Exercícios de Física - Capacitores e Circuitos

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UDF - Centro Universitário
Física 3
Prof. Gabriela Possa
Lista de exercícios
Observações:
• Lembre-se de incluir seu nome e R.G.M.;
• Embora os alunos sejam encorajados a discutirem a resolução dos problemas em grupos, a lista de
exercícios deverá ser entregue INDIVIDUALMENTE (textos e cálculos idênticos não serão avalia-
dos);
• Cuide da boa apresentação da sua lista (folhas arrancadas de caderno e sem grampo não serão
aceitas);
• Inclua todos os cálculos, explicações e diagramas necessários para a resolução dos problemas;
• A lista deverá ser resolvida à mão (documentos digitalizados não serão aceitos);
• O prazo para a entrega da lista termina no dia da primeira prova (26/04), ANTES da mesma.
Problema 1
No circuito ilustrado abaixo uma bateria de 20V está conectada a uma associação de capacitores de ca-
pacitâncias C1 = C6 = 3 µF e C3 = C5 = 2C2 = 2C4 = 4 µF . Calcule:
a) A capacitância equivalente desta combinação de capacitores.
b) A carga armazenada pelo capacitor equivalente.
c) A diferença de potencial e a carga acumulada pelo capacitor 1.
d) A diferença de potencial e a carga acumulada pelo capacitor 2.
e) A diferença de potencial e a carga acumulada pelo capacitor 3.
677PROBLEMS
PART 3
HALLIDAY REVISED
potential difference V of the battery. The density of conduction elec-
trons in the copper plates is 8.49 ! 1028 electrons/m3. The vertical
scale is set by ds" 1.00 pm, and the horizontal scale is set by Vs" 20.0
V.What is the ratio C/A?
sec. 25-4 Capacitors in Parallel and in Series
•8 How many 1.00 mF capacitors must be connected in parallel to
store a charge of 1.00 C with a potential of 110 V across the 
capacitors?
•9 Each of the uncharged capaci-
tors in Fig. 25-27 has a capacitance
of 25.0 mF. A potential difference
of V " 4200 V is established when
the switch is closed. How many
coulombs of charge then pass
through meter A?
•10 In Fig. 25-28, find the equivalent capacitance of the combination.
Assume that C1 is 10.0mF,C2 is 5.00mF,and C3 is 4.00mF.
are (c) V1 and (d) q1 of capacitor 1, (e) V2 and (f) q2 of capacitor 2,
and (g) V3 and (h) q3 of capacitor 3?
V
A
C C C
Fig. 25-27 Problem 9.
Fig. 25-28 Problems 10 and 34.
V
C1
C3
C2
Fig. 25-31 Problem 15.
V
+
–
C2C5
C3
C4
C6
C1
C3C2
C1
V
(a) (b)
qs
0
1
V (V)
Vs
2
3q 
( 
 C
)
µ
Fig. 25-32 Problem 16.
••17 In Fig. 25-29, a potential difference of V" 100.0 V is ap-
plied across a capacitor arrangement with capacitances C1" 10.0 mF,
C2" 5.00 mF, and C3" 4.00 mF. If capacitor 3 undergoes electrical
breakdown so that it becomes equivalent to conducting wire, what is
the increase in (a) the charge on capacitor 1 and (b) the potential dif-
ference across capacitor 1?
••18 Figure 25-33 shows a circuit section of four air-filled
capacitors that is connected to a larger circuit.The graph below the
section shows the electric potential V(x) as a function of position x
•11 In Fig. 25-29, find the
equivalent capacitance of the
combination. Assume that C1 "
10.0 mF, C2 " 5.00 mF, and C3 "
4.00 mF.
••12 Two parallel-plate capaci-
tors, 6.0 mF each, are connected in
parallel to a 10 V battery. One of
the capacitors is then squeezed so
that its plate separation is 50.0% of
its initial value. Because of the squeezing, (a) how much additional
charge is transferred to the capacitors by the battery and (b) what
is the increase in the total charge stored on the capacitors?
••13 A 100 pF capacitor is charged to a potential
difference of 50 V, and the charging battery is disconnected. The
capacitor is then connected in parallel with a second (initially
uncharged) capacitor. If the potential difference across the first
capacitor drops to 35 V, what is
the capacitance of this second ca-
pacitor?
••14 In Fig. 25-30, the battery has a
potential difference of V " 10.0 V
and the five capacitors each have a
capacitance of 10.0 mF. What is the
charge on (a) capacitor 1 and (b) ca-
pacitor 2?
••15 In Fig. 25-31, a 20.0 V battery is connected across
ILWSSM
ILW
V
C2
C3
C1
Fig. 25-29 Problems 11,
17, and 38.
