Eletromagnetismo

Física

•

UFAL

Wallace Calixto

08/10/2018

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Magnetismo
Magnetite
Magnesia
Campo magnético
Não há monopolos (“cargas”...) magnéticos!
Campo magnético
Campo magnético
William Gilbert, De Magnete, 1600
Campo magnético
Bússola chinesa?
(dinastia Han: 206 A.C.-220 D.C.)
4
Se o cilindro fosse muito longo o campo no interior era aproximadamente uniforme e no
exterior era aproximadamente nulo. As linhas do campo fecham-se pelo lado de fora do
“cilindro” de correntes. De resto, as linhas de força do campo magnético fecham-se
sempre! Repare-se que as linhas de força do campo criado pelo solenóide se
assemelham às criadas pela barra magnética (ver Fig. 22.3).
Fizemos uma apresentação e descrição muito qualitativa dos campos produzidos
por correntes. Faremos uma descrição quantitativa nas próximas aulas.
A Terra é, ela própria, um íman gigantesco! O campo magnético terrestre
deve-se a correntes no interior do nosso planeta. Estas correntes eléctricas produzem um
campo magnético semelhante ao que seria produzido por um íman. A Fig. 22.5 mostra
as linhas de força do campo magnético terrestre.

Figura 22.5

Num dado local uma agulha magnética suspensa do seu centro de gravidade − ou seja,
uma bússola − aponta sempre na mesma direcção: o pólo norte da agulha aponta para o
pólo sul magnético terrestre e o pólo sul da agulha aponta para o pólo norte magnético
terrestre (recordar que pólos de “sinais” contrários atraem-se − ver Fig. 22.2). Repare-se
que os pólos magnéticos da Terra não coincidem com os pólos geográficos. A linha
pólo Norte – pólo Sul (pólos geográficos) cruza-se com a linha pólo sul – pólo norte
(pólos magnéticos) no centro da Terra, fazendo um ângulo de cerca de 8º entre si, como
mostra a Fig. 22.6. O pólo sul magnético está actualmente próximo do pólo Norte
(geográfico). Mas nem sempre foi assim. Devido a movimentos das correntes no
interior da Terra, o campo magnético terrestre vai variando ao longo do tempo.
Quem utiliza a bússola tem de conhecer o ângulo que fazem entre si a direcção
que une o ponto de observação e o Norte geográfico e a direcção que une o mesmo
ponto e o sul magnético. Este ângulo denomina-se declinação magnética do lugar. O
conhecimento da declinação magnética é crucial para se obter a direcção do norte
geográfico a partir da indicação dada pela bússola, já que a agulha da bússola aponta
para o pólo sul magnético. Esta declinação, para além de variar no tempo, porque o
campo magnético terrestre varia ao longo do tempo, varia de local para local. Ou seja, a
Declinação magnética: Coimbra: 3º47’ W
Rio de Janeiro: 21º43’ W
Campo magnético
Pólo Norte geográfico
Pólo “Norte” magnético (2005)
... desloca-se a 41 km/ano para a Sibéria
Força sobre uma carga em movimento na
presença de um campo magnético
⇥v
⇥B
⇥F
q
⇧F = q⇧v � ⇧B
⇥v = 0 F = |q| vB sin �
⇥v ⇥B//
⇥F = 0
⌃F = q
�
⌃E + ⌃v � ⌃B
⇥
1 T (Tesla)=1 Ns/(Cm)
=1 N/(Am ) Lei de Laplace
Força de Lorentz
Força sobre um condutor
percorrido por uma corrente
4
dv


B

A
I
F


Figura 23.4

Vamos considerar que a corrente eléctrica é unicamente devida a electrões. A corrente I
é da esquerda para a direita e, por isso, a velocidade de deriva dos electrões é da direita
para a esquerda. Como a carga dos electrões é negativa a força sobre cada carga, de
acordo com (23.1) é para cima, como se indica na Fig. 23.4. O valor da força sobre cada
electrão é Bevd . Designemos por n o número de electrões de condução por unidade de
volume. Assim, no troço de circuito considerado, cujo volume é A , a força resultante é

BnevAF d= . (23.7)

Recordemos da 17ª aula que a intensidade de corrente é neAvI d= . Portanto, a força é

BIF = (23.8)

No exemplo estudado a corrente e o campo magnético são perpendiculares. Se não
fossem perpendiculares a força sobre o troço rectilíneo de circuito seria dada por

BIF



×= , (23.9)

onde o vector 

é o elemento rectilíneo de corrente com o sentido desta. A intensidade
da força é θsinBIF = , sendo θ o ângulo que o elemento rectilíneo do circuito faz
com o campo magnético.

Campo criado por uma linha infinita de corrente

Vimos o efeito de campos magnéticos sobre cargas e correntes sem nos termos
preocupado com a origem desses campos. Mas sabemos já que a origem destes campos
está em magnetes permanentes ou em correntes. Que os campos magnéticos e as
correntes estavam relacionados era uma evidência experimental depois de o
dinamarquês Hans Christian Oersted se ter apercebido, em 1820, que a passagem de
corrente num circuito fazia rodar a agulha de uma bússola exactamente como quando
dela se aproximava um íman. No mesmo ano, Biot e Savart, já conhecedores da
descoberta de Oersted, estudaram o campo magnético produzido por uma corrente
rectilínea e muito extensa (que, na prática, se podia considerar infinita). Embora não
pudessem medir com precisão a corrente no fio, podiam mantê-la constante e
concluíram que a intensidade do campo, B, era inversamente proporcional à distância
ao fio, d. Mais tarde verificar-se-ia, também experimentalmente, que o campo era
evdBnAlF =
⇧J = ne⇧vd I = JA = Anevd
= IlB ⇧F = I⇧l � ⇧B
Força sobre um condutor
percorrido por uma corrente
Now consider an arbitrarily shaped wire segment of uniform cross section in
a magnetic field, as shown in Figure 29.9. It follows from Equation 29.3 that the
magnetic force exerted on a small segment of vector length ds in the presence of a
field B is
(29.4)
where d FB is directed out of the page for the directions of B and ds in Figure 29.9. We
can consider Equation 29.4 as an alternative definition of B. That is, we can define the
magnetic field B in terms of a measurable force exerted on a current element, where
the force is a maximum when B is perpendicular to the element and zero when B is
parallel to the element.
To calculate the total force FB acting on the wire shown in Figure 29.9, we integrate
Equation 29.4 over the length of the wire:
(29.5)
where a and b represent the end points of the wire. When this integration is carried
out, the magnitude of the magnetic field and the direction the field makes with the
vector ds may differ at different points.
We now treat two interesting special cases involving Equation 29.5. In both cases,
the magnetic field is assumed to be uniform in magnitude and direction.
Case 1. A curved wire carries a current I and is located in a uniform magnetic field B,
as shown in Figure 29.10a. Because the field is uniform, we can take B outside the
integral in Equation 29.5, and we obtain
(29.6)
But the quantity ds represents the vector sum of all the length elements from a to b.
From the law of vector addition, the sum equals the vector L!, directed from a to b.
Therefore, Equation 29.6 reduces to
(29.7)
From this we conclude that the magnetic force on a curved current-carrying wire
in a uniform magnetic field is equal to that on a straight wire connecting the
end points and carrying the same current.
FB " I L! ! B
!ba
FB " I "!b
a
d s# ! B
FB " I !b
a
d s ! B
d FB " I ds ! B
902 C H A P T E R 2 9 • Magnetic Fields
B
ds
I
Figure 29.9 A wire segment of
arbitrary shape carrying a current I
in a magnetic field B experiences a
magnetic force. The magnetic
force on any segment ds is I ds ! B
and is directed out of the page. You
should use the right-hand rule to
confirm this force direction.
(b)
d s
B
I
I
b
a
d s
L′
B
(a)
Figure 29.10 (a) A curved wire carrying a current I in a uniform magnetic field.
The total magnetic force acting on the wire is equivalent to the force on a straight wire
of length
L! running between the ends of the curved wire. (b) A current-carrying loop
of arbitrary shape in a uniform magnetic field. The net magnetic force on the loop
is zero.
d~F = Id~s⇥ ~B
= I ds tˆ⇥ ~B
Now consider an arbitrarily shaped wire segment of uniform cross section in
a magnetic field, as shown in Figure 29.9. It follows from Equation 29.3 that the
magnetic force exerted on a small segment of vector length ds in the presence of a
field B is
(29.4)
where d FB is directed out of the page for the directions of B and ds in Figure 29.9. We
can consider Equation 29.4 as an alternative definition of B. That is, we can define the
magnetic field B in terms of a measurable force exerted on a current element, where
the force is a maximum when B is perpendicular to the element and zero when B is
parallel to the element.
To calculate the total force FB acting on the wire shown in Figure 29.9, we integrate
Equation 29.4 over the length of the wire:
(29.5)
where a and b represent the end points of the wire. When this integration is carried
out, the magnitude of the magnetic field and the direction the field makes with the
vector ds may differ at different points.
We now treat two interesting special cases involving Equation 29.5. In both cases,
the magnetic field is assumed to be uniform in magnitude and direction.
Case 1. A curved wire carries a current I and is located in a uniform magnetic field B,
as shown in Figure 29.10a. Because the field is uniform, we can take B outside the
integral in Equation 29.5, and we obtain
(29.6)
But the quantity ds represents the vector sum of all the length elements from a to b.
From the law of vector addition, the sum equals the vector L!, directed from a to b.
Therefore, Equation 29.6 reduces to
(29.7)
From this we conclude that the magnetic force on a curved current-carrying wire
in a uniform magnetic field is equal to that on a straight wire connecting the
end points and carrying the same current.
FB " I L! ! B
!ba
FB " I "!b
a
d s# ! B
FB " I !b
a
d s ! B
d FB " I ds ! B
902 C H A P T E R 2 9 • Magnetic Fields
B
ds
I
Figure 29.9 A wire segment of
arbitrary shape carrying a current I
in a magnetic field B experiences a
magnetic force. The magnetic
force on any segment ds is I ds ! B
and is directed out of the page. You
should use the right-hand rule to
confirm this force direction.
(b)
d s
B
I
I
b
a
d s
L′
B
(a)
Figure 29.10 (a) A curved wire carrying a current I in a uniform magnetic field.
The total magnetic force acting on the wire is equivalent to the force on a straight wire
of length L! running between the ends of the curved wire. (b) A current-carrying loop
of arbitrary shape in a uniform magnetic field. The net magnetic force on the loop
is zero.
~F = I
Z b
a
d~s
!
⇥ ~B= I~L0 ⇥ ~B
Força sobre um condutor
percorrido por uma corrente
Now consider an arbitrarily shaped wire segment of uniform cross section in
a magnetic field, as shown in Figure 29.9. It follows from Equation 29.3 that the
magnetic force exerted on a small segment of vector length ds in the presence of a
field B is
(29.4)
where d FB is directed out of the page for the directions of B and ds in Figure 29.9. We
can consider Equation 29.4 as an alternative definition of B. That is, we can define the
magnetic field B in terms of a measurable force exerted on a current element, where
the force is a maximum when B is perpendicular to the element and zero when B is
parallel to the element.
To calculate the total force FB acting on the wire shown in Figure 29.9, we integrate
Equation 29.4 over the length of the wire:
(29.5)
where a and b represent the end points of the wire. When this integration is carried
out, the magnitude of the magnetic field and the direction the field makes with the
vector ds may differ at different points.
We now treat two interesting special cases involving Equation 29.5. In both cases,
the magnetic field is assumed to be uniform in magnitude and direction.
Case 1. A curved wire carries a current I and is located in a uniform magnetic field B,
as shown in Figure 29.10a. Because the field is uniform, we can take B outside the
integral in Equation 29.5, and we obtain
(29.6)
But the quantity ds represents the vector sum of all the length elements from a to b.
From the law of vector addition, the sum equals the vector L!, directed from a to b.
Therefore, Equation 29.6 reduces to
(29.7)
From this we conclude that the magnetic force on a curved current-carrying wire
in a uniform magnetic field is equal to that on a straight wire connecting the
end points and carrying the same current.
FB " I L! ! B
!ba
FB " I "!b
a
d s# ! B
FB " I !b
a
d s ! B
d FB " I ds ! B
902 C H A P T E R 2 9 • Magnetic Fields
B
ds
I
Figure 29.9 A wire segment of
arbitrary shape carrying a current I
in a magnetic field B experiences a
magnetic force. The magnetic
force on any segment ds is I ds ! B
and is directed out of the page. You
should use the right-hand rule to
confirm this force direction.
(b)
d s
B
I
I
b
a
d s
L′
B
(a)
Figure 29.10 (a) A curved wire carrying a current I in a uniform magnetic field.
The total magnetic force acting on the wire is equivalent to the force on a straight wire
of length L! running between the ends of the curved wire. (b) A current-carrying loop
of arbitrary shape in a uniform magnetic field. The net magnetic force on the loop
is zero.
~F = I
✓I
d~s
◆
⇥ ~B = 0
Força sobre uma espira de corrente
2
A força que a corrente 1 exerce sobre a corrente 2, por unidade de comprimento desta
corrente, é, portanto,

