Ejercicios Resueltos Electricidad y Magnetismo (Garrido - Narrias)

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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

G-5"# 6)- -, ,)'%& '-"5E+&.$" -# )*% $.&$)*9-&-*$.% @ #) &%2." -# R =

√1 2

·a

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!

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T~ = −T sin(θ)ˆi + 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

! ;', $'&),1%*# %01'$3#$ A'+#& 1'0'+#& 5)' q =

mg tan(θ) B E

C#+# θ ≪ 1 1'0'+#& 5)' tan(θ) ≈ θB D#$ ,# 1%01#@ (#*'+#& '&/$363$ ,% /%$E% '0 -)0/340 *', 13'+(# /#+#7

q(t) =

mgθ(t) E

;'$3A%+#& /#0 $'&('/1# %, 13'+(# #61'0'+#&7

dq mg dθ = dt E dt F)'E#@ /#+#

dq dt

= α@ *'&('G%0*#

dθ dt

$'&),1%7

αE dθ = dt mg

!

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(1)

! "#$%& '(# )%*#$%& +#,-./%0-+ #, ,-+1% ℓ .%0 ,-& */&2-0./- d/2 #0 3(0./40 *#, 501(,% θ /0*/.-*% #0 ,- 61(+- -02#+/%+ )%+ ,- +#,-./40 2+/1%0%$72+/.-8

d/2 = cos(θ) ℓ =⇒ ℓ = (d/2) cos(θ) 9%$% &# )+%*(.# (0 )#'(#:% *#&),-;-$/#02% *# ,- .-+1- *# )+(# A?-,(-0*% #0 B C &# 2/#0# '(#8

−16kqQx F~ = (2) d3 D%+ ,- )+/$#+- ,#@ *# 0#E2%0 &# 2/#0# '(# F~ = m~a> A0 #&2- &/2(-./40 ,- )-+2F.(,- &# $(#?# 2 &% I- 3+#.(#0./- -01(,-+ &#+5 #02%0.#&8 r kqQ w=4 md3 A, )#+/%*% *# %&./,-./40 &#+5 #02%0.#& =⇒

π 2π = T = w 2

s

md3 kqQ

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!

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F+ 0#62#$($.( ($ y '(* 0+62# (*70.)&0# %()? Ey = E cos(θ)@ 9* 6#'-*# '(* 0+62# (*70.)&0#B E B (% 2#) '(E$&0&1$3

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

;'3$ *4%96 '$.& $, 9.6#0*+#6 96. *4#4 *4.84 ,43$.4,1 5

~ =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

)

d2 − 3 2qk (1 − (1 + ) 2) r2 r2 ;5 8$5$.4,( ,4 )05*+>5 (1 + x)α '$ 90$#$ 49.6F+%4. 96. 34A,6.( *045#6 x ≈ 0 4E =

(1 + x)α = 1 + αx B6%6 d ≪ r( '$ 3+$5$ /0$ dr ≈ 0 A 96. ,6 34536 ( dr )2 ≈ 01 5 6:3$5+#4 94.4 $, *4%96( '$ 3+$5$ /0$E 2 ~ = 3qkd ˆj E r4

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!

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

dE =

kdq R2

6'* !8-(! (# $! (#*&/(!( (# %!"3! λ2 .'(#+'& #&%"/:/" dq = λ2 ds7 ('*(# ds #& -* .#,-#?' )"'A' (#$ !"%'0 1&)# )"'A' (# !"%' $' .'(#+'& #&%"/:/" #* =-*%/'* (#$ !*3-$' ,-# $' "#%'""#5 ds = Rdθ

!

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

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!"# $%&

~ x3 = kλ2 ˆi 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)ˆi = ( )i 2 x 2R

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

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

!

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"#$%#&'()* λ4 +

2 λ4 = (2λ1 + 6λ2 ) 3 ,##-%.'/'()* .*$ 0'.*1#$ )# λ1 2 λ2 3 λ4 )#4(5)*$ '(6#15*1-#(6#2 #(7*(61'-*$ #. 0'.*1 )# .' 7'18' 9:$7')'+ 12 2 q ′ = q(1 + ) 3 π ;5 7*($5)#1'-*$ π ≈ 3 +

q′ =

10 q 3

!

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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

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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

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dF~21 =⇒ dx2

=

=

=

=

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Z

+∞

1

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!

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~ = dE

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

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

~ = dE

kλdx(−xˆi + dˆj) 3

(x2 + d2 ) 2

E$0(4)+$'# '(%'( −∞ + +∞B

~ = E

Z

+∞

−∞

kλdx(−xˆi + dˆj) (x2

+

3

d2 ) 2

= kλ((−

Z

+∞

−∞

x (x2

+

Z ˆ ) i + ( 3

d2 ) 2

+∞

−∞

d (x2

3

+ d2 ) 2

)ˆj)

! "# $%&'()& f (x) =

x 3

(x2 +d2 ) 2

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-#. 20,# 5#10. 67 80. 02.0 1#409 1# $%&'()& g(x) =

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d 3

(x2 +d2 ) 2

4*+4* −∞ # +∞ +*.: *1 40;1* 4* 1# (&2*3.#1 4* 3 4*+4* 6 # +∞?