Fig. 25-30 Problem 14.
C2
+
– V
C1
••16 Plot 1 in Fig. 25-32a gives the charge q that can be stored on ca-
pacitor 1 versus the electric potential V set up across it. The vertical
scale is set by qs = 16.0 mC, and the horizontal scale is set by Vs = 2.0 V.
Plots 2 and 3 are similar plots for capacitors 2 and 3, respectively.
Figure 25-32b shows a circuit with those three capacitors and a 6.0 V
battery.What is the charge stored on capacitor 2 in that circuit?
Fig. 25-33 Problem 18.
12
x
x
4
1 2 3
2 V
5 V
V
V 
(V
)
capacitors of capacitances C1 " C6 " 3.00 mF and C3 " C5 "
2.00C2 " 2.00C4 " 4.00 mF. What are (a) the equivalent capac-
itance Ceq of the capacitors and (b) the charge stored by Ceq? What
halliday_c25_656-681v2.qxd 23-11-2009 14:32 Page 677
1
Problema 2
Na figura abaixo, a corrente na resistência 6 é i6 = 1, 40 A e as resistências são R1 = R2 = R3 = 2, 0 Ω,
R4 = 16, 0 Ω, R5 = 8, 0 Ω e R6 = 4, 0 Ω. Qual é a fem da fonte ideal?
••41 In Fig. 27-41, !1 ! 3.00 V, !2 ! 1.00 V, R1 ! 4.00 ", R2 !
2.00 ",R3 ! 5.00 ", and both batteries are ideal.What is the rate at
which energy is dissipated in (a) R1, (b) R2, and (c) R3? What is the
power of (d) battery 1 and (e) battery 2?
••42 In Fig. 27-52, an array of n par-
allel resistors is connected in series
to a resistor and an ideal battery. All
the resistors have the same resis-
tance. If an identical resistor were
added in parallel to the parallel ar-
ray, the current through the battery
would change by 1.25%. What is the
value of n?
••43 You are given a number of 10 " resistors, each capable of
dissipating only 1.0 W without being destroyed. What is the mini-
mum number of such resistors that you need to combine in series
••36 In Fig. 27-47, !1 ! 6.00
V, !2 ! 12.0 V, R1 ! 100 ", R2 !
200 ", and R3 ! 300 ". One point
of the circuit is grounded (V ! 0).
What are the (a) size and (b) direc-
tion (up or down) of the current
through resistance 1, the (c) size
and (d) direction (left or right) of
the current through resistance 2,
and the (e) size and (f) direction of the current through resistance
3? (g) What is the electric potential at point A?
and 10 V. The plots in Fig. 27-43b give the currents through the
two batteries as a function of !2. The vertical scale is set by is !
0.20 A. You must decide which plot corresponds to which battery,
but for both plots, a negative current occurs when the direction of
the current through the battery is opposite the direction of
that battery’s emf. What are (a) emf !1, (b) resistance R1, and (c)
resistance R2?
••33 In Fig. 27-44, the current in resistance 6 is i6 ! 1.40 A
and the resistances are R1 ! R2 ! R3 ! 2.00 ", R4 ! 16.0 ",
R5 ! 8.00 ", and R6 ! 4.00 ". What is the emf of the ideal
battery?
••37 In Fig. 27-48, the resistances
are R1 ! 2.00 ", R2 ! 5.00 ", and
the battery is ideal. What value of R3
maximizes the dissipation rate in
resistance 3?
••38 Figure 27-49 shows a section
of a circuit. The resistances are R1 !
2.0 ", R2 ! 4.0 ", and R3 ! 6.0 ",
and the indicated current is i ! 6.0
A. The electric potential difference
between points A and B that connect
the section to the rest of the circuit is
VA# VB! 78 V. (a) Is the device rep-
resented by “Box” absorbing or pro-
viding energy to the circuit, and (b)
at what rate?
••39 In Fig. 27-50, two batter-
ies of emf ! ! 12.0 V and internal
resistance r ! 0.300 " are connected
in parallel across a resistance R. (a) For what
value of R is the dissipation rate in the resistor
a maximum? (b) What is that maximum?
••40 Two identical batteries of emf ! ! 12.0
V and internal resistance r ! 0.200 " are to
be connected to an external resistance R, ei-
ther in parallel (Fig. 27-50) or in series
(Fig. 27-51). If R ! 2.00r, what is the current i
in the external resistance in the (a) parallel
and (b) series arrangements? (c) For which
arrangement is i greater? If R ! r/2.00, what
is i in the external resistance in the (d) paral-
lel and (e) series arrangements? (f) For which
arrangement is i greater now?
