ8@,1,%6,-#$&%-(%)&%K)-5*+&5&#$&

q int=

ρV

=ρ2x

A

N&&5*+,F,#'(&)$&=,+(%&#<O>(2$&#&5()A

2E

A=

ρ2xA

ǫ 0

E=

ρx ǫ 0

P',',+,)-5&$%H,=&5()/0&&+1,5*(&)$,&#'-%&11-3#

i;*(%+($,#$()&$-&#&/0&A

~ Eρ

=ρx ǫ 0

i,si

0<

x<

a

!

•x

>a

"#$%&'#(#(#)*+,-.&%/,0+#$,%,-,%#*)12,23#%)&%4,0+#-,5)/#3%6,$%#5#23##5-,/)5/,

#2#-',/$&#-7'3%)'&3,-'&/&5#/+#53%,#2-,5)*+)#23#8*+%,9

:-)*+,-0+##-',5&,23#%)&%#;)53)%<=+>&5&-&#2-,5',%,5-,3#%,-#545#%<9φ

=2E

A?

@,',%*,)23#%)&%5#%<-,',%*,0+##53,(#5(#−

a<

x<

aA$&%-&3,23&9q i

nt=

ρ2aA?:$-)',2(&

-,-#4(#*,+559

φ=

2EA

=ρ2aA

ǫ 0

E=

ρa ǫ 0

B-',/$&#53,#2()%#'')12

i$&%-&3,23&9

Eρ

=ρa ǫ 0i,

six

>a

C&3,%0+##-',/$&#2#53,%#*)12#5'&253,23#?

!"#

$%

&'()*+,)%

$+%-.),-%

$%+

'"

)"/)"+"

&/0(+,)"

1E

σ2 2

"#$,%,%#/&5#-#5$,')&#2

D%#*)&2#5A#-)23#%)&%(#-,',5',%,#5.7%)',E%FGH4#-#;3#%)&%

E%IGH4',-'+-,/&5#-',/$&#-7'3%)'&#2',(,+2&+3)-)J,2(&-,-#4(#*,+559

•r

<R

K,(&-,5)/#3%6,#5.7%)',4-,(#25)(,(

5+$#%8'),-

σ#5'&253,23#A5#3#2(%<0+##-',/$&

#-7'3%)'&#53,%<#2()%#'')12%,(),-?L&25)(#%,%#'&/&5+$#%8')#*,+55),2,+2,',5',%,#5.7%)',

(#%,()&

r<

RE5+$#%8')#"H3,-'&/&/+#53%,-,8*+%,9

D

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

B-=+>&,3%,M75(##53,5+$#%8')#5#%<

φ=

∫

S~ E·n

dSA$#%&#-M#'3&%

n#5$,%,-#-&,-',/N

$&E,/O&5%,(),-#5H4$&%-&0+#

~ E·n

=E

4$&%-&3,23&

φ=

E∫

SdS

=E

4πr2?

P&%&3%&-,(&A-,',%*,)23#%)&%#52+-,4,0+#5&-&Q,4',%*,#2-,',5',%,9q i

nt=

0?P&%-,-#4

(#*,+555#3#2(%<0+#

φ=

E4πr2

=q i

nt

ǫ 0=

0

E=

0,si

r<

R

•r

>R

L&25)(#%,/&5A,-)*+,-0+##2

#-',5&,23#%)&%A+2,',5',%,#5.7%)',(#%,()&

r>

R'&/&

5+$#%8')#*,+55),2,?R,>&-&5/)5/&5,%*+/#23&5(#5)/#3%6,,23#%)&%M#/&50+##-',/$&

#-7'3%)'&5#%<%,(),-4$&%-&3,23&A$,%,-#-&,-M#'3&%2&%/,-4(##53,/,2#%,9

φ=

∫

S

~ E·n

dS

=E

∫

SdS

=E

4πr2

@,',%*,)23#%2,

q int5#%<-,',%*,'&23#2)(,#23&(,-,5+$#%8')#A-,'+,-5#&O3)#2#/+-3)$-)N

',2(&#-<%#,3&3,-$&%-,(#25)(,(5+$#%8'),-(#',%*,9q i

nt=

σ4πR

2?

P&%-,-#4(#*,+55A5#3#2(%<0+# φ

=E

4πr2

=q i

nt

ǫ 0=

σ4πR

2

ǫ 0

K#5$#>,2(&#-',/$&#-7'3%)'&5#3)#2#0+# E

=σR

2

ǫ 0r2

B53#',/$&#53,%<#2()%#'')12%,(),-A$&%-&3,23&9

~ Eσ

=σR

2

ǫ 0r2

r,si

r>

R

K&2(#

r#5#-M#'3&%+2)3,%)&#2()%#'')12%,(),-?

:Q&%,0+##2'&23%,/&5-&5(&5',/$&5#-7'3%)'&5$&(#/&5#2'&23%,%#-',/$&%#5+-3,23#

5+$#%$&2)72(&-&5?P,%,#53&A'&25)(#%,%#S%#*)&2#59

TH

0<

x<

aTTH

a<

x<

a+

2R

TTTH

x>

a+

2R

TH

0<

x<

a9B2#53,%#*)12-&5',/$&5#-7'3%)'&55&29

~ Eρ

=ρx ǫ 0i4

~ Eσ

=σR

2

ǫ 0r2r?

:Q&%,3#2#/&5

0+##;$%#5,%

r4

r(#.&%/,'&2M#2)#23#$,%,$&(#%5+/,%,/O&5',/$&5?L&25)(#%#/&5-,

5)*+)#23#8*+%,9

U#/&50+#5#'+/$-)%<0+#

a+

R=

x+

r'&2-&0+#%#5+-3,

r=

a+

R−

x?:(#/<5AM#/&5

0+#

r=−

i4$&%-&3,23&$&(#/&5#5'%)O)%#-',/$&#-7'3%)'&'&/&9

~ Eσ

=−

σR

2

ǫ 0(a

+R−

x)2

i

!

"##$%&'&(#)&*#+,&'-.#+/,%)0,.)#$1+%&(%##(#$%&)#203(#$

~ E=

~ Eρ+

~ Eσ*#$4#,0)

~ E=

(ρx ǫ 0−

σR

2

ǫ 0(a

+R−

x)2

)i

556

a<

x<

a+

2R78(#$%&)#203(#+,&'-.+.&-.)%&$.+.

Eρ9&:1##+,&'-.-).41,04.-.)

+&,&$,&&#$;#)0,&#$(1+.#(#+0(%#)0.)#++&<=.)+.%&(%.7

~ E=

~ Eρ

=ρa ǫ 0i,

sia

<x

<a

+2R

5556

x>

a+

2R8(#$%&)#20.*&'>.$,&'-.$#$%&)?(#(40)#,,03(9$#(%04.

i<=&)&.>%#(#)

+&#@-)#$03(4#

~ Eσ#(;1(,03(4#@*,.($04#)&'.$+&$0210#(%#A21)&7

B#'.$:1#$#,1'-+#+&)#+&,03(

x=

a+

R+

r*,.(+.:1#)#$1+%&:1#

r=

x−

a−

R<C4#'?$*

r=

i<"##$%&;.)'&*%#(4)#'.$:1#7 ~ Eσ

=σR

2

ǫ 0(x−

a−

R)2

i

D1#2.*-.)-)0(,0-0.4#$1-#)-.$0,03(*#+,&'-.#+/,%)0,.#(#$%&)#203($#)?7

~ E=

(ρa ǫ 0

+σR

2

ǫ 0(x−

a−

R)2

)i,

six

>a

+2R

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

F

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

~ E=

(ρx ǫ 0−

σR

2

ǫ 0(a

+R−

x)2

)i0

<x

<a

ρa ǫ 0i

a<

x<

a+

2R

(ρa ǫ 0

+σR

2

ǫ 0(x−

a−

R)2

)ix

>a

+2R

!"#$%&'()

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

R/,"#&'#$%&+&'$&',+(0+1"23)-.(%,+$

ρ4−

ρ3#%*"()'$56"$,'#.("$&'+)7+$'$*'(+$'$.8#+3#+&%$.+#,%+)'#"(93'

2R5:'+

~ d'21',."(93'1+&'2,'#.("&'2+'$*'(+;"$%.%1++2,'#.("&'2+'$*'(+#'0+.%1+5<(3'7'93''2

,+);"'2-,.(%,"'#2+%#.'($',,%=#&'2+$'$*'(+$'$,"#$.+#.'4'#,3'#.('$31+2"(5

*"$+,-./0

>'2;("72')+?@/$+7')"$93''2,+);"'2-,.(%,"'#'2%#.'(%"(&'3#+'$*'(+)+,%A+&'&'#$%&+&

&',+(0+3#%*"()'

ρ'$

~ E(r

)=

ρr

3ǫ 0·r5

>'2+$%)'.(B+&'2+&%$.(%73,%=#/+;(%"(%$+7')"$93'+&')8$;+(+

R≤

r'2,+);",3);2'

~ E(~r

)=

E(r

)r5:%7%'#

#"'$#','$+(%";+(+2+('$"23,%=#

&'2;("72')+,"#",'('2,+);"

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

!"#$%&'(')"$,")"$3;'(D,%'&'%#.'0(+,%=#3#+,8$,+(+'$*-(%,+&'(+&%"

r>

R5E'#')"$

93'

∮

Ω

~ E·n

dS

=Q

int

ǫ 0

E(r

)4πr2

=4πR

3ρ

3ǫ0

=⇒

~ E(~r

)=

R3ρ

3ǫ 0

r2r

<"(.+#."/.'#')"$93''2,+);"'2-,.(%,"0'#'(+&";"(2+'$*'(+)+,%A+'$

~ E(r

)=

ρr

3ǫ 0·r

r<

RR

3ρ

3ǫ 0

r2r

R≤

r

<"#0+)"$'2"(%0'#

&',""(&'#+&+$'#

'2,'#.("&'2+'$*'(+&'&'#$%&+&

&',+(0+

ρ5F$B/

.'#')"$93'2"$,+);"$'2-,.(%,"$'#2"$%#.'(%"('$&',+&+'$*'(+'$.8#&+&"$;"(

~ E1(r

)=

ρ 3ǫ0·~r

para

|~r|<

R

~ E2(r

)=

−ρ 3ǫ 0·(

~r−

~ d)

para

|~r−

~ d|<

R

6"$1',."('$'#2+%#.'($',,%=#&'+)7+$'$*'(+$$+.%$*+,'#$%)32.8#'+)'#.'