Z

+∞

g(x)dx = 2

Z

+∞

g(x)dx

0

−∞

80. 10 2#&209 1# *=-.*+()& -#.# *1 '#,-0 +* .*4%'* #

~ = 2kdλ E

Z

+∞

(x2

0

@* 2(*&* A%* 80. 10 2#&20

R

1 3

(x2 +d2 ) 2

=

1 +

3

d2 ) 2

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 ·

2kλ 2kλ2 dθ = R + R sin(θ) 1 + sin(θ)

M&2*3.#&40 4*+4* θ = 0 # θ = π 0;2*&*,0+ 1# $%*.G# 202#1 +0;.* *1 #1#,;.* +*,('(.'%1#.7 8#.# *+20 %2(1(G#,0+ *1 C*'C0 4* A%*?

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

!

Z

1 sin(θ) − 1 = 1 + sin(θ) cos(θ)

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

F = 4kλ2

!"#$%&' ( !" #! $%&'' !"#$%&' () !" #"$%" &'!('") q > 0 *+(" $,-*"-" &,$ '!" +'&*$.#/* #*$$"-" 0,$1"-" &,$ '! 1"!(, #2!/#, -* $"-/, R 3 ")('$" H 4 3 '!" +'&*$.#/* +*1/*+05$/#" #,!#5!($/#" #,! )" #"$%"4 +*%6! +* ,7+*$8" *! )" .%'$"9 :")#')* *) ;''* ))"1"1,+ DE +*$B /%'") " )" +'1" -* ),+ ;''* &"+*! &,$ )" +*1/*+0*$" C>'* ))"1"1,+ Se E 3 ),+ ;''* &"+*! &,$ *) 1"!(, #2!/#,C>'* ))"1"1,+ Sc E9 F+(, +* &'*-* *+#$/7/$ #,1,@ φ = φe + φc =⇒ φc = φ + φe

(2)

G* *+(" 1"!*$"4 *!#,!($"!-, *) 8"),$ -* φe $*+,)8*1,+ *) &$,7)*1" /!1*-/"("1*!(*9 φe +* #")#')" #,1,@ φe =

Z

Se

~ · dS ~ E

F) #"1&, *)5#($/#, &$,-'#/-, &,$ )" #"$%" q *+ E = 4πǫ1 Rq rˆ9 G,!-* rˆ *+ *) 8*#(,$ '!/("$/, *! -/$*##/2! $"-/")9 F) 8*#(,$ dS~ ("17/5! *+(" *! -/$*##/2! $"-/")4 3" >'* dS~ = nˆ dS 3 nˆ *+ &"$")*), " rˆ #,1, 1'*+($" )" .%'$"@ 0

HI

2

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

!

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

~ · dS ~= E

1 q 1 q rˆ · n ˆ dS = dS 2 4πǫ0 R 4πǫ0 R2

.(%#/*0()& $# &1%2#(# #3 4-5& 60*0 30 $#72#$8#*09 Z Z Z 1 q 1 q ~ ~ dS φe = E · dS = dS = 2 4πǫ0 R2 Se Se Se 4πǫ0 R R R :0 2(%#/*03 Se dS ;&**#$6&()# 03 +*#0 )# 30 $#72#$8#*0< =(%&(;#$ Se dS = 2πR2 < "# #$%0 70(#*0 #3 4-5& $#*+9

φe =

1 q q · 2πR2 = 2 4πǫ0 R 2ǫ0

(3)

"# >?@'>A@ B > @ $# %2#(# C(037#(%# ,-#9

φc =

q 2ǫ0

!"#$%&' ()

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*"$+,-./0

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!

Qint ~ ·n E ˆ dS = ǫ0 Ω

I

!" # Qint #$ %& '&()& #"'#((& & *!( %& $+*#(,'-# Ω. /$0& 1!(2& -"0#)(&% # %& 3#4 # 5&+$$ $# *+# # #6*(#$&( #" 1!(2& -1#(#"'-&% +$&" ! #% 0#!(#2& # %& -7#()#"'-& 8%! '+&% $# #9& &% %#'0!(:; !+# #% '&2*! #%?'0(-'! $#& "+%! #"0(! #% '-%-" (!. +# (2πR)hσ σ ~ ·n E ˆ dS = E(r)(2πr)h = =⇒ E(r) = ǫ ǫ 0 0 Ω

A!( 0&"0!;

I

~ E(r) =

J!0# >+# #% $&%0! # -$'!"0-"+- & ,'-#: #$ + ǫσ0 .



σ ǫ0

~0


R r

  R r

r '+ #L3.)1:

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! """# 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& 08

*"$+,-./0

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

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

!

"#$%&$#'( ($ %#)*+ ($,%-'.%+ (/ 0+1 '(2.+/(13 "&#/0+ 0 < x < a 4 x > a5

• 0& 5&-& #2 -,5 ',%,5 -,3#%,-#5 4 5#% a ǫ0

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~ σ = σR22 rˆ> JA'*( )/=/6'~ ρ = ρx ˆi ; E L: 0 < x < aB "= /-)( */@3I= #'- 2(60'- /#,2)*32'- -'=B E ǫ0 ǫ0 r &$ 40$%)0>1,0.(#$ 4# ,&)2& $#)?7

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

!

~ = E

 ρx   ( ǫ0 −   ( ρa + ǫ0

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

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 > r1 , r2 4 "% -56-) )# .(&#.$-& - $,-7+) '# /& .-68# .(&'/.$(, 9'#)2,#.%-68#: ;/# )%,7# ).%#) '# 8-) #)*#,-) .(&'/.$(,-) #)

V1 = k

q1 q2 , V2 = k r1 r2

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