lista-1

Prévia do material em texto

Universidade Federal do Recôncavo da Bahia 
 
GCET823 - Física III 
Lista 1 
(Atualizada em novembro de 2021) 
Carga elétrica e Lei de Coulomb 
 
1. Duas esferas condutoras iguais, mantidas fixas, se atraem mutuamente com uma força eletrostática de 0,108 N 
quando a distância entre os centros é de 50,0 cm. As esferas são ligadas por um fio condutor de diâmetro desprezível. 
Quando o fio é removido, as esferas se repelem com uma força de 0,0360 N. Supondo que a carga total das esferas era 
inicialmente positiva, determine: (a) a carga negativa inicial de uma das esferas; (b) A carga positiva inicial da outra 
esfera. 
2. Três partículas são mantidas fixas sobre um eixo 𝑥. A partícula 1, de carga 𝑞!, está em 𝑥	 = 	−𝑎; a partícula 2, de 
carga 𝑞", está em 𝑥	 = 	+𝑎. Determine a razão 𝑞!/𝑞" para que a força eletrostática a que está submetida a partícula 3 
seja nula: (a) se a partícula 3 estiver no ponto 𝑥	 = 	+0,500𝑎; (b) Se a partícula 3 estiver no ponto 𝑥	 = 	+1,50𝑎. 
 
3. A figura mostra quatro esferas condutoras iguais, que estão separadas por 
grandes distâncias. A esfera 𝑊 (que estava inicialmente neutra) é colocada 
em contato com a esfera 𝐴 e depois as esferas são novamente separadas. Em 
seguida a esfera 𝑊 é colocada em contato com a esfera 𝐵 (que possuía 
inicialmente uma carga de −21𝑒) e depois as esferas são novamente 
separadas. Finalmente, a esfera 𝑊 é colocada em contato com a esfera 𝐶 (que possuía inicialmente uma carga de +48𝑒), 
e depois as esferas são novamente separadas. A carga final da esfera 𝑊 é +18𝑒. Qual era a carga inicial da esfera 𝐴? 
 
4. Na figura, a partícula 1, de carga +1µC, e a particula 2, de carga 
−3µC, são mantidas a uma distância 𝐿	 = 	10,0 cm uma da outra sobre 
o eixo 𝑥. Determine (a) a coordenada 𝑥 (b) a coordenada 𝑦 de uma 
partícula 3 de carga desconhecida 𝑞# para que a força total exercida 
sobre ela pelas particulas 1 e 2 seja nula. 
 
5. Uma casca esférica não-condutora, com raio interno de 4,0 cm e um raio externo de 6,0 cm, possui uma distribuição 
de cargas não-homogênea. A densidade volumétrica de carga 𝜌 é a carga por unidade de volume, medida em coulombs 
por metro cúbico. No caso dessa casca, 𝜌	 = 	𝑏/𝑟 , onde 𝑟 é a distância em metros a partir do centro da casca e 𝑏 = 	3,0 
µC/m². Qual é a carga total da casca? 
6. Calcule o número de coulombs de carga positiva que estão presentes em 250 cm³ de água (neutra). (Sugestão: um 
átomo de hidrogênio contém um próton; um átomo de oxigênio contém oito prótons.) 
 
7. Na figura, duas pequenas esferas condutoras de mesma massa 𝑚 e mesma carga 𝑞 
estão penduradas em fios não condutores de comprimento 𝐿. Suponha que o ângulo 𝜃	é 
tão pequeno que a aproximação tan𝜃 ≈ sen	𝜃 pode ser usada. (a) Mostre que a 
distância de equilíbrio entre as esferas é dada por 
		𝑥 = C
𝑞"𝐿
2𝜋ɛ$𝑚𝑔
G
!/#
 
(b) Se 𝐿	 = 120 cm, 𝑚	 = 10 g e 𝑥	 = 5.0 cm, qual é o valor de |𝑞| ? 
 
 
 
8. Três cargas pontuais estão localizadas em um arco de círculo, como 
mostra a figura. Determine a força elétrica que seria exercida sobre uma 
carga pontual de −5,00 nC posicionada em 𝑃. 
 
 
 
 
9. Quatro partículas carregadas estão localizadas nos vértices de um quadrado 
de lado 𝑎, conforme mostra a figura. Determine a força exercida sobre a 
partícula 𝑞. 
 