|~r−

~ d|<

R4|~r|<

R5F$B/;"('2;(%#,%;%"&'$3;'(;"$%,%=#/'2,+);"'2-,.(%,"'#2+%#.'($',,%=#

&'+)7+$'$*'(+$'$

~ E(~r

)=

~ E1(~r

)+

~ E2(~r

)=

ρ 3ǫ 0·~r−

ρ 3ǫ 0·(

~r−

~ d)

=ρ 3ǫ 0·~ d

'2,3+2'$,"#$.+#.'/.+2,")"$'93'(B+;("7+(5

G

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

!"#$%&'(

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

!"#$%&'()

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

σ>

0"!'1+*+*&2&3

%&&.(+.$4"

x=

056"

ax#&&".(&"-'+("+.+'/+)("-(+*−

q<

05

+76".(&"-'&&*)!-&".$+*&*8.-'$.!#!9'&&*&2&3:&"-'&*+.+'/+−

q<

0:&*!'$/&"

.!!'%&"+%!;5

97<"+)+'-=.(*+%&1+#+

m:.+'/+−

e<

0#&(9$.+&"&*)("-!1&%$!&"-'&−

q:;:#&%&2+

*$9'&5> !"?(8&"&'/=+.$"8-$.+**&/+*+.+'/++*)*+"!@A$/"!'&&0&.-!#/'+B$-+.$!"+*&#75

*"$+,-./0

+7 !"!.&1!#&*.+1)!&*8.-'$.!/&"&'+%!)!'&*)*+"!:*+.+'/+−

q#!9'&&*&2&3?(&*!#

("&5C+*&#.+1)!##!"

~ E1

=σ 2ǫ 0

x

~ E2

=1

4πǫ 0

−q

(a−

x)2·−

x=

1

4πǫ 0

q

(a−

x)2·x

D#=E&*.+1)!&*8.-'$.!&"-'&;:−

qA?(&&#-FE%$/+1!#E&"&*)("-!D7&#

~ E=

~ E1+

~ E2

=

(

σ 2ǫ0

+1

4πǫ 0

q

(a−

x)2

)

·x

G&'!:+?(&

~ E=−

~ ∇VE&"-!".&#&"

ΩA*+'&/$4")&%$%+7

∂V ∂z

=0

∂V ∂y

=0−

∂V ∂x

=

(

σ 2ǫ0

+1

4πǫ 0

q

(a−

x)2

)

=⇒

V(x

)=

−(

σx

2ǫ 0

+1

4πǫ 0

q

(a−

x))

+C

976*.+1)!&*&.-'!#-F-$.!&#.!"#&'B+-$B!E+#=?(&)!%&1!#(#+'.!"#&'B+.$4"%&&"&'/=+

&"-'&&"&'/=+)!-&".$+*:.$"8-$.+5H+.+'/+−

e)+'-&%&*'&)!#!%&#%&

a/2

xI+.$+&*

)*+"!5G!'.!"#&'B+.$4"%&&"&'/=+E-&"&1!#?(&

JK

JL

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

−eV

(a/2

)=

mv

2

2−

eV(0

)

=⇒

K=

mv

2

2=

e(V

(0)−

V(a

/2))

=e

(

−1

4πǫ 0

q a+

C+

1

4πǫ 0

2q a

+σa

4ǫ 0−

C

)

=e

(

1

4πǫ 0

q a+

σa

4ǫ0

)

!

!"#$%&'()

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

r 1,r

20.-,1-)

q 1,q

2)#2-,-'-)2(,/&-1,-&'%)3

$-&.%-

d>

>r 1

,r24"%-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/&$('#

,#*#,#&.%-'#82($#&.%-8#8%&>&%$(#%1/-8-,#82($#&.%-8-.#,(-88B:#)'#.%,:V

(+∞

)=

04C)B:

#82($#&.%-8)(6,#8-))/2#,>.%#)'#8-)#)*#,-).(&'/.$(,-)#)

V1

=kq 1 r 1

,V2

=kq 2 r 2

'(&'#D#5())/2/#)$(;/##82($#&.%-8'#/&-#)*#,-&(#)-*#.$-'(2(,#8'#8-($,-9(5A)6%#&

#)-*#.$-'('#*(,5-'#)2,#.%-68#=:'-'(;/##)$A&5/05/0)#2-,-'-:#)'#.%,:d

>>

r 1,r

2:

0$-52(.(#8.-52('#/&-#)*#,-,#'%)$,%6/0#8-.-,1-'#8-($,-4E(,.(&)#,7-.%<&'#.-,1-:

)%(q

1) f

,(q 2

) f)(&8-).-,1-)#&.-'-#)*#,-/&-7#?-8.-&?-'(#8#;/%8%6,%(:$#&#5();/#

q 1+

q 2=

(q1) f

+(q

2) f

C8-8.-&?-,#8#;/%8%6,%(:#)'#.%,:./-&'('#F-'#D-6#,$,-&)*#,#&.%-'#.-,1-)#&$,#8-)#)*#,-):

$#&#5();/#8-'%*#,#&.%-'#2($#&.%-8#8+.$,%.(#&$,#-56-)#)&/8(:#)'#.%,:

V1

=V

2

k(q

1) f

r 1=

k(q

2) f

r 24π(r

1)2

(σ1) f

r 1=

4π(r

2)2

(σ2) f

r 2

=⇒

(σ2) f

(σ1) f

=r 1 r 2

@##)$(7#5();/##&1#&#,-8:8-),#1%(&#)#&8-)/2#,>.%#'#/&.(&'/.$(,.(&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-.%<&#&.(&$,-'-#&8-#./-.%<&'#.(&)#,7-.%<&'#.-,1-:(6$#&#5()

(q1) f

+(q

2) f

=q 1

+q 2

4π(r

1)2

(σ1) f

+4π(r

2)2

(σ2) f

=q 1

+q 2

4πr 1

(σ1) f

(r1+

r 2)

=q 1

+q 2

=⇒

(σ1) f

=1

4πr 1

q 1+

q 2r 1

+r 2

(σ2) f

=1

4πr 2

q 1+

q 2r 1

+r 2

IJ

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

!"#$%&'(1

@#5/#)$,#;/#8-'%).(&$%&/%'-'#&|~ E⊥(~r

)|-82-)-,'#/&-,#1%<&7-.B--($,-7-.B-2(,/&-

)/2#,>.%#.-,1-'-'#'#&)%'-')/2#,>.%-8σ(~r

):.(&

~r#&8-)/2#,>.%#:#)

σ(~r

)

ǫ 0

K(&)%'#,-&'(#)$(:#&./#&$,##8.-52(#8+.$,%.(#&8-)/2#,>.%#'#/&.(&'/.$(,#&#;/%8%6,%(

#8#.$,()$A$%.(4

*"$+,-./0

K(&)%'#,#5()/&)#.$(,5/05/02#;/#L('#8-)/2#,>.%#.#&$,-'(#&

~r:$-&2#;/#L(

;/#)#-.-)%28-&(:0)%$/#5()-$,-7+)'##88-/&.%8%&',(

Ω'#-8$/,-

0<

2h

<<

10A,#-

0<

A<

<1:$-8;/#8-)/2#,>.%#8(.,/?#2(,8-5%$-'4E(,8-8#0'#1-/)):$#&#5()

∮

Ω

~ E(~r

)·n

dS

=Q

int

ǫ 0

E#,(

Qin

t∼ =

σ(~r

)A=⇒

Qin

t

ǫ 0∼ =

σ(~r

)A

ǫ 0

M#&#5()-'#5A)

∮

Ω

~ E(~r

)·n

dS

=

∫

manto

~ E(~r

)·n

dS

+

∫

cara

su

p

~ E1(~r

)·n

1dS

+

∫

cara

inf

~ E2(~r

)·n

2dS

E('#5()#G2,#)-,#8.-52(#8+.$,%.(.(5(8-)/5-'#)/.(52(&#&$#$-&1#&.%-80&(,5-8

-8-)/2#,>.%#:#)'#.%,:

~ E=

~ E⊥

+~ E//4C'#5A):.(5(#8A,#-#)5/02#;/#L-:#85<'/8(

'#8.-52(#8+.$,%.()(6,#+8)#5-&$%#&#.-)%.(&)$-&$#:2(,8(;/#8(2('#5()-2,(G%5-,2(,

|~ E(~r

)|#&$('-#8A,#-4M-56%+&:n

1=−

n2

K(&#)$(#&5#&$#:$#&#5();/#

∫

cara

su

p

~ E1(~r

)·n

1dS

+

∫

cara

inf

~ E2(~r

)·n

2dS

=~

(E1) ⊥

A−

~(E

2) ⊥

A

=(

~(E

1) ⊥−

~(E

2) ⊥

)A

#G2,#)%<&;/#&('#2#&'#'#

h9-8%1/-8;/#

σ(~r

)Aǫ 0

=4M-56%+&$#&#5();/#

∫

manto

~ E(~r

)·n

dS→

0cu

ando

h→

0

!

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

(~

(E1) ⊥−

~(E

2) ⊥

)A=

σ(~r

)A

ǫ 0

E⊥A

=σ(~r

)A

ǫ 0

=⇒

E⊥(~r

)=

+σ(~r

)

ǫ 0

.,/')*#01'1(

2*(312&*4'(

'015-5/45*'-62&4*#&7&52*(*84'#'(&,2,49,'(

#1

5(&'45*4%'-

2,)8*'-62&452*'(#15(&'45*4'#(1-*:&*3,#12,49,'#&735#&45/153,'(-,#18'4+25'3'&,-

;*4),01''-2,)8*'-62&452*'(-,#18'4+25''#(*4),-,-,)5#),<81'##5&1=5'4,2*)8*('(&'

&,(9'(25,-,'--,%-,#2,49,##')*='4$,(:(*'#&,4$,('('015-5/45*'-'2&4*#&7&52*>?"#$%2*(@

#53'4,(3*'#&*'('-,(7-5#5#,(&'45*4%&'(')*#01''-2,)8*'-62&452*'(

~r#*/4'-,#18'4+25'

'#

~ E(~r

)=

+σ(~r

)

ǫ 0n

2*(

n'-='2&*4(*4),-'A&'45*4,-,#18'4+25''(

~r?