(a) (b)
R2R1
S
R2R1
R3
S
+
–
+
–
Fig. 27-45 Problem 34.
R1
R1
R3
R2
R2
A
B
D
C
+
–
Fig. 27-46 Problem 35.
1 2R1
R2 R3
A
+
–
+
–
Fig. 27-47 Problem 36.
+
–
R3R2
R1
Fig. 27-48 Problems 37
and 98.
Box
A B
i
R1
R2
R3
Fig. 27-49 Problem 38.
+ –
+ –
r
r
R
Fig. 27-50
Problems 39 
and 40.
+ – + –
rr
R
Fig. 27-51 Problem 40.
RR
R
n resistors
in parallel
Fig. 27-52 Problem 42.
R2R1 R5
R4 R6R3 i6
+
–
Fig. 27-44 Problem 33.
729PROBLEMS
PART 3
HALLIDAY REVISED
••34 The resistances in Figs. 27-45a and b are all 6.0 ", and the
batteries are ideal 12 V batteries. (a) When switch S in Fig. 27-45a
is closed, what is the change in the electric potential V1 across resis-
tor 1, or does V1 remain the same? (b) When switch S in Fig. 27-45b
is closed, what is the change in V1 across resistor 1, or does V1 re-
main the same?
••35 In Fig. 27-46, ! ! 12.0 V, R1 ! 2000 ", R2 ! 3000 ", and 
R3 ! 4000 ". What are the potential differences (a) VA# VB, (b)
VB # VC, (c) VC # VD, and (d) VA# VC?
halliday_c27_705-734v2.qxd 23-11-2009 14:35 Page 729
Problema 3
Um capacitor inicialmente descarregado é conectado em série a um resistor e uma fonte de fem E = 110 V ,
que possui resistência interna desprezível. Imediatamente após a ligação do circuito, a corrente que passa
no resistor é igual a 6, 5× 10−5 A. A constante de tempo do circuito é de 6, 2 s. Calcule a resistência do
resistor e a capacitância do capacitor.
Problema 4
Um elétron do tubo de imagem de um receptor de televisão está se movendo a 7, 2× 106 m/s na presença
de um campo magnético de 83, 0 mT . Determine: (a) o valor máximo e (b) o valor mínimo da força que
o campo magnético pode exercer sobre o elétron. (c) Em um certo instante, o elétron tem uma aceleração
de módulo 4, 90 × 1014 m/s2. Qual é o ângulo entre a velocidade do elétron e o campo magnético neste
instante?
Problema 5
Um elétron no ponto A da figura abaixo possui velocidade v0 igual a (1, 41× 106 m/s). Determine:
(a) o módulo, a direção e o sentido do campo magnético que obriga o elétron a descrever uma orbita
semicircular de A até B;
(b) o tempo necessário para que o elétron se desloque de A até B.
914 CHAPTER 27 Magnetic Field and Magnetic Forces
particle is measured to be 
(a) Calculate all the components of the velocity of the
particle that you can from this information. (b) Are there compo-
nents of the velocity that are not determined by the measurement
of the force? Explain. (c) Calculate the scalar product What
is the angle between and 
27.9 .. A group of particles is traveling in a magnetic field of
unknown magnitude and direction. You observe that a proton mov-
ing at in the experiences a force of
in the and an electron moving at
in the experiences a force of 
in the y-direction. (a) What are the magnitude and
direction of the magnetic field? (b) What are the magnitude and
direction of the magnetic force on an electron moving in the
at
Section 27.3 Magnetic Field Lines and Magnetic Flux
27.10 . A flat, square surface with side length 3.40 cm is in the 
xy-plane at . Calculate the magnitude of the flux through 
this surface produced by a magnetic field
.
27.11 . A circular area with a radius of 6.50 cm lies in the
What is the magnitude of the magnetic flux through this
circle due to a uniform magnetic field (a) in the 
-direction; (b) at an angle of from the -direction; (c) in
the direction?
27.12 . A horizontal rectangular surface has dimensions 2.80 cm
by 3.20 cm and is in a uniform magnetic field that is directed at an
angle of above the horizontal. What must the magnitude of
the magnetic field be in order to produce a flux of 
through the surface?
27.13 .. An open plastic soda bottle with an opening diameter of
2.5 cm is placed on a table. A uniform 1.75-T magnetic field directed
upward and oriented from vertical encompasses the bottle. What
is the total magnetic flux through the plastic of the soda bottle?
27.14 .. The magnetic field 
in a certain region is 0.128 T,
and its direction is that of the 
-axis in Fig. E27.14. (a) What
is the magnetic flux across the
surface abcd in the figure? 
