Physics 121 Honors
Common Exam 1
Name:
Spring 2003
Section:
Closed book exam. Only one 8.500 × 1100 formula sheet (front and back side) can be used. Calculators are
allowed. Use the scantron forms (pencil only!) for the multiple choice problems. Circle the answers on the
examination sheet as well, and return it together with the scantron form. Use the back of these pages, or
attach your own pages with solutions for problems that require calculations.
The multiple-choice problems are 5 points each. Partial credit (up to 3 points) will be given for multiplechoice problems, if you provide a detailed solution on the examination sheets. The work-out problems are
10 points each. The nominal number of points is 100.
Clearly print your first and last name and indicate your section number on both the scantron form and the
examination sheet.
Good luck!
Multiple-Choice Problems
Problem 1: Two tiny conducting balls of identical mass m and identical charge q = 3.20 × 10−8 C hang
from nonconducting threads of length L = 140 cm. The angle in the equilibrium position as indicated in the
figure is θ = 2.8◦ . What is the mass m of the conducting balls?
A) m = 1 g
Fe = T sin θ mg = T cos θ
Fe
x
B) m = 2 g
⇒ tan θ =
and sin θ =
mg
2L
C) m = 10 g
2
q
⇒m =
4πε0 x2 g tan θ
D) m = 20 g
q2
=
= 0.001 kg
E) m = 40 g
16πε0 gL2 tan θ sin2 θ
√
Problem 2: The radius of the nucleus of a Gold atom is about rAu ≈ 1.4 × 10−15 3 A m, where A = 197 u
is the atomic mass of Gold. What is the magnitude of the repulsive electrostatic force between two protons
separated by rAu ?
A) F = 0.60 N
F =
B) F = 3.5 N
C) F = 8.4 N
D) F = 120 N
E) F = 1.8 × 105 N
=
1 e2
2
4πε0 rAu
e2
= 3.47 N
4πε0 (1.4 × 10−15 m)2 A2/3
Problem 3: Three point charge lie along the x-axis. The positive charge q1 = 12.0 µ C is at x1 = 1.50 m,
the positive charge q2 = 5.20 µ C is at the origin, and the resultant force on the negative charge q3 is zero.
What is the x-coordinate of q3 ?
A) −0.90 m
B) −0.60 m
C) +0.60 m
D) +0.90 m
1 q2 q3
1
q1 q3
=
2
4πε0 x
4πε0 (x1 − x)2
√
√
⇒ q2 (x1 − x) =
q1 x
√
q2
x = √
√ x1 = +0.595 m
q1 + q2
E) +2.40 m
Problem 4: What is the direction and magnitude of the electric field due to the three point charges at point
P as shown in the figure?
A) Enet = 2q/(πε0 a2 ) @ − 45◦
1 +2q
4πε0 d 2
1 2q
=
4πε0 12 a2
q
=
πε0 a2
E =
B) Enet = q/(πε0 a2 ) @ − 45◦
C) Enet = q/(4πε0 a2 ) @ + 45◦
D) Enet = q/(2πε0 a2 ) @ + 45◦
E) Enet = q/(πε0 a2 ) @ + 45◦
Problem 5: A nonuniform electric field given by ~E = 2.0xî − 5y2 jˆ pierces the Gaussian cube shown in the
figure. E is measured in [N/C] and x in [m]. What is the electric flux Φtop through the top face of the cube?
A) −80 Nm/C2
B) −40 Nm/C2
C) +40 Nm/C2
D) +80 Nm/C2
E) +405 Nm/C2
I
Φ =
~E d~A = −EA
= −5y2 a2 = −80 Nm2 /C
Problem 6: Two identical beads each have mass m = 10.0 g. When placed in a hemispherical bowl of
radius R = 78.5 cm with frictionless, nonconducting walls, the beads move and at equilibrium they are a
distance R apart. What is the charge q on each bead?
A) 1.6 µ C
B) 2.0 µ C
R
R
C) 2.8 µ C
m
D) 3.7 µ C
m
R
Fe = N cos θ and mg = N sin θ
mg 4πε0 R2 mg
⇒ tan θ =
=
Fe
q2
r
4πε0 mg
⇒q =
R = 1.97 µ C
tan 60◦
E) 4.5 µ C
Problem 7: An electric dipole consists of two charges +3.0 µ C and −3.0 µ C separated by 0.64 mm. The
dipole is immersed in a uniform electric field of magnitude E = 2.4 × 106 N/C along the negative x-axis
and the dipole moment points along the positive y-axis. What is the magnitude and direction of the torque
on the dipole?
A) 4.6 × 10−3 Nm (counter-clockwise)
B) 4.6 × 10−3 Nm (clockwise)
C) 0
D) 7.2 Nm (counter-clockwise)
E) 7.2 Nm (clockwise)
~τ = ~p × ~E
τ = pE sin 90◦ = p E = q d E = 4.6 × 10−3 Nm
Problem 8: The figure shows four spheres, each with charge +Q uniformly distributed through its volume.
A point P, as shown in the figure, has the same distance from the center of the spheres. Rank the spheres
according to the magnitude of the electric field that they produce at point P, greatest first!
A) (c), (d), (b), (a)
B) (b), (a), (d), (c)
C) (d), (c), (b), (a)
D) (a), (b), (c), (d)
E) (a) and (b) tie, (c), (d)
Charge distribution in (a) and (b) does
not matter as long as P is outside the
sphere, the total charge is the same,
and the charge distribution is spherically symmetric. The ranking of
(a)/(b), (c), and (d) depends only on
the total charge enclosed by a sphere
whose surface encloses P.