B

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

!"#$%&'().'&5'('(3*#,4),314,#'#;6452,#%)'&7-52,#%C1'2,#:2*(26(&452,#3'4,35*#,

:/</D,>%:3''#8'#*43'#84'25,/-',1(01'+(5&*?E,,4),314,5(&'4(,#'2,49,2*(1(,2,49,

q 0>

0?.'#18*('01'-,&5'44,<01'#'&*),2*)**459'(3'8*&'(25,-'#>'#&75(+(5&,)'(&'

,-'F,3,?

G?E,,4),314,'A&'4(,#'2*('2&,,&5'44,,&4,=6#3'1(,/,&'4$,21:,35;'4'(25,3'

8*&'(25,-'(&4'#1#/*4('#'#

V0?

,>H,-21-'-,2,49,01'#'5(312''(-,##18'4+25'#5(&'45*4:'A&'45*43'2,3,1(,3'

-,#,4),314,#?

/>H,-21-''-2,)8*'-62&452*'(&*3*#-*#81(&*#3'-'#8,25*?

2>H,-21-''-8*&'(25,-'-62&452*'(&*3*#-*#81(&*#3'-'#8,25*%:-,35;'4'(25,3'

8*&'(25,-'(&4'-,#,4),314,#?

GG?.'2*4&*254215&,-,/,&'4$,<2*('A5I(

354'2&,,&5'44,>?J'85&,-*#2,-21-*#,(&'45*4'#

5(&'-59'(&')'(&'?

GGG?.'3'#2*('2&,-,,4),314,'A&'4(,3'-,&5'44,%:-1'9*#',2'42,1(,2,49,

q>

0C,#&,

1(,35#&,(25,

c>

b3'-2'(&4*3'-2*(3'(#,3*4?K'253,#5-,,225I(3'-,2,49,0)*35+2,

*(*L

3>-,2,49,&*&,-3'2,3,1(,3'-,#,4),314,#

'>-,3'(#53,33'2,49,'('--,#

;>-,;1(25I(8*&'(25,-:-,35;'4'(25,3'8*&'(25,-'(&4''--,#?

MA8-501'/4'=')'(&'#1#4'#81'#&,#?

*"$+,-./0

G?M-84*/-'),&5'('#5)'&4$,'#;6452,2*(2'(&4*3'2**43'(,3,#'-2'(&4*3'-,#27#2,4,#

'#;6452,#%3'/53*,01'-,&5'44,'#&75(+(5&,)'(&'-'F,(,<4'2*43,401'-,&5'44,#'85'(#,

2*)*1(

#1)53'4*:

;1'(&'5(+(5&,3'2,49,>?N*3,#-,#3'(#53,3'#3'2,49,#'47(

1(5;*4)'#?K,3,-,#5)'&4$,'#;6452,%'-2,)8*:8*&'(25,-'-62&452*#3'8'(3'47(#I-*3'

-,35#&,(25,,-2'(&4*3'-,#27#2,4,#:'-2,)8*,81(&,47'('-#'(&53*3'

r%'#3'254%

V=

V(r

):

~ E(~r

)=

E(r

)r?

,>.',(

q 1,q

2,q

3,q

4-,#2,49,#3'-,##18'4+25'#5(&'45*4:'A&'45*43'-,#,4),314,#

5(&'4(,:'A&'4(,%4'#8'2&5=,)'(&'?N'(')*#01'

q 0=

q 1+

q 2?

"8-52,(3*'-&'*4'),3'O,1##2*(1(,#18'4+25''#;6452,2*(26(&452,:'(&4'-,#

#18'4+25'#3'-,845)'4,,4),314,%&'(')*#

∮

~ E·n

dS

=0

=q 1 ǫ 0

=⇒

q 1=

0=⇒

q 2=

q 0

3*(3'

~ E=

~ 081'#'-2,)8*'-62&452*'('-5(&'45*43'1(2*(312&*4'('015-5/45*

'-'2&4*#&7&52*'#(1-*?

!

"#$%&'()*'+*,'-$.-*,-/')-0'122&*(1('21#-,3&%--245,%&'&*(&5(.,%&'6-(.,-

$'221#-,3&%-2)-$'2-71()'',/')1,'8.-(-/*2

∮

~ E·n

dS

=0

=q 1

+q 2

+q 3

ǫ 0=⇒

q 1+

q 2+

q 3=

0=⇒

q 3=−

q 2=−

q 0

!"#$%&'(')&*)+()",)(-).%')/%*00$(&*&%0*!)-1$#))023-#$%$(&$3&,-#$%')

-%'#(

r>

b45*)$(&,)&6%"%0)6*&'%%-.%'*-%7,)&).(0

∮

~ E·n

dS

=E

(r)4

πr2

=q 1

+q 2

+q 3

+q 4

ǫ 0=⇒

~ E(r

)=

q 44πǫ 0

r r2

8(-(,-%!%-,)70%9).(05*)"%'#2)-)&$#%)&,-)"%%-.%'*-%):,)-&%4"%,#)--%;)&

)"#&1&#,(<)0

V07)0,()0

V(b

)−

V(+∞

)=

V(b

)=

V0

=−

∫

b

+∞

~ E·d

~r

V0

=−

∫

b

+∞

q 44π

ǫ 0

dr r2

V0

=q 4

4πǫ 0

1 b=⇒

q 4=

4πǫ 0

bV0

'(&'),(.%.(0*&$%.#&(-%'#%"0(9-)"%#&,)6-%"')"=&)%>

9<?#+#'%.(0)")0!%$#()&+%-#%0-)6#(&)0>@('%0"%00*!)-1$#)0')#&,)6-%$#A&5*)

,(.%-).(00)-B&)023-#$%04$(&$3&,-#$%0%"%0%-.%'*-%0>

#<

r≤

a;-)6#A&#&,)-#(-4.),%"')"%%-.%'*-%#&,)-&%<

C#)"-%'#(')"%0*!)-1$#)')#&,)6-%$#A&)0

r≤

a∮

~ E·n

dS

=E

(r)4

πr2

=q 1 ǫ 0

=0

=⇒

~ E=

~ 0

##<

a<

r<

b;-)6#A&)&,-)%-.%'*-%0<

D0)+#')&,)5*)):#0,)$%.!()"3$,-#$()&)0,%-)6#A&7!*)0E%4*&%'#2)-)&$#%

')!(,)&$#%")&,-)%-.%'*-%0>@)&).(0)&)0,)$%0(5*)

∮

~ E·n

dS

=E

(r)4

πr2

=q 0 ǫ 0

=⇒

~ E(r

)=

q 04πǫ 0

r r2

###<

r=

b;.),%"')"%%-.%'*-%):,)-&%<

D&)0,)!*&,(7

~ E(b

)=

~ 07!*)0)"$%.!()"3$,-#$()&)"#&,)-#(-')*&.),%"

)00#).!-)&*"(>

#+<

r>

b;-)6#A&):,)-#(-%"%0%-.%'*-%0<

@%.9#3&%5*=)0)+#')&,)5*)):#0,)*&$%.!()"3$,-#$(7!*)0E%4*&%'#2)-F

)&$#%')!(,)&$#%")&,-)"%%-.%'*-%):,)-&%4)"#&1&#,(;!*&,(')-)2)-)&$#%

')"!(,)&$#%"<> 0=7,)&).(05*)

∮

~ E·n

dS

=E

(r)4

πr2

=q 4 ǫ 0

=4πǫ 0

bV0

ǫ 0=⇒

~ E(r

)=

bV0

r2·r

GH

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

I)0*.#)&'(7)"$%.!()"3$,-#$()0

~ E(r

)=

~ 0r≤

aq 0

4πǫ 0·

r r2

a<

r<

b~ 0

r=

bbV

0

r2·r

b≤

r

$<@%.9#3&'#+#'%.(0)")0!%$#()&+%-#%0-)6#(&)0>

#<

a≤

r≤

b

@)&).(0

V(r

)=

−∫

~ E·d

r+

C

=−

∫

q 04πǫ 0

dr r2

+C

=q 0

4πǫ 0·1 r

+C

'(&'))"$%.#&(')#&,)6-%$#A&*0%'(2*)-%'#%">

8%-%'),)-.#&%-J#.!(&).(0"%$(&'#$#A&')9(-')>C%9).(05*)

V(b

)=

V07

!(-,%&,(

C=

V0−

q 04πǫ 0·1 b7)&,(&$)0

V(r

)=

q 04π

ǫ 0

(

1 r−

1 b

)

+V

0

8(-,%&,(7,)&).(05*)"%'#2)-)&$#%')!(,)&$#%")&,-)"%0%-.%'*-%0)0

V(a

)−

V(b

)=

q 04π

ǫ 0

(

1 a−

1 b

)

##<

r≤

aD"!(,)&$#%")&

r=

a;%-.%'*-%#&,)-&%<)0

V(a

)=

q 04πǫ 0

(

1 a−

1 b

)

+V

0

J(.()"$%.!()"3$,-#$()&)0,%-)6#A&)0&*"(7)&,(&$)0,('%"%-)6#A&')9)

,)&)-)".#0.(!(,)&$#%";')9)0)-)5*#!(,)&$#%"<7!(-"(5*)

V(r≤

a)

=q 0

4πǫ 0

(

1 a−

1 b

)

+V

0

###<

b≤

r

V(r

)=

−∫

~ E·d

r+

C‘

=−

∫

q 44πǫ 0

dr r2

+C

‘

=q 4

4πǫ 0·1 r

+C

‘

=bV

0

r+

C‘

!