d
IIF
π
µ
2
210
=

. (24.4)

A força que a corrente 2 exerce por unidade de comprimento da corrente 1 tem
exactamente este valor mas o sentido oposto à força de 1 sobre a unidade de
comprimento de 2. As forças 12F

e 21F

representadas na Fig. 24.1 são um par
acção-reacção.
É interessante notar que a força entre duas correntes paralelas com o mesmo
sentidos é atractiva. A força seria repulsiva se as correntes tivessem sentidos opostos.

Momento magnético e força sobre uma espira de corrente

Consideremos uma corrente I que circula numa espira condutora quadrada, de lado  , e
que a espira está colocada numa região do espaço onde existe um campo magnético
uniforme (Fig. 24.2). Cada lado do quadrado fica sujeito a uma força perpendicular a
esse mesmo lado e ao campo magnético, pois, como sabemos, BIF



×= .

A
B
C
D
I
I
I
I
I
I
I
I
B

B
 µ 


  F

F

−
A,B
C,D
A B
CD
I
I
I
I
ADF

BCF

F



θ
O
µ θ
M

B


Figura 24.2
⇥B
I
Quadrado de lado �
2
A força que a corrente 1 exerce sobre a corrente 2, por unidade de comprimento desta
corrente, é, portanto,

d
IIF
π
µ
2
210
=

. (24.4)

Momento magnético e força sobre uma espira de corrente

Consideremos uma corrente I que circula numa espira condutora quadrada, de lado  , e
que a espira está colocada numa região do espaço onde existe um campo magnético
uniforme (Fig. 24.2). Cada lado do quadrado fica sujeito a uma força
perpendicular a
esse mesmo lado e ao campo magnético, pois, como sabemos, BIF



×= .

A
B
C
D
I
I
I
I
I
I
I
I
B

B
 µ 


  F

F

−
A,B
C,D
A B
CD
I
I
I
I
ADF

BCF

F



θ
O
µ θ
M

B


Figura 24.2
A B
CD
�
⇥M = 0
�
⇥F = 0
�
⇥F = 0�
⇥M �= 0
Lados AD
e BC
Lados AB
e DC
2
A força que a corrente 1 exerce sobre a corrente 2, por unidade de comprimento desta
corrente, é, portanto,

d
IIF
π
µ
2
210
=

. (24.4)

Momento magnético e força sobre uma espira de corrente

A
B
C
D
I
I
I
I
I
I
I
I
B

B
 µ 


  F

F

−
A,B
C,D
A B
CD
I
I
I
I
ADF

BCF

F



θ
O
µ θ
M

B


Figura 24.2
⌅M = ⌅⇤� ⌅F
M = ⇧ I⇧B sin � = I⇧2B sin �
⇧µ = IAeˆ
Ep = �⌅µ · ⌅B
⌅M = ⌅µ� ⌅B
Momento magnético⎨
dipolar
Força sobre uma espira de corrente
S ECT I O N 29 . 3 • Torque on a Current Loop in a Uniform Magnetic Field 905
Now suppose that the uniform magnetic field makes an angle ! " 90° with a line
perpendicular to the plane of the loop, as in Figure 29.14. For convenience, we assume
that B is perpendicular to sides ! and ". In this case, the magnetic forces F1 and F3
exerted on sides # and $ cancel each other and produce no torque because they pass
through a common origin. However, the magnetic forces F2 and F4 acting on sides !
and " produce a torque about any point. Referring to the end view shown in Figure
29.14, we note that the moment arm of F2 about the point O is equal to (b/2) sin !.
Likewise, the moment arm of F4 about O is also (b/2) sin !. Because F2 # F4 # IaB, the
magnitude of the net torque about O is
where A # ab is the area of the loop. This result shows that the torque has its
maximum value IAB when the field is perpendicular to the normal to the plane of the
loop (! # 90°), as we saw when discussing Figure 29.13, and is zero when the field is
parallel to the normal to the plane of the loop (! # 0).
A convenient expression for the torque exerted on a loop placed in a uniform
magnetic field B is
(29.9)
where A, the vector shown in Figure 29.14, is perpendicular to the plane of the loop and
has a magnitude equal to the area of the loop. We determine the direction of A using the
right-hand rule described in Figure 29.15. When you curl the fingers of your right hand
in the direction of the current in the loop, your thumb points in the direction of A. As
we see in Figure 29.14, the loop tends to rotate in the direction of decreasing values of !
(that is, such that the area vector A rotates toward the direction of the magnetic field).
The product I A is defined to be the magnetic dipole moment ! (often simply
called the “magnetic moment”) of the loop:
(29.10)
The SI unit of magnetic dipole moment is ampere-meter2 (A $m2). Using this definition,
we can express the torque exerted on a current-carrying loop in a magnetic field B as
(29.11)
Note that this result is analogous to Equation 26.18, " # p # E, for the torque exerted
on an electric dipole in the presence of an electric field E, where p is the electric
dipole moment.
" $ ! # B
! # I A
" # I A # B
# IAB sin !
# IaB ! b2 sin !" % IaB ! b2 sin !" # IabB sin !
& # F2
b
2
sin ! % F4
b
2
sin !
F2
F4
O
B
A
b
2
– sin θ
b
2
–
θ θ
θ
!
"×
Active Figure 29.14 An end view of the loop in
Figure 29.13b rotated through an angle with
respect to the magnetic field. If B is at an angle !
with respect to vector A, which is perpendicular to
the plane of the loop, the torque is IAB sin ! where
the magnitude of A is A, the area of the loop.
At the Active Figures link
at http://www.pse6.com, you
can choose the current in the
loop, the magnetic field, and
the initial orientation of the loop
and observe the subsequent
motion.
Torque on a current loop in a
magnetic field
Magnetic dipole moment of a
current loop
Torque on a magnetic moment
in a magnetic field
A
I
µ
Figure 29.15 Right-hand rule
for determining the direction of
the vector A. The direction of the
magnetic moment ! is the same as
the direction of A.
eˆ
A
Carga positiva num campo
magnético uniforme
qvB =
mv2
r
r =
mv
qB
T =
2�r
v
fc =
qB
2�m
2
Se a força magnética se mantiver perpendicular à velocidade, o movimento é circular
uniforme (o módulo da velocidade mantém-se constante). É este o caso representado na
Fig. 23.2 em que uma partícula de carga positiva descreve uma trajectória circular (se a
partícula tivesse carga negativa o sentido do movimento seria o oposto).