 
 
10. Uma carga de 𝑞! = −1,0	µC está na origem. Uma segunda carga, de 𝑞" = 2,0	µC, está em 𝑥 = 0 e 𝑦 = 0,10 m e 
uma terceira, de 𝑞# = 4,0	µC, está em 𝑥 = 0,20 m e 𝑦 = 0. Determine as forças que atuam sobre cada uma das cargas. 
 
Respostas 
1. (a) −1,00	𝜇C; (b) 3,00	𝜇C 
 
2. (a) 9; (b) 25 
 
3. −6𝑒 
 
4. (a) −14 cm; (b) 0 
 
5. 3,8 × 10&' C 
 
 
6. (a) 1,3 × 10( C 
 
7. (b) 2,4 × 10&' C 
 
8. 8, 98 × 10&)	N à esquerda 
 
9. *+
!
,!
(3,06�̂� + 5,06𝚥)̂ 
 
10. �⃗�! = (0,90𝚤̂ + 1,8𝚥̂) N; �⃗�" = (−1,3�̂� − 1,2𝚥̂) N; 
 �⃗�# = (0,39𝚤̂ − 0,64�̂�) N 
 
 
 
718 Chapter 23 Electric Fields
 27. Two equal positively 
charged particles are at 
opposite corners of a trap-
ezoid as shown in Figure 
P23.27. Find symbolic 
expressions for the total 
electric field at (a) the 
point P and (b) the point P 9.
 28. Consider n equal positively charged particles each of 
magnitude Q /n placed symmetrically around a circle 
of radius a. (a) Calculate the magnitude of the elec-
tric field at a point a distance x from the center of the 
circle and on the line passing through the center and 
perpendicular to the plane of the circle. (b) Explain 
why this result is identical to the result of the calcula-
tion done in Example 23.8.
 29. In Figure P23.29, determine the point (other than 
infinity) at which the electric field is zero.
1.00 m
!2.50 mC 6.00 mC
"!
Figure P23.29
 30. Three charged particles are at the corners of an equi-
lateral triangle as shown in Figure P23.15. (a) Calcu-
late the electric field at the position of the 2.00-mC 
charge due to the 7.00-mC and 24.00-mC charges. 
(b) Use your answer to part (a) to determine the force 
on the 2.00-mC charge.
 31. Three point charges are located on a circular arc as 
shown in Figure P23.31. (a) What is the total electric 
field at P, the center of the arc? (b) Find the elec-
tric force that would be exerted on a 25.00-nC point 
charge placed at P.
S
Q/C
M
W
perpendicular bisector of the 
two fixed charges a distance x 
from the midpoint between those 
charges (Fig. P23.20). (a) Show 
that if x is small compared with 
d, the motion of 2Q is simple 
harmonic along the perpendicu-
lar bisector. (b) Determine the 
period of that motion. (c) How 
fast will the charge 2Q be mov-
ing when it is at the midpoint 
between the two fixed charges if 
initially it is released at a distance 
a ,, d from the midpoint?
 21. Two identical conducting small spheres are placed with 
their centers 0.300 m apart. One is given a charge of 
12.0 nC and the other a charge of 218.0 nC. (a) Find 
the electric force exerted by one sphere on the other. 
(b) What If? The spheres are connected by a conduct-
ing wire. Find the electric force each exerts on the 
other after they have come to equilibrium.
 22. Why is the following situation impossible? Two identical 
dust particles of mass 1.00 mg are floating in empty 
space, far from any external sources of large gravi-
tational or electric fields, and at rest with respect to 
each other. Both particles carry electric charges that 
are identical in magnitude and sign. The gravitational 
and electric forces between the particles happen to 
have the same magnitude, so each particle experiences 
zero net force and the distance between the particles 
remains constant.
Section 23.4 Analysis Model: Particle in a Field (Electric)
 23. What are the magnitude and direction of the electric 
field that will balance the weight of (a) an electron and 
(b) a proton? (You may use the data in Table 23.1.)
 24. A small object of mass 3.80 g and charge 218.0 mC 
is suspended motionless above the ground when 
immersed in a uniform electric field perpendicular to 
the ground. What are the magnitude and direction of 
the electric field?
 25. Four charged particles are at the corners of a square 
of side a as shown in Figure P23.25. Determine (a) the 
electric field at the location of charge q and (b) the 
total electric force exerted on q.
" "
""
a
aa
a
q
3q 4q
2q
Figure P23.25
 26. Three point charges lie along a circle of radius r at 
angles of 308, 1508, and 2708 as shown in Figure P23.26. 
Find a symbolic expression for the resultant electric 