"#$%

V(b

)=

V0

=⇒

C‘=

0

=⇒

V(r

)=

bV0

r

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

V(r

)=

q 04πǫ 0

(

1 a−

1 b

)

+V

0r≤

aq 0

4πǫ 0

(

1 r−

1 b

)

+V

0a

<r

<b

bV0

rb≤

r

4456.1%$0%1*$1(*02$.2720#$82-#',#1*$-921#$

V0

=0-

.22$)2,($2#:0#$+2#'02$;2.)*')%

/%0#+1*2.<(#.20*#$$25=+1%+'#1(#+1*2-+%927$;12)/%>(#$2,#.2'#?(+,22$)2,($2-

/(#''@.%#:*'0#12)/%1(2+,%92A(+2,*>#$#+1*2,#/%0#+1*2.5&#1(#$,#<(#

~ E=−

~ ∇V5

B%'1;.1(.%''%+*,3+0*1%'2.%'2+0#$*%$#'-2'872'02*+0$%,(1*$

V0

=0A1%+'*,#$2$.%

,*19%2+0#$*%$)#+0#5C##'02>%$)2-0#+#)%'

12$?2D

q 1=

0

q 2=

q 0

q 3=

−q 0

q 4=

0

12)/%D

r≤

a:

~ E=

~ 0

a<

r<

b:

~ E=

q 04πǫ 0·

r r2

b≤

r:

~ E=

~ 0

/%0#+1*2.D

r≤

a:

V(r

)=

q 04πǫ 0

(

1 a−

1 b

)

a<

r<

b:

V(r

)=

q 04πǫ 0

(

1 r−

1 b

)

b≤

r:

V(r

)=

0

E.2,*>#$#+1*2,#/%0#+1*2.#+0$#2$)2,($2'#'

V(a

)−

V(b

)=

q 04πǫ 0

(

1 a−

1 b

)

-.2

)*')2<(##+#.12'%2+0#$*%$5

44456.,#'1%+#102$.22$)2,($2,#.20*#$$2-'*+21#$12$2F+.212$?2

q>

0-.2'12$?2'#+.2'

2$)2,($2'+%'(>$#+)%,*G121*@+5

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

,H6.21#$12$.212$?2<-.212$?20%02.,#12,22$)2,($2'*?(#'*#+,%.2)*')2-

/%$1%+'#$I21*@+,#12$?2#.310$*125J#2)%'<(3/2'21%+

q 1,q

2,q

3,q

45"%$.#A,#

K2(''-0#+#)%'<(#

∮

~ E·n

dS

=0

=q 1 ǫ 0

=⇒

q 1=

0=⇒

q 2=

q 0

,%+,##.L(M%#'1#$%/(#'#.12)/%,#+0$%,#(+1%+,(10%$#+#<(*.*7$*%'*#)/$#

#'+(.%N.2'(/#$G1*#,#*+0#?$21*@+

>(##'>3$*12-1%+13+0$*122.2'2$)2,($2'A

,#+0$%,#.2/$*)#$22$)2,($2H5

∮

~ E·n

dS

=0

=q 1

+q 2

+q 3

ǫ 0=

q 0+

q 3ǫ 0

=⇒

q 3=−

q 0=⇒

q 4=

0

"%$02+0%-.2'12$?2'#+.2''(/#$G1*#',#.2'12'12$%+#'+%12)7*2+N.%<(##$2,#

#'/#$2$'#H5

#HB2,#+'*,2,,#12$?2,#.2'(/#$G1*#,#.22$)2,($2#:0#$+2'#)%,*G12-/(#'.2

12$?2

q*+0$%,(1#(+22'*)#0$82#+#.#'/21*%-A.2'12$?2'#+.%')#02.#'N1%+O

,(10%$#'H#'0;+.*7$#',#)%I#$'#N.2,#+'*,2,,#12$?2,#.2'(/#$G1*##:0#$+2'#

$#,*'0$*7(A#,#02.>%$)2,#2+(.2$#.12)/%/$%,(1*,%/%$

q#+#.*+0#$*%$,#.2

2$)2,($2#:0#$*%$H5"#$%.2',#+'*,2,#',#.2''(/#$G1*#'*+0#$+2'+%'(>$#+12)O

7*%5P*'@.%1%+'*,#$2)%'2$)2,($2'-.212$?2,#.22$)2,($2#:0#$+2'#$;20$28,2

/%$.212$?2

q>

0-A.2,#+'*,2,'(/#$G1*2.0#+,$;(+);:*)%N,#12$?2+#?20*I2H

#+.2/2$0#,#.2#'>#$2);'1#$12+22

q5

>HB22$)2,($2#:0#$+2'#/%+,$;2(+/%0#+1*2.,*'0*+0%,#1#$%N'*'*?(#1%+'*,O

#$;+,%'#

V(+∞

)=

0H5=+#>#10%-V

(b)

=−

∫

b +∞

~ E·d

r6=

0-/(#'

~ E6=

0A+%92A

'*)#0$82<(#/(,*#$22+(.2$.2*+0#?$2.5B2,*>#$#+1*2,#/%0#+1*2.#+0$#2$)2,($2'

'#?(*$;'*#+,%.2)*')2N#'0%'#,#'/$#+,#,#.9#19%<(#.2?#%)#0$82,#.2'2$O

)2,($2'+%9212)7*2,%-/%$.%<(#'(12/21*02+1*2+%9212)7*2,%-A02)/%1%

9212)7*2,%.212$?2#+#..2'H5

!

!"#$%&'()

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

λ1-+).#

L23$4*($5)(%*6#5($4&+-

(-745)&4#($5#'#(-(%6+4&#8*(-+)#'(+2

*"$+,-./0

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

V(~r

)=

∫

Ω

dq

4πǫ 0|~r−

~r 1|

3$(%5(4+%#<%&~r

=xx

+yy

+zz1

~r 1=

x1x<4#$

0≤

x1≤

L<%#$-+%6#%&4&#$(%'(-6*$5#1

'(-+,+)&--+)(%6(45&,+0($5(<5($(0#%8*(

V(~r

)=

∫

L

0

λdx

1

4πǫ 0√

(x1−

x)2

+y

2+

z2

3$4#$5)(0#%

∫

dx

√x2+

129&x

=ta

n(θ

)<($5#$4(%

∫

dx

√x

2+

1=

∫

sec2

(θ)d

θ√

tan

2(θ

)+

1=

∫

sec(

θ)dθ

=ln

(sec

(θ)+

tan(θ

))=

ln(x

+√

x2+

1)=

arc

cosh

(x)

3=5($'&($'#(%5#+

∫

dx1

√(x

1−

x)2

+y2+

z2)(%*-5+>,()?@8*(-#A

∫

dx

1√

(x1−

x)2

+y

2+

z2

=ln

(x1−

x+

√

((x

1−

x)2

+y

2+

z2)

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

,y,z

)=

∫

L

0

λdx

1

4πǫ 0√

(x1−

x)2

+y

2+

z2

=λ

4πǫ 0

∫

L

0

dx

1√

(x1−

x)2

+y

2+

z2

=λ

4πǫ 0·l

n(x

1−

x+

√

(x1−

x)2

+y

2+

z2)∣ ∣ ∣

L 0

=λ

4πǫ 0·l

n

(

L−

x+

√

(L−

x)2

+y

2+

z2

−x

+√

x2+

y2+

z2

)

!"#$%&'12

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

+A3-6#5($4&+-(-745)&4#($5#'#(-(%6+4&#> !"#EF%(4##)'($+'+%6#-+)(%<1*%(-+-(1'(

-#%4#%($#%6+)+(=6)(%+)-+%'&%5+$4&+%)(-(,+$5(%($5+-(%4##)'($+'+%A2

:A3-4+06#(-745)&4#($5#'#(-(%6+4&#2

G

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

4AH+($().?+6#5($4&+-'(-'&6#-#>(%5#<6+)+*$'&6#-#($.($()+-A2

*"$+,-./0

+AI#)(-6)&$4&6&#'(%*6()6#%&4&C$<(-6#5($4&+-(-745)&4#($(-6*$5#I

(%5;'+'#6#)

V=

q

4πǫ 0

(

1 r +−

1 r −

)

I#)-+-(1'(-4#%($#<5($(0#%8*(

r2 +=

r2+

d2−

2rd·c

os(

π 2−

θ)

=r2

+d

2−

2rd·s

en(θ

)

=⇒

1 r +=

1√

r2+

d2−

2rd·s

en(θ

)

=1 r·

1√

1+

(d r)2−

2d r·s

en(θ

)

I()#(%5+0#%4+-4*-+$'#(-6#5($4&+-6+)+

r>

>d<6#)-#8*(

d r<

<11

1 r +=

1 r·

1√

1+

(d r)2−

2d r·s

en(θ

)

∼ =1 r·(

1−

1 2

(

(

d r

)

2

−2d r·s

en(θ

)))

=1 r·(

1−

1 2

(

d r

)

2

+d r·s

en(θ

))

B%*,(J<5($(0#%6+)+

r −

!

r2 −=

r2+

d2−

2rd·c

os(

π 2+

θ)

=r2

+d

2+

2rd·s

en(θ

)

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

1 r −∼ =

1 r·(

1−

1 2

(

d r

)

2

−d r·s

en(θ

))

0&',&12$$,3&.$*4/),3)')

r>

>d'$52,.)

V(r

,θ)

=q

4πǫ 0

(

1 r +−

1 r −

)

∼ =q

4πǫ 0

r·(

(

1−

1 2

(

d r

)

2

+d r·s

en(θ

))

−(

1−

1 2

(

d r

)

2

−d r·s

en(θ

)))

=q

4πǫ 0

r·(

2d r·s

en(θ

))

=2qd

4πǫ 0·se

n(θ

)

r2

=p

4πǫ 0·se

n(θ

)

r2

V(r

,θ)

=1

4πǫ 0·~p·r r2

#&*#$

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

89:)8$(&512$

~ E=−∇

V=−

∂V ∂r

r−

1 r

∂V ∂θ

θ

;5<=.$*$(&512$

∂V ∂r

=−

1

2πǫ 0

pse

n(θ

)

r3

1 r

∂V ∂θ

=1

4πǫ 0

pco

s(θ)

r3

=⇒

~ E(r

,θ)

=p

4πǫ 0

r3

(

2se

n(θ

)r−

cos(

θ)θ)

49>&*5/#$'$(&5,)5/-2/$*.$?-2')7@)#&12$,)'$-/A*&423)#)3&'$,#/3&,&$5)4&.)#)=

3&#$(&5.&()'

V(+∞

)=

04&(&3&.$*4/),#$'$%$'$*4/)7

;,.')$',)54)'-)5#$5#$$,/*?*/.&"%&'()'$,#/3&,&=,)$*$'-<))5&4/)#)),#/3&,&$5

BC

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

U=

qV(x

+dx,y

+dy,z

+dz)−

qV(x

,y,z

)

=q(V

(x+

dx,y

+dy,z

+dz)−

V(x

,y,z

))

=qd

V

=q

(

∂V ∂x

dx

+∂V ∂y

dy

+∂V ∂z

dz

)

=∂V ∂x

px

+∂V ∂y

py

+∂V ∂z

pz

=~p·~ ∇

V

U=

−~p·~ E

!