v

v

v

F

F

F

B


Figura 23.2

Vejamos qual é a frequência deste movimento. A força magnética que é, em módulo,
qvBF = , é a força centrípeta que origina o movimento circular uniforme. Como
sabemos das disciplinas de Elementos de Física / Física Geral I a força centrípeta vale
rmv /2 , em que r é o raio da trajectória. Assim, de

r
v
mqvB
2
= (23.3)

obtemos o raio da trajectória:

qB
mv
r = . (23.4)

Esta expressão mostra que o raio é directamente proporcional ao momento linear da
partícula e inversamente proporcional à intensidade do campo magnético e à carga. O
período do movimento circular uniforme é
v
rT π2= e, portanto, a frequência, que é o
inverso do período, é

m
qBf
π2c
= . (23.5)

O índice “c” significa “ciclotrão”: a frequência (23.5) é um parâmetro importante em
aceleradores de partículas que têm esta denominação. Da expressão anterior ressaltam
Carga positiva num campo
magnético uniforme
3
duas importantes conclusões: a frequência é a mesma para todas as partículas para as
quais a razão mq / seja a mesma; a frequência não depende da velocidade da partícula.
O movimento de uma partícula que penetre numa região onde exista um campo
magnético uniforme com uma velocidade que não seja perpendicular a B vai ter um
movimento helicoidal. Consideremos a situação em que a velocidade se pode decompor
em dois vectores em direcções perpendiculares, ⊥+= vvv

|| , em que ||v
 é paralelo a B

e
⊥v

é perpendicular a B

. No plano perpendicular ao campo magnético o movimento é,
como já vimos, circular uniforme. Segundo a direcção do campo, o movimento é
rectilíneo e uniforme pois a força magnética é nula segundo essa direcção. Da
combinação destes dois movimentos resulta um movimento helicoidal como se mostra
na Fig. 23.3.

v

⊥v

||v

B


Figura 23.3

No caso de coexistirem campos eléctricos e magnéticos numa mesma região do espaço
a partícula fica sujeita a uma força eléctrica e a uma força magnética cuja resultante
é
( )BvEqF  ×+= , (23.6)

denominada força de Lorentz, da qual existem numerosas aplicações. Com o auxílio de
campos eléctricos e magnéticos é possível guiar feixes de partículas carregadas.

Força sobre um condutor percorrido por uma corrente

Quando um fio condutor percorrido por uma corrente é colocado numa região
onde existe um campo magnético, em geral fica sujeito a uma força. A Fig. 23.4 mostra
um troço rectilíneo de um condutor, de comprimento  , percorrido por uma corrente I,
colocado numa região onde existe um campo magnético uniforme de valor B que aponta
“para lá”.