Problem 9: A spherical, conducting shell has a total charge of Q = −8 µ C. After a point charge of +16 µ C
is placed at the center of the spherical shell, what is the charge on the outer surface of the shell?
A) +16 µ C
B) +8 µ C
C) 0 µ C
Q = −8 µ C and q = +16 µ C
⇒ Qin = −q = −16 µ C
⇒ Qout = Q − Qin = +8 µ C
D) −8 µ C
E) −16 µ C
Problem 10: A spherical shell has a uniform volume charge density ρ = 3.33 × 10−6 C/m3 between a =
20 cm and b = 45 cm and ρ = 0 inside the spherical shell (r < a). What is the magnitude of the electric
field a distance c = 30 cm from the center of the shell?
A) E = 1.3 × 105 N/C
B) E = 8.3 × 104 N/C
C) E = 6.0 × 105 N/C
D) E = 2.6 × 104 N/C
E) E = 1.9 × 104 N/C
¢
4 ¡
qenc = ρ V = ρ π c3 − a3
3
1 qenc
E =
4πε0 c2
¢
ρ ¡ 3
3
=
c
−
a
3ε0 c2
= 2.65 × 104 N/C
Work-Out Problems
Please indicate your final answers and make sure that all the steps of your solution are shown!
Problem A: A rod of length l has a uniform positive charge per unit length λ and a total charge of +Q.
a) Calculate the electric field ~E(P) at point P that is located along the long axis of the rod and a distance
a from one end!
b) Provide a sketch and clearly indicate your coordinate system!
c) What is the electric field if the point P is far away from the end of the rod (a À l)?
1
dq
4πε0 (l + a − x)2
Z l
Z l
Z l
1
dq
λ
dx
λ
dx
⇒E =
=
=
2
2
4πε0 0 (l + a − x)2
0 4πε0 (l + a − x)
0 4πε0 (l + a − x)
·
¸l
·
¸
·
¸
·
¸
λ
1
λ
1
1
λ
l
1
q
=
=
−
=
=
4πε0 (l + a − x) 0 4πε0 a l + a
4πε0 a(l + a)
4πε0 a(l + a)
1
a À l ⇒ E ∼ 2 (point charge)
a
dE
= dEx =
Problem B: A proton (m p = 1.67 × 10−27 kg) is projected in the positive x-direction into a region of a
uniform electric field ~E = (−4.75 × 105 N/C)î. The proton travels 5.48 cm before coming to rest. Neglect
gravity in parts (a) to (d)!
a) Calculate the acceleration vector of the proton!
b) What was the initial speed of the proton?
c) Do you have to consider relativistic effects?
d) Determine the time it takes the proton to come to rest!
e) How would gravity affect the trajectory of the proton?
~F
eE
= 4.55 × 1013 m/s2
a=
m
p
v20 + 2a(x − x0 ) ⇒ v = 2a(x − x0 ) = 2.23 × 106 m/s
c ⇒ (relativistic effects negligible)
2(x − x0 )
1
⇒ t=
= 4.9 × 108 s
2 (v0 + v)t
v
Fe ⇒ (gravity negligible)
= q~E = +e~E = m~a
v2 =
v ¿
x − x0 =
Fg ¿
⇒
Problem C: Two curved plastic rods, one of charge +q and the other of charge −q, form a circle of radius R
in an x, y-plane. The positively charged arc spans 240◦ and the negative one 120◦ . The charge is uniformly
distributed on both rods.
a) Provide a sketch that includes a coordinate system and draw the electric field lines!
b) What are the magnitude and direction of the electric field ~E at a point P located at the center of the
circle?
i+120◦ 1.73λ
λ h
1.73q
sin θ
=
= 4 2 2
◦
4πε0 r
4πε0 r 4( 3 )π ε0 r
−120
◦
i
h
+60
λ
1.73λ
1.73q
E− =
=
sin θ
= 2 2 2
◦
4πε0 r
4πε0 r 4( 3 )π ε0 r
−60
~E = −(E + ) jˆ − (E − ) jˆ = 1.73q jˆ
8 π 2 ε0 r 2
E+ =
λ=
λ=
q
2(2π r/3)
q
2π r/3
Problem D: A long, straight wire is surrounded by a hollow metal cylinder whose axis coincides with that
of the wire. The wire has a charge per unit length of λ , and the cylinder has a net charge per unit length of
2λ . From this information, use Gauss’s law to find
a) the charge per unit length on the inner surface of the cylinder,
−λ
b) the charge per unit length on the outer surface of the cylinder,
+3λ
c) the electric field inside the cylinder,
E=
λ
2πε0 r
(radially outward)
d) the electric field outside the cylinder, and
E=
3λ
2πε0 r
(radially outward)
e) the electric field in the interior of the metal cylinder.
E =0
f) Provide a plot of the electric field E as a function of the distance r from the central axis.
Electric Field Strength [N/C]
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
Radial Distance [ m ]
0.4
0.5
Problem E: The picture shows the basic set-up of the Millikan experiment, which is used to measure the
elementary charge.
a) Describe in a few words the basic idea of the Millikan experiment!
b) Explain the expression that the charge of an object is quantized!
c) How could today’s technology help to improve the measuring technique?
d) In the simplified version of the Millikan experiment, we assume that the
oil drops are stationary, i. e., they are hovering in mid-air. How would
you have to rewrite Newton’s 2nd law, if the oil drops are not stationary,
i. e., they are moving downwards?
e) What factors have to be taken into account that could adversely influence the measurement?
Before you turn in your quiz, please double-check you multiple-choice answers. Are they marked
correctly on the scantron form? Don’t forget to use the correct units in the work-out problems!