!"#$%&'()

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

b12*(3&($(*$+'&%3)&4*0&-$'(0+),+*$&.#)5(

ρ(r

)=

ρ06+)+7+)(,&-$

'(3()5&$+'+6#)

a<

r<

b8'($%&'+'

$*7+6+)+

r<

a9:(3()5&$((7

6#3($0&+7(7(03)#%3;3&0#($3#'#(7(%6+0&#9

*"$+,-./0

'1<#2*(=+)(5#%6)&5()#%();0+70*7+)(70+56#(7/03)&0#6)#'*0&'#6#)(%3+'&%3)&4*0&-$

'(0+),+($3#'#(7(%6+0&#80#$(%3()(%*73+'##43($')(5#%(76#3($0&+7*3&7&>+$'#?

Vp

=−

∫

p ∞~ E·d

~ ℓ

"#$%&'()+$'#(76#3($0&+7&,*+7+0()#($(7&$@$&3#9

A76)#47(5+3&($(%&5(3)B+(%./)&0+86#)7#3+$3#%(3($');2*(

~ E=

Er1

'#$'(

r(%(7C(03#)

*$&3+)&#($'&)(00&-$)+'&+79

"#$%&'()+)(5#%D)(,&#$(%?

EF

r<

aEEF

a<

r<

bEEEF

r>

b

•21

r<

a0

"#$%&'()#0#5#%*6()@0&(,+*%%&+$+*$0+%0+)#$(%./)&0#'()+'&#

r<

aG%*6()@0&(HF9H(3&($(

2*((7C(03#)$#)5+7+(%3+%*6()@0&((%6+)+7(7#+

r9:((%3+5+$()+%(3($');2*((70+56#

~ E(%6+)+7(7#+7C(03#)$#)5+7n'(7+%*6()@0&(H6#)7#2*(

~ E·n

=E

86#)7#3+$3#?

φ=

∫

S

~ E·n

dS

=E

∫

SdS

=E

4πr2

I#)#3)#7+'#7+0+),+($

(7&$3()&#)'(7+%*6()@0&((%$*7+8+2*((%3+0#$3($&'+($

(7

0#$'*03#)?q i

nt=

09J67&0+$'#7+<(8'(K+*%%%(3($');2*(?

E4πr2

=0

L6#)7#3+$3#

E=

01%&)M+9

•221

a<

r<

b0"#$%&'()#0#5#%*6()@0&(,+*%%&+$+*$+0+%0+)+(%./)&0+'()+'&#)G%*6()@0&(HF0#5#5*(%3)+

7+@,*)+?

J7&,*+72*(($(70+%#+$3()&#)%(3($');2*(

~ E·n

=E

86#)7#3+$3#?

N

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

φ=

∫

S

~ E·n

dS

=E

4πr2

<+0+),+&$3()&#)

q int%(6*('(0+70*7+).;0&75($3(8+2*(7+'($%&'+''(0+),+(%0#$%3+$3(9

H(3($');2*(

q int=

ρ0V9V

(%(7C#7*5($($3)(7+(%.()+'()+'&#

r87+(%.()+'()+'&#

a0#$

7#2*()(%*73+?

q int=

ρ04 3π(r

3−

a3)

J67&0+$'#7+7(8'(,+*%%?

E4πr2

=ρ0

4 3ǫ 0

π(r

3−

a3)

"#$7#2*(%(#43&($(?

E=

ρ0(r

3−

a3)

3r2

ǫ 0

"#5#0#5($3+5#%+$3()&#)5($3((%3(0+56#(%3+($'&)(00&-$)+'&+7r1

6#)7#3+$3#?

~ E=

ρ0(r

3−

a3)

3ǫ 0

r2r

•2221

r>

b0

I)#0('&($'#'(&,*+75+$()+2*(($

7+)(,&-$

EEFC()(5#%2*(7+0+),++75+0($+'+($

7+

'&%3)&4*0&-$2*()#'(++70#$'*03#)%();?

q int=

4 3πρ0(b

3−

a3)

:((%3+5+$()++67&0+$'#7+7(8'(,+*%%3($')(5#%2*(?

E4πr2

=ρ0(b

3−

a3)

3rǫ

0

E=

ρ0(b

3−

a3)

3ǫ 0

r2

O(03#)&+75($3(2*('+(P6)(%+'#0#5#?

~ E(r

)=

ρ0(b

3−

a3)

3ǫ 0

r2r

!

"#$%&'#()*+#,-.&/*#,0-12'-*#(1*)*#,#$/.-'*2#$%,1.

~ E=

0r

<a

ρ0(r

3−

a3)

3ǫ 0

r2

ra

<r

<b

ρ0(b

3−

a3)

3ǫ 0

r2

rr

>b

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>

bA*&*/*)#&*$=#2

~ E·d

~ ℓ=

Er·d

~ ℓ=

Edr6

7##$1.&.(#2.#,/*1#(-'.,#,0-12'-*$#2BC

V(r

)=−

∫

r ∞~ E·d

~ ℓ=−

∫

r ∞E

dr

=−

∫

r ∞

ρ0(b

3−

a3)

3ǫ 0

r2)d

r

D(1#92.()*$#*81'#(#5%#C

V(r

)=

ρ0(b

3−

a3)

3ǫ 0

r,

sir

>b

34*2.5%#2#&*$-.,-%,.2#,/*1#(-'.,.%(.)'$1.(-'.

r+)*()#

a<

r<

b6E*2)#@('-'<(

1#(#&*$5%#

V(r

)=−

∫

r ∞~ E·d

~ ℓ

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

r+#(,.$

12.:#-1*2'.$)#$)#'(@('1*4.$1.

b&.$,.12.:#-1*2'.)#$)#

b4.$1.

r6

V(r

)=−

∫

b ∞~ E·d

~ ℓ+−

∫

r

b

~ E·d

~ ℓ

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

r=

bC

V(b

)=−

∫

b ∞~ E·d

~ ℓ=

ρ0(b

3−

a3)

3ǫ 0

b

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

−∫

r

b

~ E·d

~ ℓ=

−∫

r

b

ρ0(r

3−

a3)

3ǫ 0

r2dr

=−

ρ0

3ǫ 0·(∫

b ∞rd

r−

a3

∫

r

b

1 r2dr)

=−

ρ0

3ǫ 0·(

(r2 2

∣ ∣ ∣ ∣

r b

)−

a3(−

1 r

∣ ∣ ∣ ∣

r b

))

=ρ0

3ǫ 0·(

b2−

r2

2+

a3(r−

b)

rb)

G

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

7##$1.&.(#2.+#,/*1#(-'.,#(

r2#$%,1.$#2

V(r

)=

ρ0

3ǫ 0

(b3−

a3

b+

b2−

r2

2+

a3(r−

b)

rb),

sia

<x

<b

E.2.,.2#9'<(

r<

a)#$-*&/*(#&*$,.12.:#-1*2'.)#$)##,'(@('1*4.$1.2#(!12.:#-1*2'.$6

H#()2#&*$5%##,/*1#(-'.,#$

V(r

)=−

∫

r ∞~ E·d

~ ℓ=−

∫

b ∞~ E·d

~ ℓ+−

∫

a

b

~ E·d

~r+−

∫

r

a

~ E·d

~ ℓ

3,$#2(%,*#,-.&/*#,0-12'-*/.2.

r<

a1#()2#&*$5%#

−∫

r

a

~ E·d

~ ℓ=

0

I)##$1.&.(#2.#,/*1#(-'.,2#$%,1.$#2C

V(r

)=

ρ0

3ǫ 0

(b3−

a3

b+

b2−

a2

2+

a3(a−

b)

ab

),si

r<

a

!

!"#$%&'()

"#$%&'()((*%&%+(,-'-'#($*-./0)-12(+&($($34&*&$')#%,05*-)/#%670(4#%64-'-0$#'(

)-'&#

r5'($%&'-'(%'(4-)/-%%08().4&-*(%4#$%+-$+(%

σ5−

σ9

'*:$40($+)((*4-,8#(*;4+)&4#%#<)(*-*=$(-

AB6>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#%+()&#),($+(F5(G-*0-)($

r=

b9A-'#

>0(*#%4&*&$')#%%#$,05*-)/#%5(%+H$4-)/-'#%4#$'($%&'-'%08().4&-*0$&B#),(8#'(,#%

'(%8)(4&-)*-%4#$'&4&#$(%'(<#)'(50+&*&E-)*#%-)/0,($+#%'(%&,(+)=-9:*4-,8#(*;4+)&4#

>0(8)#'04(4-'-4&*&$')#(%+-)H($'&)(44&@$)-'&-*5'((%+-,-$()-0+&*&E-$'#4#,#%08().4&(

/-0%%&-$-0$4&*&$')#'(*-)/#

L5)-'&#

r

φ=

∫

S

~ E·n

dS

=q i

nt

ǫ 0(1

)

A(%-))#**-,#%*-&$+(/)-*4#,#

∫

S

~ E·n

dS

=E

∫

SdS

=E

2πrL

(2)

"#,#*-'($%&'-''(4-)/-%08().4&-*(%

σ4#$%+-$+(D%(7-4('(&/0-*,-$()-8-)-8-)-(*

4&*&$')#F6*-4-)/-'(*4&*&$')#%()H%&,8*(,($+(

q int=

σV

=σ2πaL

(3)

I

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

J((,8*-E-$'#D3F5DKF($DLF6#<+($(,#%>0(

E=

σa

rǫ0

(4)

:G-*0-$'#($

r=

d+($(,#%>0(

E=

σa

dǫ 0

M7#)-%08()8#$&($'#(*4-,8#(*;4+)&4#8)#'04&'#8#)4-'-4&*&$')#%#<)((*80$+#8('&'#%(

'(<(4#$%&'()-)*-%&/0&($+(4#$./0)-4&@$1

"#,#5-,($4&#$-,#%(*4-,8#(*;4+)&4#(%)-'&-*658#)*#+-$+#(%+-)H$($*-'&)(44&@$

&$'&4-'-($*-./0)-9"#,#*#%,@'0*#%'(*#%4-,8#%8)#'04&'#%8#)-,<#%4&*&$')#%%#$

&/0-*(%($(*80$+#8('&'#6*-%4#,8#$($+(%($

y%(-$0*-)-$58#)*#+-$+#(*4-,8#(*;4+)&4#

(%+-)H($'&)(44&@$

i5+($')H,#'0*#

2E

y=

2Eco

s(π 3)6

(%'(4&)

E=

σa

dǫ 0

#*C-)-4-*40*-)*-'&B()($4&-'(8#+($4&-*($+)(*#%'#%4&*&$')#%'(<(,#%4-*40*-)(*4-,8#

(*;4+)&4#%#<)(*-*=$(->0(0$(--,<#%4&*&$')#%9"#$%&'()-)((*#)&/($($(*4($+)#'(*4&*&$')#

'(*-&E>0&()'-64#,#,0(%+)-*-./0)-1

C-)-

0<

x<

D(*4-,8#(*;4+)&4#%&(,8)((%+-)H($'&)(44&@$

i 9"#,#7(,#%G&%+#-$+()&#)?