Seletor de velocidadesS ECT I O N 29 . 5 • Applications Involving Charged Particles Moving in a Magnetic Field 911
Velocity Selector
In many experiments involving moving charged particles, it is important that the particles
all move with essentially the same velocity. This can be achieved by applying a combina-
tion of an electric field and a magnetic field oriented as shown in Figure 29.23. A uniform
electric field is directed vertically downward (in the plane of the page in Fig. 29.23a), and
a uniform magnetic field is applied in the direction perpendicular to the electric field
(into the page in Fig. 29.23a). If q is positive and the velocity v is to the right, the magnetic
force q v ! B is upward and the electric force q E is downward. When the magnitudes of
the two fields are chosen so that qE ! qvB, the particle moves in a straight horizontal line
through the region of the fields. From the expression qE ! qvB, we find that
(29.17)
Only those particles having speed v pass undeflected through the mutually perpendic-
ular electric and magnetic fields. The magnetic force exerted on particles moving at
speeds greater than this is stronger than the electric force, and the particles are
deflected upward. Those moving at speeds less than this are deflected downward.
The Mass Spectrometer
A mass spectrometer separates ions according to their mass-to-charge ratio. In one
version of this device, known as the Bainbridge mass spectrometer, a beam of ions first
passes through a velocity selector and then enters a second uniform magnetic field B0
that has the same direction as the magnetic field in the selector (Fig. 29.24). Upon
entering the second magnetic field, the ions move in a semicircle of radius r before
striking a detector array at P. If the ions are positively charged, the beam deflects
upward, as Figure 29.24 shows. If the ions are negatively charged, the beam deflects
v !
E
B
Bin
+
E
Source
Slit
–
(a)
++++++
––––––
v
(b)
+ q
qv × B
qE
× × × × × × ×
× × × × × × ×
× × × × × × ×
× × × × × × ×
× × × × × × ×
× × × × × × ×
× × × × × × ×
Active Figure 29.23 (a) A velocity selector. When a positively charged particle is
moving with velocity v in the presence of a magnetic field directed into the page and an
electric field directed downward, it experiences a downward electric force qE and an
upward magnetic force q v ! B. (b) When these forces balance, the particle moves in a
horizontal line through the fields.
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
r
P
Bin
Velocity selector
E v
B0, in
×
×
×
××
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
q
Detector
array
Active Figure 29.24 A mass spectrometer. Positively charged particles are sent first
through a velocity selector and then into a region where the magnetic field B0 causes
the particles to move in a semicircular path and strike a detector array at P.
At the Active Figures link
at http://www.pse6.com, you
can adjust the electric and
magnetic fields to try to
achieve straight line motion for
the charge.
At the Active Figures link
at http://www.pse6.com, you
can predict where particles will
strike the detector array.
Fonte de
cargas
Fenda
S ECT I O N 29 . 5 • Applications Involving Charged Particles Moving in a Magnetic Field 911
Velocity Selector
In many experiments involving moving charged particles, it is important that the particles
all move with essentially the same velocity. This can be achieved by applying a combina-
tion of an electric field and a magnetic field oriented as shown in Figure 29.23. A uniform
electric field is directed vertically downward (in the plane of the page in Fig. 29.23a), and
a uniform magnetic field is applied in the direction perpendicular to the electric field
(into the page in Fig. 29.23a). If q is positive and the velocity v is to the right, the magnetic
force q v ! B is upward and the electric force q E is downward. When the magnitudes of
the two fields are chosen so that qE ! qvB, the particle moves in a straight horizontal line
through the region of the fields. From the expression qE ! qvB, we find that
(29.17)
Only those particles having speed v pass undeflected through the mutually perpendic-
ular electric and magnetic fields. The magnetic force exerted on particles moving at
speeds greater than this is stronger than the electric force, and the particles are
deflected upward. Those moving at speeds less than this are deflected downward.
The Mass Spectrometer
A mass spectrometer separates ions according to their mass-to-charge ratio. In one
version of this device, known as the Bainbridge mass spectrometer, a beam of ions first
passes through a velocity selector and then enters a second uniform magnetic field B0
that has the same direction as the magnetic field in the selector (Fig. 29.24). Upon
entering the second magnetic field, the ions move in a semicircle of radius r before
striking a detector array at P. If the ions are positively charged, the beam deflects
upward, as Figure 29.24 shows. If the ions are negatively charged, the beam deflects
v !
E
B
Bin
+
E
Source
Slit
–
(a)
++++++
––––––
v
(b)
+ q
qv × B
qE
× × × × × × ×
× × × × × × ×
× × × × × × ×
× × × × × × ×
× × × × × × ×
× × × × × × ×
× × × × × × ×
Active Figure 29.23 (a) A velocity selector. When a positively charged particle is
moving with velocity v in the presence of a magnetic field directed into the page and an
electric field directed downward, it experiences a downward electric force qE and an
upward magnetic force q v ! B. (b) When these forces balance, the particle moves in a
horizontal line through the fields.
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
r
P
Bin
Velocity selector
E v
B0, in
×
×
×
××
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
q
Detector
array
Active Figure 29.24 A mass spectrometer. Positively charged particles are sent first
through a velocity selector and then into a region where the magnetic field B0 causes
the particles to move in a semicircular path and strike a detector array at P.
At the Active Figures link
at http://www.pse6.com, you
can adjust the electric and
magnetic fields to try to
achieve straight line motion for
the charge.
At the Active Figures link
at http://www.pse6.com, you
can predict where particles will
strike the detector array.
Só vão “a direito” as cargas para as quais v =
E
B
Espetrómetro de massa
S ECT I O N 29 . 5 • Applications Involving Charged Particles Moving in a Magnetic Field 911
Velocity Selector
In many experiments involving moving charged particles, it is important that the particles
all move with essentially the
same velocity. This can be achieved by applying a combina-
tion of an electric field and a magnetic field oriented as shown in Figure 29.23. A uniform
electric field is directed vertically downward (in the plane of the page in Fig. 29.23a), and
a uniform magnetic field is applied in the direction perpendicular to the electric field
(into the page in Fig. 29.23a). If q is positive and the velocity v is to the right, the magnetic
force q v ! B is upward and the electric force q E is downward. When the magnitudes of
the two fields are chosen so that qE ! qvB, the particle moves in a straight horizontal line
through the region of the fields. From the expression qE ! qvB, we find that
(29.17)
Only those particles having speed v pass undeflected through the mutually perpendic-
ular electric and magnetic fields. The magnetic force exerted on particles moving at
speeds greater than this is stronger than the electric force, and the particles are
deflected upward. Those moving at speeds less than this are deflected downward.
The Mass Spectrometer
A mass spectrometer separates ions according to their mass-to-charge ratio. In one
version of this device, known as the Bainbridge mass spectrometer, a beam of ions first
passes through a velocity selector and then enters a second uniform magnetic field B0
that has the same direction as the magnetic field in the selector (Fig. 29.24). Upon
entering the second magnetic field, the ions move in a semicircle of radius r before
striking a detector array at P. If the ions are positively charged, the beam deflects
upward, as Figure 29.24 shows. If the ions are negatively charged, the beam deflects
v !
E
B
Bin
+
E
Source
Slit
–
(a)
++++++
––––––
v
(b)
+ q
qv × B
qE
× × × × × × ×
× × × × × × ×
× × × × × × ×
× × × × × × ×
× × × × × × ×
× × × × × × ×
× × × × × × ×
Active Figure 29.23 (a) A velocity selector. When a positively charged particle is
moving with velocity v in the presence of a magnetic field directed into the page and an
electric field directed downward, it experiences a downward electric force qE and an
upward magnetic force q v ! B. (b) When these forces balance, the particle moves in a
horizontal line through the fields.
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
r
P
Bin
Velocity selector
E v
B0, in
×
×
×
××
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
q
Detector
array
Active Figure 29.24 A mass spectrometer. Positively charged particles are sent first
through a velocity selector and then into a region where the magnetic field B0 causes
the particles to move in a semicircular path and strike a detector array at P.
At the Active Figures link
at http://www.pse6.com, you
can adjust the electric and
magnetic fields to try to
achieve straight line motion for
the charge.
At the Active Figures link
at http://www.pse6.com, you
can predict where particles will
strike the detector array.
Detetores
Seletor de velocidades
Ciclotrão
S ECT I O N 29 . 5 • Applications Involving Charged Particles Moving in a Magnetic Field 913
The Cyclotron
A cyclotron is a device that can accelerate charged particles to very high speeds. The
energetic particles produced are used to bombard atomic nuclei and thereby produce
nuclear reactions of interest to researchers. A number of hospitals use cyclotron facili-
ties to produce radioactive substances for diagnosis and treatment.
Both electric and magnetic forces have a key role in the operation of a cyclotron. A
schematic drawing of a cyclotron is shown in Figure 29.27a. The charges move inside
two semicircular containers D1 and D2, referred to as dees, because of their shape like
the letter D. A high-frequency alternating potential difference is applied to the dees,
and a uniform magnetic field is directed perpendicular to them. A positive ion
released at P near the center of the magnet in one dee moves in a semicircular path
(indicated by the dashed red line in the drawing) and arrives back at the gap in a time
interval T/2, where T is the time interval needed to make one complete trip around
the two dees, given by Equation 29.15. The frequency of the applied potential differ-
ence is adjusted so that the polarity of the dees is reversed in the same time interval
during which the ion travels around one dee. If the applied potential difference is
adjusted such that D2 is at a lower electric potential than D1 by an amount !V, the ion
accelerates across the gap to D2 and its kinetic energy increases by an amount q !V. It
then moves around D2 in a semicircular path of greater radius (because its speed has
increased). After a time interval T/2, it again arrives at the gap between the dees. By
this time, the polarity across the dees has again been reversed, and the ion is given
another “kick” across the gap. The motion continues so that for each half-circle trip
around one dee, the ion gains additional kinetic energy equal to q !V. When the
radius of its path is nearly that of the dees, the energetic ion leaves the system through
the exit slit. Note that the operation of the cyclotron is based on the fact that T is inde-
pendent of the speed of the ion and of the radius of the circular path (Eq. 29.15).