(b) What is the magnetic flux
across the surface befc? (c) What
is the magnetic flux across the
surface aefd? (d) What is the net
flux through all five surfaces that
enclose the shaded volume?
Section 27.4 Motion of Charged Particles in a
Magnetic Field
27.15 .. An electron at point 
in Fig. E27.15 has a speed of
Find (a) the
magnitude and direction of 
the magnetic field that will cause
the electron to follow the semi-
circular path from to and
(b) the time required for the
electron to move from to 
27.16 .. Repeat Exercise 27.15 for the case in which the particle
is a proton rather than an electron.
B.A
B,A
1.41 * 106 m>s. v0 A
+z
B
S
25°
4.20 * 10-4 Wb
30.0°
+y-
+z53.1°+z
B = 0.230 T
xy-plane.
10.300 T2≥n! 10.500 T2kN BS " 10.200 T2ın #z = 0
3.20 km>s?-y-direction
+10-16 N
8.50 *-z-direction4.75 km>s +y-direction,2.25 * 10-16 N +x-direction1.50 km>s
F
S
?vS
vS # FS.
10-7 N2≥n. 17.40 *FS " !13.40 * 10-7 N2ın # 27.17 . CP A 150-g ball containing excess electronsis dropped into a 125-m vertical shaft. At the bottom of the shaft,
the ball suddenly enters a uniform horizontal magnetic field that
has magnitude 0.250 T and direction from east to west. If air resist-
ance is negligibly small, find the magnitude and direction of the
force that this magnetic field exerts on the ball just as it enters the
field.
27.18 . An alpha particle (a He nucleus, containing two protons
and two neutrons and having a mass of ) traveling
horizontally at enters a uniform, vertical, 1.10-T mag-
netic field. (a) What is the diameter of the path followed by this
alpha particle? (b) What effect does the magnetic field have on the
speed of the particle? (c) What are the magnitude and direction of
the acceleration of the alpha particle while it is in the magnetic
field? (d) Explain why the speed of the particle does not change
even though an unbalanced external force acts on it.
27.19 . CP A particle with charge travels in a
circular orbit with radius 4.68 mm due to the force exerted on it by
a magnetic field with magnitude 1.65 T and perpendicular to the
orbit. (a) What is the magnitude of the linear momentum of the
particle? (b) What is the magnitude of the angular momentum of
the particle?
27.20 . (a) An nucleus (charge moving horizontally
from west to east with a speed of experiences a mag-
netic force of 0.00320 nN vertically downward. Find the magni-
tude and direction of the weakest magnetic field required to
produce this force. Explain how this same force could be caused
by a larger magnetic field. (b) An electron moves in a uniform,
horizontal, 2.10-T magnetic field that is toward the west. What
must the magnitude and direction of the minimum velocity of the
electron be so that the magnetic force on it will be 4.60 pN, verti-
cally upward? Explain how the velocity could be greater than this
minimum value and the force still have this same magnitude and
direction.
27.21 . A deuteron (the nucleus of an isotope of hydrogen) has a
mass of and a charge of The deuteron travels
in a circular path with a radius of 6.96 mm in a magnetic field with
magnitude 2.50 T. (a) Find the speed of the deuteron. (b) Find the
time required for it to make half a revolution. (c) Through what
potential difference
would the deuteron have to be accelerated to
acquire this speed?
27.22 . In an experiment with
cosmic rays, a vertical beam of par-
ticles that have charge of magnitude
and mass 12 times the proton
mass enters a uniform horizontal
magnetic field of 0.250 T and is
bent in a semicircle of diameter
95.0 cm, as shown in Fig. E27.22.
(a) Find the speed of the particles
and the sign of their charge. (b) Is
it reasonable to ignore the gravity force on the particles? (c) How
does the speed of the particles as they enter the field compare to
their speed as they exit the field?
27.23 . A physicist wishes to produce electromagnetic waves of
frequency 3.0 THz using a
magnetron (see Example 27.3). (a) What magnetic field would be
required? Compare this field with the strongest constant magnetic
fields yet produced on earth, about 45 T. (b) Would there be any
advantage to using protons instead of electrons in the magnetron?
Why or why not?
11 THz = 1 terahertz = 1012 Hz2
3e
+e.3.34 * 10-27 kg
500 km>s+8e)16O
L
S
pS
6.40 * 10-19 C
35.6 km>s 6.64 * 10-27 kg
4.00 * 108
z
x
y
50.0 cm
e
c
f
d
a
b
40.0 cm
30.0 cm
30.0 cm
Figure E27.14
10.0 cm
A B
v0
Figure E27.15
95.0 cm
B
S
Figure E27.22
2