field at the center of the circle.
W
S
S
"Q
"Q2d
45.0#45.0#
"
"
d P
P $
Figure P23.27
30°
150°
!2q
qq r
x
y
270°
" "
!
Figure P23.26
"q
"q
!Q
x
y
x
"
"
!
d
2
d
2
Figure P23.20
"
"
!
"3.00 nC
4.00 cm
4.00 cm
"3.00 nC
30.0#
30.0#
!2.00 nC
P
Figure P23.31
718 Chapter 23 Electric Fields
 27. Two equal positively 
charged particles are at 
opposite corners of a trap-
ezoid as shown in Figure 
P23.27. Find symbolic 
expressions for the totalelectric field at (a) the 
point P and (b) the point P 9.
 28. Consider n equal positively charged particles each of 
magnitude Q /n placed symmetrically around a circle 
of radius a. (a) Calculate the magnitude of the elec-
tric field at a point a distance x from the center of the 
circle and on the line passing through the center and 
perpendicular to the plane of the circle. (b) Explain 
why this result is identical to the result of the calcula-
tion done in Example 23.8.
 29. In Figure P23.29, determine the point (other than 
infinity) at which the electric field is zero.
1.00 m
!2.50 mC 6.00 mC
"!
Figure P23.29
 30. Three charged particles are at the corners of an equi-
lateral triangle as shown in Figure P23.15. (a) Calcu-
late the electric field at the position of the 2.00-mC 
charge due to the 7.00-mC and 24.00-mC charges. 
(b) Use your answer to part (a) to determine the force 
on the 2.00-mC charge.
 31. Three point charges are located on a circular arc as 
shown in Figure P23.31. (a) What is the total electric 
field at P, the center of the arc? (b) Find the elec-
tric force that would be exerted on a 25.00-nC point 
charge placed at P.
S
Q/C
M
W
perpendicular bisector of the 
two fixed charges a distance x 
from the midpoint between those 
charges (Fig. P23.20). (a) Show 
that if x is small compared with 
d, the motion of 2Q is simple 
harmonic along the perpendicu-
lar bisector. (b) Determine the 
period of that motion. (c) How 
fast will the charge 2Q be mov-
ing when it is at the midpoint 
between the two fixed charges if 
initially it is released at a distance 
a ,, d from the midpoint?
 21. Two identical conducting small spheres are placed with 
their centers 0.300 m apart. One is given a charge of 
12.0 nC and the other a charge of 218.0 nC. (a) Find 
the electric force exerted by one sphere on the other. 
(b) What If? The spheres are connected by a conduct-
ing wire. Find the electric force each exerts on the 
other after they have come to equilibrium.
 22. Why is the following situation impossible? Two identical 
dust particles of mass 1.00 mg are floating in empty 
space, far from any external sources of large gravi-
tational or electric fields, and at rest with respect to 
each other. Both particles carry electric charges that 
are identical in magnitude and sign. The gravitational 
and electric forces between the particles happen to 
have the same magnitude, so each particle experiences 
zero net force and the distance between the particles 
remains constant.
Section 23.4 Analysis Model: Particle in a Field (Electric)
 23. What are the magnitude and direction of the electric 
field that will balance the weight of (a) an electron and 
(b) a proton? (You may use the data in Table 23.1.)
 24. A small object of mass 3.80 g and charge 218.0 mC 
is suspended motionless above the ground when 
immersed in a uniform electric field perpendicular to 
the ground. What are the magnitude and direction of 
the electric field?
 25. Four charged particles are at the corners of a square 
of side a as shown in Figure P23.25. Determine (a) the 
electric field at the location of charge q and (b) the 
total electric force exerted on q.
" "
""
a
aa
a
q
3q 4q
2q
Figure P23.25
 26. Three point charges lie along a circle of radius r at 
angles of 308, 1508, and 2708 as shown in Figure P23.26. 
Find a symbolic expression for the resultant electric 
field at the center of the circle.
W
S
S
"Q
"Q2d
45.0#45.0#
"
"
d P
P $
Figure P23.27
30°
150°
!2q
qq r
x
y
270°
" "
!
Figure P23.26
"q
"q
!Q
x
y
x
"
"
!
d
2
d
2
Figure P23.20
"
"
!
"3.00 nC
4.00 cm
4.00 cm
"3.00 nC
30.0#
30.0#
!2.00 nC
P
Figure P23.31