We can obtain an expression for the kinetic energy of the ion when it exits the
cyclotron in terms of the radius R of the dees. From Equation 29.13 we know that
v " qBR/m. Hence, the kinetic energy is
(29.19)
When the energy of the ions in a cyclotron exceeds about 20 MeV, relativistic
effects come into play. (Such effects are discussed in Chapter 39.) We observe that T
increases and that the moving ions do not remain in phase with the applied potential
K " 12mv
2 "
q 2B 2R 2
2m
B
P
D1
D2
(a)
North pole of magnet
Particle exits here
Alternating ∆V
▲ PITFALL PREVENTION
29.1 The Cyclotron Is Not
State-of-the-Art
Technology
The cyclotron is important histori-
cally because it was the first
particle accelerator to achieve very
high particle speeds. Cyclotrons
are still in use in medical appli-
cations, but most accelerators
currently in research use are not
cyclotrons. Research accelerators
work on a different principle and
are generally called synchrotrons.
Figure 29.27 (a) A cyclotron consists of an ion source at P, two dees D1 and D2 across
which an alternating potential difference is applied, and a uniform magnetic field.
(The south pole of the magnet is not shown.) The red dashed curved lines represent
the path of the particles. (b) The first cyclotron, invented by E. O. Lawrence and M. S.
Livingston in 1934.
Co
ur
te
sy
o
f L
aw
re
nc
e
Be
rk
el
ey
La
bo
ra
to
ry
/U
ni
ve
rs
ity
o
f C
al
ifo
rn
ia
(b)
Tensão “alterna”
Saída da partícula
Pólo Norte do íman
Efeito Hall
Placa metálica
914 C H A P T E R 2 9 • Magnetic Fields
difference. Some accelerators overcome this problem by modifying the period of the
applied potential difference so that it remains in phase with the moving ions.
29.6 The Hall Effect
When a current-carrying conductor is placed in a magnetic field, a potential difference
is generated in a direction perpendicular to both the current and the magnetic field.
This phenomenon, first observed by Edwin Hall (1855–1938) in 1879, is known as the
Hall effect. It arises from the deflection of charge carriers to one side of the conductor
as a result of the magnetic force they experience. The Hall effect gives information
regarding the sign of the charge carriers and their density; it can also be used to
measure the magnitude
of magnetic fields.
The arrangement for observing the Hall effect consists of a flat conductor carrying
a current I in the x direction, as shown in Figure 29.28. A uniform magnetic field B is
applied in the y direction. If the charge carriers are electrons moving in the negative
x direction with a drift velocity vd, they experience an upward magnetic force FB !
q vd ! B, are deflected upward, and accumulate at the upper edge of the flat conduc-
tor, leaving an excess of positive charge at the lower edge (Fig. 29.29a). This accumula-
tion of charge at the edges establishes an electric field in the conductor and increases
until the electric force on carriers remaining in the bulk of the conductor balances the
magnetic force acting on the carriers. When this equilibrium condition is reached, the
electrons are no longer deflected upward. A sensitive voltmeter or potentiometer
connected across the sample, as shown in Figure 29.29, can measure the potential
difference—known as the Hall voltage "VH—generated across the conductor.
If the charge carriers are positive and hence move in the positive x direction (for
rightward current), as shown in Figures 29.28 and 29.29b, they also experience an
upward magnetic force q vd ! B. This produces a buildup of positive charge on the
upper edge and leaves an excess of negative charge on the lower edge. Hence, the sign
of the Hall voltage generated in the sample is opposite the sign of the Hall voltage
resulting from the deflection of electrons. The sign of the charge carriers can
therefore be determined from a measurement of the polarity of the Hall voltage.
In deriving an expression for the Hall voltage, we first note that the magnetic force
exerted on the carriers has magnitude qvdB. In equilibrium, this force is balanced by
the electric force qEH, where EH is the magnitude of the electric field due to the
charge separation (sometimes referred to as the Hall field). Therefore,
EH ! vd B
qvdB ! qEH
vd
y
vd
x
z
a
I
t
d
c
+
–
I
B
B
FB
FB
Figure 29.28 To observe the Hall effect, a magnetic field is applied to a current-
carrying conductor. When I is in the x direction and B in the y direction, both positive
and negative charge carriers are deflected upward in the magnetic field. The Hall
voltage is measured between points a and c.
d
t
Efeito Hall
− +
~FB
~v
⊗~B
~v
~FB
qvB = qEH
+++++
− − − −
~EH
�VH = EHd = vBd
�VH =
I
nqtd
Bd
�VH =
IB
nqt
A tensão de Hall pode ser usada para medir
a intensidade do campo magnético
Efeito Hall S ECT I O N 29 . 6 • The Hall Effect 915
0
× × × × × × × × ×
× × × × × × × × ×
× × × × × × × ×
× × × × × × × × ×
× × × × × × × × ×
I
I
+ + + + + + + + +
– – – – – – – – –
(a)
c
qvd × B
–
qEH
B
vd
a
∆VH
0
× × × × × × × × ×
× × × × × × × × ×
× × × × × × × ×
× × × × × × × × ×
× × × × × × × × ×
I
I
– – – – – – – – –
+ + + + + + + + +
(b)
c
qvd × B
qEH
B
vd
a
+ ∆VH
Figure 29.29 (a) When the charge carriers in a Hall-effect apparatus are negative, the
upper edge of the conductor becomes negatively charged, and c is at a lower electric
potential than a. (b) When the charge carriers are positive, the upper edge becomes
positively charged, and c is at a higher potential than a. In either case, the charge carri-
ers are no longer deflected when the edges become sufficiently charged that there is a
balance on the charge carriers between the electrostatic force qEH and the magnetic
deflection force qvB.
If d is the width of the conductor, the Hall voltage is
(29.20)
Thus, the measured Hall voltage gives a value for the drift speed of the charge carriers
if d and B are known.
We can obtain the charge carrier density n by measuring the current in the sample.
From Equation 27.4, we can express the drift speed as
(29.21)
where A is the cross-sectional area of the conductor. Substituting Equation 29.21 into
Equation 29.20, we obtain
(29.22)
Because A ! td, where t is the thickness of the conductor, we can also express Equation
29.22 as
(29.23)
where RH ! 1/nq is the Hall coefficient. This relationship shows that a properly cali-
brated conductor can be used to measure the magnitude of an unknown magnetic field.
Because all quantities in Equation 29.23 other than nq can be measured, a value for
the Hall coefficient is readily obtainable. The sign and magnitude of RH give the sign
of the charge carriers and their number density. In most metals, the charge carriers are
electrons, and the charge-carrier density determined from Hall-effect measurements is
in good agreement with calculated values for such metals as lithium (Li), sodium (Na),
copper (Cu), and silver (Ag), whose atoms each give up one electron to act as a
current carrier. In this case, n is approximately equal to the number of conducting
electrons per unit volume. However, this classical model is not valid for metals such as
iron (Fe), bismuth (Bi), and cadmium (Cd) or for semiconductors. These discrepan-
cies can be explained only by using a model based on the quantum nature of solids.
An interesting medical application related to the Hall effect is the electromagnetic
blood flowmeter, first developed in the 1950s and continually improved since then.
Imagine that we replace the conductor in Figure 29.29 with an artery carrying blood.
The blood contains charged ions that experience electric and magnetic forces like the
charge carriers in the conductor. The speed of flow of these ions can be related to the
volume rate of flow of blood. Solving Equation 29.20 for the speed vd of the ions in
∆VH !
IB
nqt
!
RHIB
t
∆VH !
IBd
nqA
vd !
I
nqA
∆VH ! EHd ! vd Bd
The Hall voltage
S ECT I O N 29 . 6 • The Hall Effect 915
0
× × × × × × × × ×
× × × × × × × × ×
× × × × × × × ×
× × × × × × × × ×
× × × × × × × × ×
I
I
+ + + + + + + + +
– – – – – – – – –
(a)
c
qvd × B
–
qEH
B
vd
a
∆VH
0
× × × × × × × × ×
× × × × × × × × ×
× × × × × × × ×
× × × × × × × × ×
× × × × × × × × ×
I
I
– – – – – – – – –
+ + + + + + + + +
(b)
c
qvd × B
qEH
B
vd
a
+ ∆VH
Figure 29.29 (a) When the charge carriers in a Hall-effect apparatus are negative, the
upper edge of the conductor becomes negatively charged, and c is at a lower electric
potential than a. (b) When the charge carriers are positive, the upper edge becomes
positively charged, and c is at a higher potential than a. In either case, the charge carri-
ers are no longer deflected when the edges become sufficiently charged that there is a
balance on the charge carriers between the electrostatic force qEH and the magnetic
deflection force qvB.
If d is the width of the conductor, the Hall voltage is
(29.20)
Thus, the measured Hall voltage gives a value for the drift speed of the charge carriers
if d and B are known.
We can obtain the charge carrier density n by measuring the current in the sample.
From Equation 27.4, we can express the drift speed as
(29.21)
where A is the cross-sectional area of the conductor. Substituting Equation 29.21 into
Equation 29.20, we obtain
(29.22)
Because A ! td, where t is the thickness of the conductor, we can also express Equation
29.22 as
(29.23)
where RH ! 1/nq is the Hall coefficient. This relationship shows that a properly cali-
brated conductor can be used to measure the magnitude of an unknown magnetic field.
Because all quantities in Equation 29.23 other than nq can be measured, a value for
the Hall coefficient is readily obtainable. The sign and magnitude of RH give the sign
of the charge carriers and their number density. In most metals, the charge carriers are
electrons, and the charge-carrier density determined from Hall-effect measurements
is
in good agreement with calculated values for such metals as lithium (Li), sodium (Na),
copper (Cu), and silver (Ag), whose atoms each give up one electron to act as a
current carrier. In this case, n is approximately equal to the number of conducting
electrons per unit volume. However, this classical model is not valid for metals such as
iron (Fe), bismuth (Bi), and cadmium (Cd) or for semiconductors. These discrepan-
cies can be explained only by using a model based on the quantum nature of solids.
An interesting medical application related to the Hall effect is the electromagnetic
blood flowmeter, first developed in the 1950s and continually improved since then.
Imagine that we replace the conductor in Figure 29.29 with an artery carrying blood.
The blood contains charged ions that experience electric and magnetic forces like the
charge carriers in the conductor. The speed of flow of these ions can be related to the
volume rate of flow of blood. Solving Equation 29.20 for the speed vd of the ions in
∆VH !
IB
nqt
!
RHIB
t
∆VH !
IBd
nqA
vd !
I
nqA
∆VH ! EHd ! vd Bd
The Hall voltage
Portadores de
carga “negativos”
Portadores de
carga “positivos”
Oersted: campo magnético também
pode ser criado por uma corrente...
3
Mas o que é um campo magnético? Na 8ª aula definimos campo eléctrico a
partir da força de Coulomb. Podemos utilizar a mesma metodologia para definir o
campo magnético já que este, em certas circunstâncias, origina uma força sobre uma
carga eléctrica. Esta relação entre o campo magnético e a força magnética exercida
sobre uma carga eléctrica será abordada na próxima aula.
Como já dissemos, também as correntes produzem campos magnéticos cujas
linhas de força podem ser representadas exactamente como as dos campos produzidos
por magnetes. Na Fig. 22.4 consideram-se três geometrias diferentes para linhas de
correntes, esquematizando-se as correspondentes linhas de força do campo magnético,
B