,($+(6(*7(47#'(>0(4-'-4&*&$')#+($/-4-)/-$0*-($%0&$+()&#)&,8*&4-)H>0((*4-,8#

(*;4+)&4#>0(8)#'0E4-$($%0&$+()&#)%(-$0*#9M7#)-64-*40*-)(,#%(*4-,8#(*;4+)&4#)(%0*?

+-$+(%#<)((*(N(

x%08()8#$&($'#*#%4-,8#%8)#'04&'#%8#)4-'-4&*&$')#1

2'&3"%$4-5!.-"3!"6,-.6"3"!%$-.$.06!".78,.%!6"

!"#"$%&'()*"+',-%#."',/-01)-"'2',)20'1)"1(',-),)2(1"&'1324,"+."1,"0%20"

~ E1

=0,

si0

<x

<a

54'1%(',-),)2(1"',-%#."',/-01)-"&'-%,-4,%1%(')64%,#%2'1%-"#","7)-)#"&'2,%.%10'

%8+(''&0%#%2'1%0'2'#"&94':

~ E1

=σa

xǫ 0

i,si

x>

a

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

;2',)20'1)"1(',-),)2(1"',-%#.".1"(4-)("."1/,&'1324,":

~ E2

=0,

sid−

a<

x<

d

<"1"01",%("+-"2&)('1%2(",%=641%#"&01%(%%20'1)"1#'20'+>'1'#"&94'%.%1%42-)'10"

x94'-4#.,%

0<

x<

d−

a+,%()&0%2-)%(',-'201"(',-),)2(1"('1'-7"%

x&'13

d−

x+(''&0%

#%2'1%',-%#."',/-01)-"1'&4,0%&'1:

~ E2

=σa

(d−

x)ǫ

0i

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

'&:

~ E=

σa

(d−

x)ǫ

0i

0<

x<

a

(σa

xǫ 0

+σa

(d−

x)ǫ

0)i

a<

x<

d−

aσa

xǫ 0

id−

a<

x<

d

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

=−

∫

d

0

~ E·d

~ ℓ

=−

∫

a

0

~ E·d

~ ℓ+−

∫

d−

a

a

~ E·d

~ ℓ+−

∫

d

d−

a

~ E·d

~ ℓ

=

∫

a

0

σa

(d−

x)ǫ

0dx

+

∫

d−

a

a(σa

xǫ 0

+σa

(d−

x)ǫ

0)d

x+

∫

d

d−

a

σa

xǫ 0

i

=(−

σa ǫ 0(l

n(d−

x)|a 0

))+

(σa ǫ 0(l

n(x

)−

ln(d−

x)|d−

aa

))+

(σa ǫ 0((

ln(x

)|d d−

a))

)

=(−

σa

ǫ 0ln

(d−

a

d))

+(2

σa

ǫ 0ln

(d−

a

a))

+(−

σa

ǫ 0ln

(d−

a

d))

=2σa

ǫ 0(l

n(d−

a

a)−

ln(d−

a

d))

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

E

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

∆V

=2σa

ǫ 0ln

(d a)

!

!"#$%&'((

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

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

σ=

σ0(1−

R r)2

'(1

σ0

>02&%,1$(

r#+$%&/+1'%+$,*13*1/('*+#4*%,)+$,#$%&'(+&*',1/)(

O5

')"1'*,1/),,#3(/,1'%+#,1,#3*1/(

P&(6),&*,7,8O

P=

R95

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

$,7+$+,1),3(&(,1

P>

*"$+,-./0

')?(4*,@+),<(&&,)0'(1&%$,)+),#3(/,1'%+#3)($*'%$(3()*1+1%##($,)+$%(

aA'+).+

/(/+#q&(6),&*,7,+*1+$%&/+1'%+

dA#*,.(2*/%#%B+1$(,&/,),&*#/+$(2$%=%$%),<(&,#$%&'(

,13,4*,C(&+1%##(&A&*<+),<(&#(&3(/,1'%+#,&3)($*'%$(&3()'+$+*1(5

"#3(/,1'%+#$,*1+1%##(#(3($,<(&'+#'*#+)*/%#%B+1$(

V=

∫

kdq

r

"#=+#()$,

r#((6/,1,<(&3()3%/+.()+&D r=

√

a2+

d2

"&/+$%&/+1'%+&,<+1/%,1,'(1&/+1/,A3()#(/+1/(3*,$,&+#%)$,#+%1/,.)+#

V=

k√

a2+

d2

∫

dq

=kq

√a

2+

d2

E@()+2'(1&%$,)+),+1%##(&$,#$%&'('(1'+).+

dq5

F,#),&*#/+$(+1/,)%()A'(1&%$,)+1$(

d=

RA

a=

r&,/,1$)04*,

dV

=kdq

√r2

+R

2

GH

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

?+'+).+

dq&,#+3($,<(&,&')%6%)'(<(

dq

=σ(r

)·d

A

I(<(#+$,1&%$+$$,'+).+$,3,1$,$,#+$%&/+1'%+

r/,1$),<(&4*,'+$++1%##($,#$%&'(4*,

'(1&%$,),<(&/,1$)0%.*+#$,1&%$+$A+4*,'+$+,#,<,1/($,J#,&/++%.*+#$%&/+1'%+

r$,#

',1/)($,#$%&'(5F,,&/+<+1,)+'(1&%$,)+),'(<(

dA

,#0),+$,*1+#+<6),$,+1'@(

drD

dA

=2π

rdr

F,,&/+<+1,)+(6/,1,<(&

dq

=2π

σ0(r−

R)d

r

"#3(/,1'%+#$,3)($*'%$(3(),&/,+1%##(&,)0,1/(1',&D

dV

=2kπσ

0(r−

R)d

r√

r2+

R2

K1/,.)+1$($,&$,

r=

0@+&/+

r=

RD

V=

2kπσ

0

∫

R

0

(r−

R)d

r√

r2+

R2

=2kπσ

0(∫

R

0

r√

r2+

R2dr−

R

∫

R

0

1√

r2+

R2dr)

=2k

πσ

0((√

r2+

R2∣ ∣ ∣

R 0)−

(ln(√

r2+

R2+

r)∣ ∣ ∣

R 0))

=2kπσ

0(R√

2−

R−

Rln

(R√

2+

R)+

Rln

(R))

=2k

πσ

0(R√

2−

R−

Rln

(R(√

2+

1))

+R

ln(R

))

=2kπσ

0(R√

2−

R−

Rln

(R)−

Rln

(√2

+1))

+R

ln(R

))

=2k

πσ

0(R√

2−

R−

Rln

(√2

+1))

=2k

πσ

0R

(√2−

1−

ln(√

2+

1))

L%'(1&%$,)+<(&

√2−

1−

ln(√

2+

1)≈

1 2),&*#/+D

V≈−

kπσ

0R

M($,<(&,&')%6%),&/,),&*#/+$(,1N*1'%O1$,#+'+).+/(/+#Q

$,#$%&'(5"&/+'+).+#+3($,<(&

'+#'*#+)%1/,.)+1$(&(6),,#$%&'($,#+,P3),&%O1

dq

=2π

σ0(r−

R)d

r

Q=

∫

R

02πσ

0(r−

R)d

r

=2π

σ0((

r2 2−

Rr)

∣ ∣ ∣ ∣

R 0

)

=−

πσ

0R

2

F,,&/+<+1,)+

V≈

Q

−πσ

0R

2−

kπσ

0R

!

V≈

kQ R

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

3,/1.'+/,-./

R1&&))/5

!"/1/&)&$&,+#1&-/02/

dq&',&2/+.%/()3&2#&)-/$*#&)6-+0.-#'&07/+0/8&,+&59#0'.$&+0:/(

&)-/$*#&)6-+0.-#/*3,+/0/&,1.0&--.;,1&)&<&1&)1.'-#8'&,+.1#=/-./&)-&,+0#8-#$#)/

-/02/1&*03&>/&'*#'.+.%/(&'+/'&$#%&07+/$>.6,=/-./&)-&,+0#1&)1.'-#5

?

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

"#$ %&'()*

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

m8-/02/

q/+/1/'*#03,/-3&01/.1&/)1&)/02#

ℓ(.,.-./)$&,+&&,0&*#'#&,)/*#'.-.;,

I@%&0A230/B5C&'3&)+/,8-/&,*#0)//--.;,1&)/

20/%&1/1/)/*#'.-.;,

II(/)-/,D/,1#3,7,23)#

θ-#,#-.1#5E9/0/43&%/)#01&)/-/02/&)

'.'+&$//)-/,D/&'+&7,23)#

θF5GH*0&'&'30&'3)+/1#&,+60$.,#'1&

ℓ(m

8θ5

+$%,-./01

G)&2.$#'&)-&0#1&)/&,&02:/20/%.+/-.#,/)&,)/*#'.-.;,

ℓ$/'/>/<#1&)/*#'.-.;,.,.-./)

@%&0A230/B5I/&,&02:/*#+&,-./)&,)/*#'.-.;,J'&075

UI

=2m

gℓ+

kq2 2ℓ

I/&,&02:/*#+&,-./)&,)/*#'.-.;,JJ

UII

=2m

gℓ(

1−

cos(

θ))+

kq2

2ℓsi

n(θ

)

"#$#)/&,&02:/'&-#,'&0%/(*#1&$#'=/-&0

UI

=U

II8*#0)#+/,+#

2m

gℓ+

kq2 2ℓ

=2m

gℓ(

1−

cos(

θ))+

kq2

2ℓsi

n(θ

)

K&'*&</,1#4L

q=±

2ℓ√

mg

sin(θ

)

k(s

ecθ−

tan(θ

))

!

