L z = m l ~
L 2 = l ( l + 1 ) ~ 2
L = p
l ( l + 1 ) ~ l m l l
l = 0 , 1 , 2 , ... inteiro
”rotação quântica”
inteiro
Aula 3
Momento de dipolo magnético Espira num campo magnético
Experiência de Stern-Gerlach
Momento de dipolo magnético
regra da mão direita
~
µ = I ~ A
carga positiva
Espira de corrente
A
ˆ n
A = A ˆ ~ n
Momento magnético e momento angular
v
r m
e
corrente = carga /tempo
i = ev 2⇡r
A
A = ⇡ r 2
µ = i A = e v
2 ⇡ r ⇡ r 2 = e v r 2 L = m v r
µ
L = e 2 m v = 2 ⇡ r
T
1
T = v
2 ⇡ r
µ
L = e 2 m
~
~ µ b = e ~
2 m magneton de Bohr
g l = 1 fator g orbital
µ
L = g l µ b
~
~
µ = g l µ b
~ L ~
carga positiva
~
µ = g l µ b
~ L ~
carga negativa
(eletron)
Momento magnético do elétron é quantizado !
µ = g l µ b
~ L = g l µ b
~
p l (l + 1) ~ = g l µ b p
l (l + 1)
l m l l
l = 0 , 1 , 2 , ... inteiro
inteiro
µ z = g l µ b
~ L z = g l µ b
~ m l ~ = g l µ b m l
<latexit sha1_base64="lgHnUPjajQbbzXRAmci/7oEQ6Nk=">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</latexit>
L = p
l ( l + 1 ) ~
Espira é um dipolo magnético
Espira gera campo magnético semelhante ao de um imã :
Dipolo magnético !
?
Força magnética sobre uma carga
em movimento
Força magnética sobre uma corrente
F = q ~v ~ ⇥ ~ B
F = I ~ ~ L ⇥ ~ B
Espira num campo magnético uniforme
(~ L no sentido da corrente)
Espira sofre um torque !
~
µ = I ~ A µ = I a b
F = I ~ ~ L ⇥ ~ B
F = I b B
B ~
↵
↵
✓
~ ✓
F
d~l
1 2
Trabalho da força F quando a espira vai de 1 a 2. :
dW = ~ F . d~l
= F dl cos↵
~ µ
a/2
= F a
2 d↵ cos↵
W =
Z ↵ 0
F a
2 cos ↵ d↵
W = F a
2 sen ↵ W = F a
2 cos ✓
F ~
l = a
2 ↵
W = F a
2 cos ✓
W = F a cos ✓
Somando a contribuição da barra inferior (3) :
F = I b B W = I a b B cos ✓ µ = I a b
W = µ B cos ✓
W = ~µ . ~ B W = U
U = ~ µ . ~ B U =
<latexit sha1_base64="C2XQOuvaF78vDJ6ocDm944oz73c=">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</latexit>µ B cos ✓
Espira tenta se alinhar como uma bússola !
~
µ ~ µ
B ~ B ~
mais energia ! menos energia !
Energia potencial de orientação : U = ~ µ . ~ B
U =
<latexit sha1_base64="C2XQOuvaF78vDJ6ocDm944oz73c=">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</latexit>µ B cos ✓
?
Espira num campo magnético não uniforme
não uniforme não uniforme
Espira num campo magnético não uniforme
B ~ B ~
q q
~v
~v
F ~ F ~
q q
F ~ F ~
q q
F ~
F ~
Campo magnético não uniforme
U = ~ µ . ~ B = µ z B z F = ~ r U = ~
ˆi @
@x + ˆj @
@y + ˆ k @
@z U
F = µ ~ z @ B z
@z k ˆ
F ~
B = B ~ z k ˆ µ = µ ~ z k ˆ
energia potencial mínima
~ µ B ~
F = g l µ b L z
~
@ B z
@z
µ z = g l µ b
~ L z
Mecânica clássica :
L z assume
valores contínuos !
Mecânica quântica :
L z = m l ~
l m l l
l = 0 , 1 , 2 , ... inteiro
inteiro
F assume
valores discretos F assume
valores contínuos
F = g l µ b L z
~
@ B z
@z
F = g l µ b m l @ B z
@z
?