. Na figura (a) representam-se as linhas de força do campo magnético produzido por
uma corrente I rectilínea e indefinida. As linhas de campo são circunferências pois o
campo tem a mesma intensidade a uma mesma distância da linha. As circunferências
representadas estão em planos horizontais e o campo magnético é perpendicular à
corrente. Naturalmente que se espera que esta intensidade vá diminuindo à medida que a
distância à linha vá aumentando, da mesma maneira que a intensidade do campo
eléctrico diminui à medida que nos afastamos da carga ou cargas que o criam. Na figura
(b) temos um anel ou espira de corrente. As linhas de força do campo magnético passam
pelo interior do anel e fecham-se por cima e por baixo deste. Sobre o eixo da espira o
campo magnético tem a direcção do próprio eixo. Também é natural esperar que quanto
mais afastado se estiver da espira mais débil o campo se vá tornando. Na figura (c)
mostram-se as linhas de força do campo magnético criado por um fio enrolado com a
forma de um cilindro (solenóide).
I
B

B

I B

B

(a)
(b)
(c)
I

Figura 22.4

Fio
3
Mas o que é um campo magnético? Na 8ª aula definimos campo eléctrico a
partir da força de Coulomb. Podemos utilizar a mesma metodologia para definir o
campo magnético já que este, em certas circunstâncias, origina uma força sobre uma
carga eléctrica. Esta relação entre o campo magnético e a força magnética exercida
sobre uma carga eléctrica será abordada na próxima aula.
Como já dissemos, também as correntes produzem campos magnéticos cujas
linhas de força podem ser representadas exactamente como as dos campos produzidos
por magnetes. Na Fig. 22.4 consideram-se três geometrias diferentes para linhas de
correntes, esquematizando-se as correspondentes linhas de força do campo magnético,
B

. Na figura (a) representam-se as linhas de força do campo magnético produzido por
uma corrente I rectilínea e indefinida. As linhas de campo são circunferências pois o
campo tem a mesma intensidade a uma mesma distância da linha. As circunferências
representadas estão em planos horizontais e o campo magnético é perpendicular à
corrente. Naturalmente que se espera que esta intensidade vá diminuindo à medida que a
distância à linha vá aumentando, da mesma maneira que a intensidade do campo
eléctrico diminui à medida que nos afastamos da carga ou cargas que o criam. Na figura
(b) temos um anel ou espira de corrente. As linhas de força do campo magnético passam
pelo interior do anel e fecham-se por cima e por baixo deste. Sobre o eixo da espira o
campo magnético tem a direcção do próprio eixo. Também é natural esperar que quanto
mais afastado se estiver da espira mais débil o campo se vá tornando. Na figura (c)
mostram-se as linhas de força do campo magnético criado por um fio enrolado com a
forma de um cilindro (solenóide).
I
B

B

I B

B

(a)
(b)
(c)
I

Figura 22.4

Anel
3
Mas o que é um campo magnético? Na 8ª aula definimos campo eléctrico a
partir da força de Coulomb. Podemos utilizar a mesma metodologia para definir o
campo magnético já que este, em certas circunstâncias, origina uma força sobre uma
carga eléctrica. Esta relação entre o campo magnético e a força magnética exercida
sobre uma carga eléctrica será abordada na próxima aula.
Como já dissemos, também as correntes produzem campos magnéticos cujas
linhas de força podem ser representadas exactamente como as dos campos produzidos
por magnetes. Na Fig. 22.4 consideram-se três geometrias diferentes para linhas de
correntes, esquematizando-se as correspondentes linhas de força do campo magnético,
B

. Na figura (a) representam-se as linhas de força do campo magnético produzido por
uma corrente I rectilínea e indefinida. As linhas de campo são circunferências pois o
campo tem a mesma intensidade a uma mesma distância da linha. As circunferências
representadas estão em planos horizontais e o campo magnético é perpendicular à
corrente. Naturalmente que se espera que esta intensidade vá diminuindo à medida que a
distância à linha vá aumentando, da mesma maneira que a intensidade do campo
eléctrico diminui à medida que nos afastamos da carga ou cargas que o criam. Na figura
(b) temos um anel ou espira de corrente. As linhas de força do campo magnético passam
pelo interior do anel e fecham-se por cima e por baixo deste. Sobre o eixo da espira o
campo magnético tem a direcção do próprio eixo. Também é natural esperar que quanto
mais afastado se estiver da espira mais débil o campo se vá tornando. Na figura (c)
mostram-se as linhas de força do campo magnético criado por um fio enrolado com a
forma de um cilindro (solenóide).
I
B

B

I B

B

(a)
(b)
(c)
I

Figura 22.4

Solenóide
Lei de Biot-Savart
4
N
S
plano da espiraIB

µ
M
M

B

µ

Figura 24.3

O íman, na presença de um campo magnético, tal como a espira, fica sujeito ao
momento (24.8). Se espira ou magnete puderem rodar, então rodam procurando alinhar
o seu µ com o campo B

.
Tal como para a interacção dipolo eléctrico − campo eléctrico, também agora há
uma energia potencial para a interacção dipolo magnético − campo magnético dada por

Bp

⋅−= µE . (24.9)

Esta expressão é a análoga, para campos magnéticos, da expressão (10.9) relativa a
campos eléctricos.

Lei de Biot-Savart

Vimos na aula anterior qual é o campo criado por uma linha rectilínea infinita de
corrente. E qual é o campo criado por uma outra corrente qualquer? Biot e Savart − que
experimentalmente obtiveram o campo magnético criado pela corrente rectilínea e
infinita também encontraram a expressão geral que permite conhecer o campo
magnético para uma linha de
corrente qualquer. A Fig. 24.4 mostra uma corrente que
percorre um fio com uma forma arbitrária. O elemento desse fio de tamanho
infinitesimal d , percorrido pela corrente I cria no ponto P, do qual está a uma distância
r, um campo magnético infinitesimal B

d .
I
Pr
r^
θ
dB
d

Figura 24.4
d �B =
µ0
4⇥
Id�⇧� rˆ
r2
�B =
�
C
d �B =
�
C
µ0
4⇥
Id�⇧� rˆ
r2
Permeabilidade magnética
µ0 = 4⇡ ⇥ 10�7 T ·m/A
Para velocidades muito
menores que c e para fios de
diâmetro reduzido quando
comparado com as dimensões
do circuito...
Campo criado por uma
linha infinita de corrente
I
P
d ~B =
µ0
4⇡
Id~`⇥ rˆ
r2
r
rˆd~l
✓
=
µ0
4⇡
Id` sin ✓
r2
kˆ
a
� ↵
r =
a
cos↵
d` = d (a tan↵) = a
d↵
cos2 ↵
sin ✓ = sin (⇡ � �) = sin� = cos↵
=
µ0
4⇡
Id↵ cos↵
a
kˆ
=
µ0
4⇡
I
a
Z ⇡
2
�⇡2
d↵ cos↵kˆ
d ~B =
µ0
4⇡
Id↵ cos↵
a
kˆ
Campo criado por uma
linha infinita de corrente
I
P
r
rˆd~l
✓
a
� ↵
~B =
Z ⇡
2
�⇡2
d ~B
=
µ0I
2⇡a
kˆ
Campo criado por uma
linha infinita de corrente
5
proporcional à corrente no fio. Conjugando os dois resultados, dIB /∝ , ou,
explicitando a constante de acoplamento,

d
IB
π
µ
2
0
= , (23.10)

onde 0µ é a permeabilidade magnética do vazio cujo valor em unidades SI é

m/A T104 70
−×= πµ . (23.11)

A unidade SI de campo magnético é o tesla (símbolo T).
Como se viu na aula anterior, as linhas de campo são circunferências (o campo
magnético só depende da distância ao fio) e a sua orientação também foi referida nessa
aula − é perpendicular à corrente. A Fig. 23.5 (ver também a Fig. 22.4 a) mostra, em
corte, as linha de força do campo magnético produzido por uma corrente rectilínea
infinita.

B

I

Figura 23.5

B =
µ0I
2⇥d
µ0 = 4⇥ � 10�7 Tm/A
Tesla: Unidade de
campo magnético
Permeabilidade
magnética
Campo magnético no eixo
de uma espira circular
6
A integração (2.11) − que não vamos fazer explicitamente − conduz ao resultado já
nosso conhecido:

eˆ
2
0
d
IB
π
µ
=

, (24.13)

onde d é a distância do ponto P à linha de corrente e ê é o vector unitário perpendicular
ao papel e que aponta “para lá”.

Campo magnético no eixo de uma espira circular

A integração (24.11) é, em geral, tecnicamente complicada de levar a cabo, sobretudo se
a linha de corrente não tiver uma forma simples. Porém, tal não acontece para uma
espira circular de corrente se se quiser conhecer o campo num ponto do seu eixo. A
Fig. 24.6 mostra uma espira circular de corrente de raio R percorrida por uma corrente I.
Qual é então o campo magnético produzido pela espira a uma altura h do plano da
espira (ponto P)?