!"#$%&'()

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

R,&2.&(&'()1.,2%.$'*.3(1&*)%4)'(./#%-&

σ56)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#

zk2#-)%&-#,'(&7&-&(2#1&*)%4)

dq1&&77)@*)7*'7)%&-#,&7+#2&(*.)7+%#1'*.1#

*#-#8

dV

=kdq

r

A#(1&

r&,7)1.,2)(*.)1&,1&

dqB),2)

zk5

=#,2&%.#%-&(2&<+)%)*)7*'7)%&7+#2&(*.)72#2)7<.(2&4%)-#,

dV

,#$%&2#1)7),'+&%?*.&5

6#-#C.-#,&(7))@'1)(2.)D<7)*)%4)

dq7)+#1&-#,&E+%&,)%*#-#&7+%#1'*2#&(2%&7)

1&(,.1)1,'+&%?*.)71&*)%4)

σ+#%&7)%&)>'&#*'+)&,)*)%4)>'&77)-)-#,

dA

@C.-#,>'&

dA

=R

2si

n(γ

)dγdθ

F)1.,2)(*.)

r&(&,2&*),#7)#$2&(&-#,'2.7.G)(1#&72&#%&-)1&7*#,&(#8

r2=

R2+

z2−

2zR

cos(

γ)

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

dq&,

dV

=kσR

2si

n(γ

)dγdθ

√

R2+

z2−

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=

∫

2π

0

∫

π

π 2

kσR

2si

n(γ

)dγdθ

√

R2+

z2−

2zR

cos(

γ)

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

V=

2kπσR

2

∫

π

π 2

sin(γ

)√

R2+

z2−

2zR

cos(

γ)

./

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

0#(#(&!1)2&(&!"13,&4&51!!#4&(-#)-+)#()#$%"&'(#),&)#61(5#7

∫

sin

γ√

A+

Bco

s(γ)d

γ

8#-&51!&)-#54$1,&2#($#4)&

u2

=A

+B

cos(

γ)9:&&!"#5#%&(#

dγ

=2udu

−B

sin(γ

)9;#!$

∫

sin

γ√

A+

Bco

s(γ)d

γ=−

2 B

∫

1du

=−

2 Bu

<151

u=

√

A+

Bco

s(γ)7

∫

sin

γ√

A+

Bco

s(γ)d

γ=−

2 B

√

A+

Bco

s(γ)

0#(#&)-#)-+)1,&*1"&%-$#)"&%&51!=+&

A=

R2

+z2>

B=−

2zR9?&&5*)@#%,1&%)#

$%"&'(#)7

∫

sin

γ√

R2+

z2−

2zR

cos(

γ)d

γ=

1 zR

√

R2+

z2−

2R

zco

s(γ)

AB1(#3*1,&51!-#)-+)#(&)$%"&'(#)!$%,$C-+)"#,&!7

V(z

)=

2kπσR

2

zR

(√

R2+

z2−

2R

zco

s(γ)∣ ∣ ∣

π π 2

)=

2kπσR

z(z

+R−

√

R2+

z2)

!AB1(#=+&"&%&51!&)*1"&%-$#)&)D-"($-1&%

zk3&)-#5*1&)D-"($-1!&*+&,&-#)-+)#(-1517

~ E=−∇

V

E&51!=+&&)*1"&%-$#)&!"#!1)1&%6+%-$1%,&)#2#($#4)&

z>*1()1"#%"1"&%,(&51!=+&&)

-#5*1&)&-"($-1&!7

~ E=−

dV dz

k

:&($2#51!

V(z

)7 dV dz

=2k

πσR

d dz(1 z(z

+R−

√

R2+

z2))

=2k

πσ(−

1

z2

(z+

R−

√

R2+

z2)+

1 z(1

+1 2

1√

R2+

z22z))

=2k

πσ(−

1 z−

R z−√

R2+

z2

z2

+1 z

+1

√R

2+

z2)

=−

2kπσR

2(√

R2+

z2−

R

z2√

R2+

z2

)

:&&!"#5#%&(#&)-#5*1&)&-"($-1&%&)*+%"1

zk!&(F7

!

~ E(z

)=

2kπσR

2(√

R2+

z2−

R

z2√

R2+

z2

)k

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

z=

034./#$5*-6%7,%1#$%&*8#-8*-/%&*

O0*/0%8%$%'(#-*$%&98*)8()*-#):*)%-1#)8*$,%#);80-.8%3<-.$#-%:#$%&'(#)*#+,-#&.%/

1#)8*$,%#)#80-.8%&#,(#1#-##&8-.5.-8%$%=

2kπσR

2(1−

R√

z2+

R2

z2

)

>(#6%70%$*-#$%&#)).$.0#8(*/1%

z→

01#)*?(/8.%/

E(z

)k@=

lım

z→

0E

(z)

=2kπσR

2lım

z→

0(1−

R√

z2+

R2

z2

)

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

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

2kπσR

2lım

z→

0(1−

R√

z2+

R2

z2

)=

2kπσR

2lım

z→

0((1

−R

√z2+

R2)′

(z2)′

)

=2k

πσR

2lım

z→

0(R

(z2+

R2)−

3 2

2z

)

=2k

πσR

2·

1

2R

2

=kπσ

F##&0*$*/#-*

~ E(0

)=

kπσk

G

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

!"#$%&'(

!"#$%&!'()

!"#$%&'()

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

r 1,r

2.,+*/+'

q 1,q

2'!0+*+%+'0&*-$+/*+$%#'1

"+$,#+

d>

>r 1

,r22 #+34+''!,&$!,"+$+"*+5)'%!-$,+46!,&$%-,"&*7%!'0*!,#+46!89-!

'#*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+!'"?!$-$+*!/#:$+,&"+%+%!6!'0+,#&80&%!3&'"&3+*,&3&0-$"&%!

*!(!*!$,#+%!60&"!$,#+6!6#$<$#"&!#/-+6+*!60&"!$,#+6+,!*&+66@8!'%!,#*8V

(+∞

)=

02A'@8

!60&"!$,#+6'&4*!6+''-0!*<,#!'%!6+'!'(!*+',&$%-,"&*+'!'

V1

=kq 1 r 1

,V2

=kq 2 r 2

%&$%!B!3&''-0-!'"&9-!!60&"!$,#+6%!-$+!'(!*+$&!'+(!,"+%&0&*!6%!6+&"*+7&3?'4#!$

!'+(!,"+%&%!(&*3+%!'0*!,#+46!;8%+%&9-!!'"?$3-.3-.'!0+*+%+8!'%!,#*8d

>>

r 1,r

28

."+30&,&!6,+30&%!-$+!'(!*+*!%#'"*#4-.!6+,+*/+%!6+&"*+2C&*,&$'!*5+,#:$%!,+*/+8

'#(q

1) f

,(q 2

) f'&$6+',+*/+'!$,+%+!'(!*+-$+5!=+6,+$=+%&!6!9-#6#4*#&8"!$!3&'9-!

q 1+

q 2=

(q1) f

+(q

2) 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

=V

2

k(q

1) f

r 1=

k(q

2) f

r 24π(r

1)2

(σ1) f

r 1=

4π(r

2)2

(σ2) f

r 2

=⇒

(σ2) f

(σ1) f

=r 1 r 2

>!!'"&5!3&'9-!!$/!$!*+686+'*!/#&$!'!$6+'-0!*<,#!%!-$,&$%-,"&*,&$3!$&*

*+%#&%!,-*5+"-*+70-$"+';,&$,!$"*+$-$+3+.&*%!$'#%+%'-0!*<,#+6%!,+*/+80&*6&,-+6!6

EF

EE

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

,+30&!6),"*#,&,!*,+%!!66+'7!$'-!G"!*#&*;!'3?'(-!*"!9-!!$*!/#&$!',&$3!$&**+%#&

%!,-*5+"-*+2

H'+$%&6+*!6+,#:$!$,&$"*+%+!$6+!,-+,#:$%!,&$'!*5+,#:$%!,+*/+8&4"!$!3&'

(q1) f

+(q

2) f

=q 1

+q 2

4π(r

1)2

(σ1) f

+4π

(r2)2

(σ2) f

=q 1

+q 2

4πr 1

(σ1) f

(r1+

r 2)

=q 1

+q 2

=⇒

(σ1) f

=1

4πr 1

q 1+

q 2r 1

+r 2

(σ2) f

=1

4πr 2

q 1+

q 2r 1

+r 2

!