Experiência de Stern-Gerlach (1922)
Medida do momento magnético de átomos de prata
átomo de prata
elétron
L = 0
Se o elétron tiver momento angular zero:
l = 0
<latexit sha1_base64="/02QXhsHZuT/EoG1o/uafY4ieAM=">AAACyHicjVHLSsNAFD2Nr1pfVZdugkVwVRIVdCMU3YirCqYt1CLJdFqHpklIJkopbvwBt/pl4h/oX3hnnIJaRCckOXPuPWfm3hskocik47wWrJnZufmF4mJpaXllda28vtHI4jxl3GNxGKetwM94KCLuSSFD3kpS7g+DkDeDwamKN295mok4upSjhHeGfj8SPcF8SZQX2se2c12uOFVHL3sauAZUYFY9Lr/gCl3EYMgxBEcESTiEj4yeNlw4SIjrYExcSkjoOMc9SqTNKYtThk/sgL592rUNG9FeeWZazeiUkN6UlDZ2SBNTXkpYnWbreK6dFfub91h7qruN6B8YryGxEjfE/qWbZP5Xp2qR6OFI1yCopkQzqjpmXHLdFXVz+0tVkhwS4hTuUjwlzLRy0mdbazJdu+qtr+NvOlOxas9Mbo53dUsasPtznNOgsVd196t7FweV2okZdRFb2MYuzfMQNZyhDo+8BR7xhGfr3EqsO2v0mWoVjGYT35b18AGEnpBK</latexit>m
<latexit sha1_base64="PfA2cC/oQWL/tCTyZ/cWlSV24RQ=">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</latexit>l = 0 F = g l µ b m l @ B z
@z F = 0
<latexit sha1_base64="QOA4KjgYg7jGEVoW04nCE7z1Plg=">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</latexit>Se o elétron tiver momento angular “um” :
m
<latexit sha1_base64="PfA2cC/oQWL/tCTyZ/cWlSV24RQ=">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</latexit>l = 0 F = g l µ b m l @ B z
@z F = 0
<latexit sha1_base64="QOA4KjgYg7jGEVoW04nCE7z1Plg=">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</latexit>l = 1
<latexit sha1_base64="HmXTmtMw4ImJO+Wtw4k7J8ZC7vw=">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</latexit>m
<latexit sha1_base64="UviE2sLmpGFOMz7cOQX6X7LEGPo=">AAACynicjVHLSsNAFD2Nr1pfVZdugkVwVZIq6EYounHhooK1hVpKkk5r6OTBZCKU4s4fcKsfJv6B/oV3ximoRXRCkjPnnnNn7r1+ysNMOs5rwZqbX1hcKi6XVlbX1jfKm1vXWZKLgDWDhCei7XsZ42HMmjKUnLVTwbzI56zlj85UvHXHRBYm8ZUcp6wbecM4HISBJ4lqRT1un9hur1xxqo5e9ixwDajArEZSfsEN+kgQIEcEhhiSMIeHjJ4OXDhIietiQpwgFOo4wz1K5M1JxUjhETui75B2HcPGtFc5M+0O6BROryCnjT3yJKQThNVpto7nOrNif8s90TnV3cb0902uiFiJW2L/8k2V//WpWiQGONY1hFRTqhlVXWCy5Lor6ub2l6okZUiJU7hPcUE40M5pn23tyXTtqreejr9ppWLVPjDaHO/qljRg9+c4Z8F1reoeVGuXh5X6qRl1ETvYxT7N8wh1nKOBpq7yEU94ti4sYY2tyafUKhjPNr4t6+EDu5eRKw==</latexit>l = 1 m l = 1
<latexit sha1_base64="Ha+pJpVPJu/B+ySop4zu8QIunYE=">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</latexit>
F = g l µ b @B z
@z
<latexit sha1_base64="U1LJD/HlV39CUZpL43kiGGrz9uA=">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</latexit>
F = +g l µ b @B z
@z
<latexit sha1_base64="wMievUn6p+Nr3UtOlZcYsP97yAQ=">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</latexit>