I
R
hr
r^
P
α
dB
dB cosα
B
d
α

Figura 24.6

Cada troço da espira de comprimento d contribui αcosdB para o campo, o qual vai ter
a direcção vertical (as projecções das contribuição de elementos da espira
diametralmente opostos segundo a horizontal anulam-se exactamente). Ora, por um
lado, de (24.11), e atendendo a que 1sin =θ ,

�
C
d⇤ = 2�R
cos� =
R
r
=
R�
R2 + h2
r2 = R2 + h2
B =
µ0I
2
R2
(R2 + h2)3/2
dB cos� =
µ0I
4⇤
R
(R2 + h2)3/2
d⌥
dB =
µ0
4⇥
Id⇧
r2
Força entre duas correntes
paralelas
1
24ª aula

Sumário:

Força entre duas correntes paralelas. Momento magnético e força sobre uma espira de
corrente. Lei de Biot-Savart. Campo magnético no eixo de uma espira circular

Força entre duas correntes paralelas

A experiência de Oersted veio mostrar que uma corrente eléctrica podia exercer
uma acção sobre um magnete. No mesmo ano de 1820, Ampère mostrou
experimentalmente que também duas correntes interagiam, exercendo forças uma sobre
a outra. Vimos na última aula as características da força que um campo magnético
exerce sobre uma corrente rectilínea de comprimento  e vimos também as
características de um campo magnético criado por uma linha de corrente rectilínea
infinita. Vamos agora aplicar estes dois resultados à determinação da força que uma
linha de corrente rectilínea infinita exerce sobre outra linha de corrente também
rectilínea e infinita paralela à primeira (Fig. 24.1). A corrente 1I cria um campo que, à
distância d é dado, em módulo, de acordo com a expressão (23.10), por

d
IB
π
µ
2
10
1 = . (24.1)

A força sobre a porção de comprimento 2 da corrente 2 é, de acordo com (23.9),

12221 BIF



×= . (24.2)

Como os dois vectores neste produto vectorial são perpendiculares, o módulo desta
força é

d
IIBIF
π
µ
2
210
212221  == . (24.3)

d 1B

2

I2
I1
12F

21F


Figura 24.1
B1 =
µ0I1
2⇥d
⇧F21 = I2⇧l2 � ⇧B1
F21 = I2l2B1 = l2
µ0I1I2
2⇥d
F
l
=
µ0I1I2
2⇥d
Força atractiva
Força repulsiva
Força entre duas correntes
paralelas
Definição de “Ampère”... ou de “Coulomb”...
Lei de Ampère
1

25ª aula

Sumário:

Lei de Ampère. Indução magnética e lei de Faraday. Aplicação: gerador de corrente
alternada

Lei de Ampère

A Lei de Biot-Savart é geral mas a sua aplicação prática pode ser trabalhosa pois
envolve um integral. Vamos estabelecer uma expressão, equivalente à lei de Boit-
Savart, que permite obter os campos magnéticos a partir de uma distribuição de
correntes. A lei de Ampère pode ser útil em certas circunstâncias, designadamente por
os cálculos envolvidos serem mais simples. Esta lei desempenha para as correntes um
papel semelhante ao da lei de Gauss (11ª aula) para cargas estacionárias quando
pretendemos determinar o campo eléctrico que produzem.
Sabemos já que o campo magnético criado por uma corrente I, rectilínea e
infinita, é, em módulo,

r
IB
π
µ
2
0
= (25.1)

e tem a direcção e sentido indicado na Fig. 25.1.

Figura 25.1

O campo é tangente à circunferência num plano perpendicular à corrente, com centro
num ponto da corrente e raio r. A expressão (25.1) pode escrever-se como
B =
µ0I
2⇥r
B (2⇥r) = µ0I
⇧B · d⇥tˆ
�
C
⌅B · d⌅⇥
�
C
Lei de Ampère
2

IrB 0)2( µπ = . (25.2)

O primeiro membro é o produto do campo pelo comprimento da circunferência. Como o
campo é tangente à circunferência, esse primeiro membro pode ser visto como o
produto escalar do campo B

pelo elemento de linha orientado tˆdd ""

= , onde "d é o
comprimento do arco infinitesimal e tˆ o versor tangente a esse arco infinitesimal, ou
seja " d⋅B , integrado para toda a circunferência. Quer dizer o primeiro membro de
(25.2) é a circulação do campo B ao longo da linha fechada C que é a circunferência, o
que nos permite então escrever

IB 0C d µ=⋅ "

. (25.3)

Esta equação é a lei de Ampere e, embora tenha sido aqui obtida para um caso
particular, é uma lei geral. Tomando a mesma corrente, e deformando o contorno
circular, como se mostra na Fig. 25.2, obtém-se sempre para a circulação do campo
magnético o mesmo valor I0µ .

I
B

I
t^
t^ C
C'

Figura 25.2

Mas para se ter o segundo membro em (25.3) diferente de zero é preciso que a corrente I
seja “enlaçada” pelo contorno C (notar que este contorno é uma linha imaginária, não é
um circuito eléctrico!). Por exemplo, na situação representada no lado esquerdo da
Fig. 25.3, o campo magnético tem circulação nula ao longo de C pois nenhuma das
correntes fica enlaçada pelo contorno. Do lado direito, apesar de haver duas correntes,
só uma contribui
para a circulação do campo magnético ao longo de C. Mas atenção!
Ambas as correntes produzem campo magnético! Só que a circulação do campo
produzido pela corrente 1 é nula ao longo da linha fechada C. Para o lado direito da
Fig. 25.3 vem 20C d IB µ=⋅ "

.
�
C
⌃B · d⌃⌅ = µ0I
Que corrente é esta?
É a soma de todas as correntes
envolvidas pelo contorno C
Lei de Ampère
3
I2
I1
C
I2
I1
C

Figura 25.3

Como referimos no início da aula, pode estabelecer-se uma analogia entre a lei de Gauss
para cargas eléctricas e correspondentes campos eléctricos − que se traduz
matematicamente por  =⋅S 0/d εQSE

, onde Q é a carga total delimitada pela superfície
fechada S −, e a lei de Ampère para correntes e correspondentes campos magnéticos −
que se traduz matematicamente por  =⋅C 0d IB µ"

, onde I é a soma das correntes
enlaçadas pela linha fechada C (que não tem de ser plana!).
Tal como para a lei de Gauss, a utilidade da lei de Ampère torna-se evidente
quando há simetrias na distribuição de correntes tais que a circulação  ⋅C d"

B se torne
fácil de obter, tal como no caso da corrente infinita em que aquele integral se reduz ao
produto do campo pelo comprimento da linha pois o campo tem valor constante e é
sempre tangente à linha. Também no caso da lei de Gauss as situações mais simples de
lidar eram aquelas em que o fluxo  ⋅S dSE

se reduzia ao produto da grandeza do campo
eléctrico pela área da superfície.

Indução magnética e lei de Faraday

Já dissemos em aulas anteriores − mas não é demais repetir! − que foi a
descoberta de Oerstead que permitiu concluir que as correntes eléctricas criam campos
magnéticos.
O inglês Michael Faraday imaginou então que o contrário pudesse acontecer, ou
seja que os campos magnéticos poderiam gerar correntes em circuitos. Ora, se
colocarmos um magnete junto de um circuito onde se intercalou um amperímetro, como
se mostra na Fig. 25.4, podemos facilmente constatar que não há registo de passagem de
corrente no circuito.