!"#$%&'()

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

a<

b/+(,(,0#/(,%+&1($

V12

V23

'($0(%/+4&5(,/(6

'*7&1%.1(1&%&'8&-(%&-&($)('&6

#*9:#'&/'&(5#$-($-((1+,;,+/#.,&%&'8&

Q-+$/'+<.+-&.,+)#'5(5(,/($#<'(1&$.0(';%+(

-(.,&($)('&-('&-+#

c>

b3%#,%*,/'+%&%#,

1&$&,/('+#'($6=7.>,/#/'&<&?#($,(%($&'+#

'(&1+@&'0&'&%#1#%&'-+%:&%&'8&A

+"$,-./01

"(;,+5#$%#5#

Q12

Q21&$%&'8&$-(1&($)('&%#,0#/(,%+&1V

22

V23'($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<

b6B&%&'8&(,%(''&-&0#'($/&$.0(';%+(($1&&0#'/&-&0#'1&

%&$%&'&-('&-+#

a3($-(%+'3Q

10#'1#/&,/#C

∫

S

~ E·n

dS

=Q

1

ǫ 0

D('#

∫

S~ E·n

dS

=E

4πr26"(($/&5&,('&(1%&50#(1(%/'+%#EF.(2&:(5#$4+$/#F.(/+(,(

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

~ E=

Q1

4πǫ 0

r2r

9:#'&31&-+)('(,%+&-(0#/(,%+&1(,/'(&5<&$$.0('%+($($C

V2−

V1

=−

∫

a

b

Q1

4πǫ 0

r2dr

=Q

1

4πǫ 0

(1 b−

1 a)

7#,($/#30#-(5#$#</(,('1&%&'8&

Q1C

Q1

=4π

ǫ 0(V

2−

V1)

ab

b−

a

9:#'&3%&1%.1&'(5#$(1%&50#(1(%/'+%#&.,&-+$/&,%+&

r>

b./+1+@&,-#1(2-(H&.$$6I1(8+5#$

.,&%#5#$.0(';%+(8&.$$+&,&.,&%&$%&'&-('&-+#

r>

b20'#%(-(5#$-(+8.&1)#'5&6B#

F.(-(<(5#$%#,$+-('&'(,($/(%&$#3($F.(1&%&8&(,%(''&-&0#'($/&$.0(';%+(($

Q1+

Q26

"(($/&5&,('&4(5#$F.(C

E4πr2

=Q

1+

Q2

ǫ 0

!J

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

~ E=

Q1+

Q2

4πǫ 0

r2

7#,($/#30#-(5#$%&%.1&'(10#/(,%+&1(,

bC

V2

=−

∫

b ∞

Q1+

Q2

4πǫ 0

r2dr

=Q

1+

Q2

4πǫ 0

b

"(($/&5&,('&31&%&'8&-(%&$%&'&(K/('+#'$('>C

Q2

=4π

ǫ 0b(

V1+

(V2−

V1)

ab

a−

b)

#*D&'&%&1%.1&'(1/'&<&?#/#/&13%#,$+-('&'(5#$F.(1&$%&'8&$F.(%#50#,(,1&%&$%&'&-(

'&-+#

c>

b1&$($/&5#$/'&2(,-#-($-((1+,;,+/#:&$/&.,&-+$/&,+%&

c671&'&5(,/(3(1/'&<&?#

,(%($&'+#0&'&1&%&-&%&'8&

dq%#,/(,+-&(,1&%&$%&'&$('&(15+$5#6I$/(/'&<&?#$(%&1%.1&

%#5#

dW

=dqV

(c)

"(($/&5&,('&3(1/'&<&?#/#/&1$('>$+501(5(,/(

W=

QV

(c)

6B#F.(/(,(5#$F.(:&%('($%&1%.1&'(10#/(,%+&1(,

r=

cF.(($-+'(%/&5(,/(

V(c

)=

Q1+

Q2

4πǫ 0

c

9$+3

W=

Q(Q

1+

Q2)

4πǫ 0

!

!"#$%&'()

"#$%&'%&'#'%$%($#)&'%&*+,-&.*#'%&*+&$.#)&'%/-0,$12','%3*%$$+#',)*%

a4b0

c4#$%3$&.)5

6'/$+.$4%)$+,*

a<

b<

c78+)&)'1/$+.$1'&'%&'#')+.$#)*#$%.',$%&'#2','41',$1/$,)*3*%$$

-+'&'#2'.*.'1+$2'.)6'−

Q01'$9.$#+',$&'#2'.*.'13*%).)6'

+Q7

*'+:+&-$+.#$$13*.$+&)'1$1$&.#)&*$+&','-+',$1'%&'%&'#'%&*+,-&.*#'%7

*#+;)1'%&'%&'#'%)+.$#)*#0$9.$#)*#%*+&*+$&.','%/$,)'+.$-+'1'/<#$=-$$%.'')%1',*

'13'%'#3*#1'&'%&'#'&$+.#'17>?-'1$%'@*#'$13*.$+&)'1$1$&.#)&*,$&','-+',$1'%&'%5

&'#'%A>?-'1$%1'&'%&'#'$+&','-+',$1'%&'%&'#'%A

,"$-./"0

B

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

!"#$%&'(1

C+'&'%&'#'$%(D#)&'&*+,-&.*#',$#',)*

a0$%3$%*#δ≫

a&*+.)$+$&'#2'+$.'

Q7;$,)%.#)<-0$

-+'&'#2'

q$+$16*1-/$+)+.$#)*#,$1&'%&'#*+,$#',)*

aE-+')%1'+.$$+1'3'#.$)+.$#)*#,$1

&'%&'#*+)/3),$=-$$%.',$+%),',,$&'#2'%$3'%$'1&*+,-&.*#F7G*%,)&$+=-$$1&'/3*

$1D&.#)&*$+$1)+.$#)*#,$1&'%&'#*+$%.',',*3*#

~ E=

K(r a)4

r

H*+,$

K$%-+&*+%.'+.$3*#,$.$#/)+'#0

r$%$16$&.*#-+).'#)*#',)'17;$3),$$+&*+.#'#I

'+J',$+%),',,$&'#2'

ρ(r

)$+$16*1-/$+)+.$#)*#,$1&'%&'#*+7

#+J'%,$+%),',$%,$&'#2'%%-3$#K&)'1$%'1)+.$#)*#0'1$9.$#)*#,$1&'%&'#*+7

.+:13*.$+&)'1$1$&.#*%.L.)&*$+.*,*$1$%3'&)*7

,"$-./203

'+J'1$0,$2'-%%%$3-$,$$%&#)<)#$+%-(*#/'/'%&*/-+&*/*I

φ=

∫

S

~ E·n

dS

=q i

nt

ǫ 0

;)+$/<'#2*4%$3-$,$-.1)M'#.'/<)$+1'(*#/',)($#$+&)'1,$1'1$0,$N'-%%=-$$%.'<1$&$I

~ ∇·~ E

=p(r

)

ǫ 0

?*/*6$/*%4$1&'/3*$1$&.#)&*$%.'%*1*$+(-+&)*+,$

r4,$$%.'/'+$#'.$+$/*%

~ ∇·~ E

=1 r2

∂ ∂r(r

2E

(r))

=1 r2

∂ ∂r(r

2K

(r a)4

)

=1 r2

∂ ∂r(K

r6 a4

)

=6K

r3

a4

H$$%.'/'+$#'41',$+%),',,$&'#2'$+$1)+.$#)*#,$1&'%&'#*+%$#LI

ρ(r

)=

6ǫ0K

r3

a4

!

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

K23&*)+$,45)0&1&%6&-$-&0&0+&1)/&(&)/)0

./-)%.$%()01&,1&%$/),

q'7$%0$-&/-$,)8&&15+70.%0&%)0&1.$/9

q=

∫

ρ(r

)dV

=4π

∫

a

0

6ǫ 0

Kr3

a4

4πr2

dr

=24

ǫ 0K

a4

r6 6

∣ ∣ ∣ ∣

a 0

)

=4π

a2ǫ 0

K

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

K=

q

4πǫ 0

a2

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

ρ(r

)=

3 2

qr3

πa

6

!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&')01$/(51-$%./(51)5/&1&%6&−

q)/)0./-)%.$%()01$/(51-$%2:)),-&+&/)%&'

0&()/,.(&(()1&%6&)/0&,57)%=1.)./-)%.$%()01$/(51-$%),9

σin

t=−

q

4πa

2

>$/),-$?5-.0.@&/($0&1$/,)%8&1.$/()0&1&%6&8)+$,45))/0&,57)%=1.))A-)%.$%,)-)/(%&

5/&1&%6&

Q+

q2>$/),-)%),50-&($7$()+$,1&0150&%0&()/,.(&(()1&%6&,57)%=1.&0)A-)%.$%9

σex

t=

Q+

q

4πa

2

:$/()#)+$,(),7%)1.&($

δ2"!B&%&,&*)%7$-)/1.&0)/-$($)0),7&1.$()*)+$,,&*)%)01&+7$)0C1-%.1$2B&%&0&%)6.$/

./-)%.$%?&-)/)+$,)01&+7$)0)1-%.1$9

~ Ein

t=

q

4πǫ 0

a2(r a)4

r,para

r<

a

D/)0)A-)%.$%()01$/(51-$%-)/)+$,-)/)+$,45))01&+7$)0)1-%.1$),,.+70)+)/-)9

~ Eex

t=

Q+

q

4πǫ 0

r2r,

sir

>a

",.)07$-)/1.&07&%&5/&%)6.$/)/45)

r>

a),9

V(r

)=−

∫

~ Eex

td~r

=−

Q+

q

4πǫ 0

∫

r ∞

1 r2dr

=Q

+q

4πǫ 0

r

"#$%&'7&%&0&%)6.$./-)%.$%Er

<aF)07$-)/1.&00$1&0150&+$,1$+$9

G

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

V=−

∫

~ E·d

~r=−

∫

b ∞~ E

ext·d

~r+−

∫

r

b

~ Ein

t·d

~r,

sir

>a

V(r

)=

−∫

~ E·d

~r

=−

∫

b ∞~ E

ext·d

~r+−

∫

r

b

~ Ein

t·d

~r

=Q

+q

4πǫ 0

a−

q

20πa

6ǫ 0

(r5−

a5)

",.',)-.)/)45)

V(r

)=

Q+

q

4πǫ 0

a−

q

20πa

6ǫ 0

(r5−

a5),

sir

<a

!

!"#$%&'()

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

ρ(r

),#$5&''$)*,'0%.#$

5&''1,#/.1,-.2&%.30)-*/&%''1,#/#/*)*-

V(r

)=

qe−

λr

4πr2

ǫ 0

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&'

~ E=−∇

V

A*/'(*1'0%*0,-#-=B%.$('0,''$%#()*'$+%,-.%*C

~ E=−

∂V ∂r

r=

q

4πǫ 0

r2e−

λr(1

+λr)

r

D1,'-'1&$,#/*0*1)'-(.,''0%*0,-#-1.0/.E%&$,#/$#/'01./#//'%#-4#8D1,*1'@#%'&,.$.<#0/*

$#=*-(#/.='-'0%.#$/'$#$':/'F#&11C

~ ∇·~ E

=ρ ǫ 0

9'0'(*15&''$%#()*'1-#/.#$:)*-$*,#0,*

~ E=

Er7

/''1,#(#0'-#

~ ∇·~ E

=1 r2

∂ ∂r(r

2E

(r))

=1

4πǫ 0

q r2(−

λe−

λr(1

+λr)

+λe−

λr)

=−

1

4πǫ 0

q rλ

2e−

λr

A*-$*,#0,*

ρ(r

)=−

1 4π

q rλ

2e−

λr

G

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