0
A
NS
I=0

Figura 25.4
3
I2
I1
C
I2
I1
C

Figura 25.3

Indução magnética e lei de Faraday

0
A
NS
I=0

Figura 25.4
�
C
⌅B · d⌅⇥ = 0
�
C
⌃B · d⌃⌅ = µ0I2
Semelhante à lei de Gauss...
É preciso saber qual o sinal das correntes para as
somar: usa-se a regra da mão direita!
C tem de ser fechado mas não precisa de ser plano
Lei de Ampère
present in the wire, all the needles point in the same direction (that of
the Earth’s magnetic field), as expected. When the wire carries a strong, steady
current, the needles all deflect in a direction tangent to the circle, as in Figure
30.9b. These observations demonstrate that the direction of the magnetic field
produced by the current in the wire is consistent with the right-hand rule described
in Figure 30.4. When the current is reversed, the needles in Figure 30.9b also
reverse.
Because the compass needles point in the direction of B, we conclude that
the lines of B form circles around the wire, as discussed in the preceding section.
By symmetry, the magnitude of B is the same everywhere on a circular path
centered on the wire and lying in a plane perpendicular to the wire. By varying the
current and distance a from the wire, we find that B is proportional to the
current and inversely proportional to the distance from the wire, as Equation 30.5
describes.
Now let us evaluate the product B !ds for a small length element ds on the
circular path defined by the compass needles, and sum the products for all elements
over the closed circular path.2 Along this path, the vectors ds and B are parallel
at each point (see Fig. 30.9b), so B !ds " B ds. Furthermore, the magnitude
of B is constant on this circle and is given by Equation 30.5. Therefore, the sum of
the products B ds over the closed path, which is equivalent to the line integral
of B !ds, is
where !ds " 2#r is the circumference of the circular path. Although this result was
calculated for the special case of a circular path surrounding a wire, it holds
for a closed path of any shape (an amperian loop) surrounding a current that exists
in an unbroken circuit. The general case, known as Ampère’s law, can be stated
as follows:
" B !ds " B " ds " $0I2#r (2# r) " $0I
934 CHAPTE R 3 0 • Sources of the Magnetic Field
▲ PITFALL PREVENTION
30.2 Avoiding Problems
with Signs
When using Ampère’s law, apply
the following right-hand rule.
Point your thumb in the direction
of the current through the amper-
ian loop. Your curled fingers then
point in the direction that you
should integrate around the loop
in order to avoid having to define
the current as negative.
Ampère’s law
The line integral of B !ds around any closed path equals $0I, where I is the total
steady current passing through any surface bounded by the closed path.
(30.13)" B !d s " $0I
Ampère’s law describes the creation of magnetic fields by all continuous current
configurations, but at our mathematical level it is useful only for calculating the
magnetic field of current configurations having a high degree of symmetry. Its use is
similar to that of Gauss’s law in calculating electric fields for highly symmetric charge
distributions.
Quick Quiz 30.4 Rank the magnitudes of !B !ds for the closed paths in
Figure 30.10, from least to greatest.
×
1 A
5 A
b
a
d
c
2 A
Figure 30.10 (Quick Quiz 30.4)
Four closed paths around three
current-carrying wires.
2 You may wonder why we would choose to do this. The origin of Ampère’s law is in nineteenth
century science, in which a “magnetic charge” (the supposed analog to an isolated electric
charge) was imagined to be moved around a circular field line. The work done on the charge was
related to B !ds, just as the work done moving an electric charge in an electric field is related to
E !ds. Thus, Ampère’s law, a valid and useful principle, arose from an erroneous and abandoned
work calculation!
I
c
~B · d~`I
a
~B · d~`I
d
~B · d~`I
b
~B · d~`
= 6µ0
= 4µ0
= 3µ0
= �µ0
I
~B · d~`=?
I
1
~B · d~`= µ0I
Campo magnético criado por
uma corrente
retilínea infinita
No exterior do fio:
B2⇡r = µ0I
B =
µ0I
2⇡r
No interior do fio:
I 0 =
Z
S
~J · nˆdS = J⇡r2
S ECT I O N 3 0 . 3 • Ampère’s Law 935
A long, straight wire of radius R carries a steady current I
that is uniformly distributed through the cross section of
the wire (Fig. 30.12). Calculate the magnetic field a
distance r from the center of the wire in the regions r ! R
and r " R.
Solution Figure 30.12 helps us to conceptualize the wire
and the current. Because the wire has a high degree of
symmetry, we categorize this as an Ampère’s law problem.
For the r ! R case, we should arrive at the same result
we obtained in Example 30.1, in which we applied
the Biot–Savart law to the same situation. To analyze the
problem, let us choose for our path of integration circle 1
in Figure 30.12. From symmetry, B must be constant in
magnitude and parallel to ds at every point on this circle.
Because the total current passing through the plane of the
Quick Quiz 30.5 Rank the magnitudes of !B #ds for the closed paths in
Figure 30.11, from least to greatest.
a
b
c
d
Figure 30.11 (Quick Quiz 30.5) Several closed paths near a single current-carrying wire.
Example 30.4 The Magnetic Field Created by a Long Current-Carrying Wire
circle is I, Ampère’s law gives
(for r ! R) (30.14)
which is identical in form to Equation 30.5. Note how
much easier it is to use Ampère’s law than to use the
Biot–Savart law. This is often the case in highly symmetric
situations.
Now consider the interior of the wire, where r " R .
Here the current I $ passing through the plane of circle 2 is
less than the total current I. Because the current is uniform
over the cross section of the wire, the fraction of the current
enclosed by circle 2 must equal the ratio of the area %r 2
enclosed by circle 2 to the cross-sectional area %R 2 of the
wire:3
Following the same procedure as for circle 1, we apply
Ampère’s law to circle 2:
" B #d s & B(2%r) & '0I $ & '0 # r 2R2 I$
I $ &
r 2
R 2
I

I $
I
&
%r 2
%R 2

'0I
2%r
B &
" B #ds & B " ds & B(2%r) & '0I
Figure 30.12 (Example 30.4) A long, straight wire of radius R
carrying a steady current I uniformly distributed across the
cross section of the wire. The magnetic field at any point can be
calculated from Ampère’s law using a circular path of radius r,
concentric with the wire.
2
R
r
1 I
ds
3 Another way to look at this problem is to realize that the current enclosed by circle 2 must
equal the product of the current density J & I/%R2 and the area %r 2 of this circle.
~` I
2
~B · d~`= µ0I 0
r
I = J⇡R2
I 0 = I
r2
R2
B2⇡r = µ0I
0
B =
µ0Ir
2⇡R2
Campo magnético criado por
uma corrente retilínea infinita936 CHAPTE R 3 0 • Sources of the Magnetic Field
due to an infinite sheet of charge does not depend on
distance from the sheet. Thus, we might expect a similar
result here for the magnetic field.
To evaluate the line integral in Ampère’s law, we
construct a rectangular path through the sheet, as in Figure
30.15. The rectangle has dimensions ! and w, with the sides
of length ! parallel to the sheet surface. The net current in
the plane of the rectangle is Js!. We apply Ampère’s law over
the rectangle and note that the two sides of length w do not
contribute to the line integral because the component of B
So far we have imagined currents carried by wires of small
cross section. Let us now consider an example in which a
current exists in an extended object. A thin, infinitely large
sheet lying in the yz plane carries a current of linear current
density Js. The current is in the y direction, and Js represents
the current per unit length measured along the z axis. Find
the magnetic field near the sheet.
Solution This situation is similar to those involving Gauss’s
law (see Example 24.8). You may recall that the electric field
A device called a toroid (Fig. 30.14) is often used to create an
almost uniform magnetic field in some enclosed area. The
device consists of a conducting wire wrapped around a ring
(a torus) made of a nonconducting material. For a toroid
having N closely spaced turns of wire, calculate the magnetic
field in the region occupied by the torus, a distance r from
the center.
Solution To calculate this field, we must evaluate !B !ds
over the circular amperian loop of radius r in the plane of
Figure 30.14. By symmetry, we see that the magnitude of the
field is constant on this circle and tangent to it, so B !ds "
B ds. Furthermore, the wire passes through the loop N times,
so that the total current through the loop is NI. Therefore,
the right side of Equation 30.13 is #0NI in this case.
Ampère’s law applied to the circle gives
(30.16)
This result shows that B varies as 1/r and hence is nonuniform
in the region occupied by the torus. However, if r is very
large compared with the cross-sectional radius a of the torus,
then the field is approximately uniform inside the torus.
For an ideal toroid, in which the turns are closely spaced,
the external magnetic field is close to zero. It is not exactly
zero, however. In Figure 30.14, imagine the radius r of the
amperian loop to be either smaller than b or larger than c. In
either case, the loop encloses zero net current, so !B !ds " 0.
We might be tempted to claim that this proves that B " 0, but
it does not. Consider the amperian loop on the right side
of the toroid in Figure 30.14. The plane of this loop is perpen-
dicular to the page, and the toroid passes through the loop. As
charges enter the toroid as indicated by the current directions
in Figure 30.14, they work their way counterclockwise around
the toroid. Thus, a current passes through the perpendicular
amperian loop! This current is small, but it is not zero. As a
result, the toroid acts as a current loop and produces a weak
external field of the form shown in Figure 30.7. The reason
that !B !ds " 0 for the amperian loops of radius r $ b and
r % c in the plane of the page is that the field lines are perpen-
dicular to ds, not because B " 0.
#0NI
2&r
B "
" B !d s " B " ds " B(2&r) " #0NI
Figure 30.14 (Example 30.5) A toroid consisting of many
turns of wire. If the turns are closely spaced, the magnetic field
in the interior of the torus (the gold-shaded region) is tangent
to the dashed circle and varies as 1/r. The dimension a is the
cross-sectional radius of the torus. The field outside the toroid
is very small and can be described by using the amperian loop
at the right side, perpendicular to the page.
Figure 30.13 (Example 30.4) Magnitude of the magnetic field
versus r for the wire shown in Figure 30.12. The field is propor-
tional to r inside the wire and varies as 1/r outside the wire.
R
r
B ∝ 1/r
B ∝ r
B
B
c
a
ds
I
I
r
b
Example 30.5 The Magnetic Field Created by a Toroid
Example 30.6 Magnetic Field Created by an Infinite Current Sheet
(30.15)
To finalize this problem, note that this result is similar
in form to the expression for the electric field inside a
uniformly charged sphere (see Example 24.5). The magni-
tude of the magnetic field versus r for this configuration
is plotted in Figure 30.13. Note that inside the wire, B: 0
as r: 0. Furthermore, we see that Equations 30.14 and
30.15 give the same value of the magnetic field at r " R ,
demonstrating that the magnetic field is continuous at the
surface of the wire.
(for r $ R)# #0I2&R 2 $ rB "
B =
µ0I
2⇡rB =
µ0Ir
2⇡R2
Solenóide
30.4 The Magnetic Field of a Solenoid
A solenoid is a long wire wound in the form of a helix. With this configuration, a
reasonably uniform magnetic field can be produced in the space surrounded by the
turns of wire—which we shall call the interior of the solenoid—when the solenoid
carries a current. When the turns are closely