Problem 1 (25 pts).
(a: 5 pts) A student claimed that the formula for the electric field outside a cube of edge length L, carrying a
uniformly distributed charge Q, at a distance x from the center of the cube, was
Q L
E = ----- ------ . Explain how you know that this formula cannot be correct.
ε0 x
The units are not correct.
(b: 5 pts) A thin plastic rod of length 4.5 m has been rubbed with a wool cloth, and has gained a net charge
–7
of 3 ×10 coulombs, distributed uniformly over its surface. A cubical block of copper with side length 0.07 m
is placed with its center 0.12 m from the center of rod, as shown in the diagram below. Note that the rod is
very long, and only part of it is shown on the diagram.
On the diagram, show the distribution of charges in and on the copper, using the conventions discussed in
the textbook.
------
0.12 m
0.07 m
+++++
(c: 5 pts) On the diagram above draw an arrow representing the electric field at the center of the copper
block, due only to the charges in and on the copper.
(d: 10 pts) What is the magnitude of the electric field vector you drew in part (c)? Show your work, and
explain.
10,000 N/C
Problem 1 ( 12 pts)
(a: 4 pts) The magnitude of the electric field due to a small dipole is measured to be 250 N/C at a location
on the axis of the dipole, 2 cm from the center of the dipole. Approximately what is the magnitude of the
electric field due to the dipole at a location 6 cm from the dipole, along the same axis? Show your work.
9.26 N/C
(b: 4 pts) The magnitude of the electric field due to a uniformly positively charged disk of radius 4 m is measured to be 3000 N/C at a location in the air 6 mm from the center of the disk, along a line perpendicular to
the disk. Approximately what is the magnitude of the electric field due to the disk at a location 9 mm from
the center of the disk, on the same line? Show your work.
3000 N/C
(c: 4 pts) The magnitude of the electric field due to a uniformly negatively charged thin plastic rod of length
2 m is measured to be 970 N/C at a location near the middle of the rod, at a location 8 cm from the rod, on
a line perpendicular to the rod. Approximately what is the magnitude of the electric field at a location 4 cm
from the rod, on the same line? Show your work.
1940 N/C
Problem 2 ( 12 pts)
At each of the the locations marked by a letter (A, B, C) in the diagram below, draw an arrow representing
the net electric field at that location. A longer arrow should represent a field with larger magnitude.
A
B
C
2A
Problem 4 (35 pts).
–6
A glass disk carries a positive charge of 3.5 ×10 distributed uniformly over its surface. Its center is at the ori–8
gin, and its radius is R = 7 m. A small hollow plastic ball, of radius 0.02 m, carrying a charge of – 2.0 ×10 C
uniformly distributed over its surface, is located with its center at ⟨ – 0.07, – 0.04, 0⟩ m.
(a: 23 pts) Calculate the net electric field at location
⟨ – 0.05, 0.1, 0⟩ . Show every step in your calculation explicitly. Give
your answer as a vector.
y
<-3800, -8900, 0> N/C
R (not to scale
x
(b: 5 pts) Draw the electric field vector you calculated on the diagram, with its tail at the observation location.
(c: 2 pts) Explicitly state any approximations or assumptions you
made in your calculation.
The disk is very large so its electric field is approximately horizontal and
its size is about
.
(c: 5 pts) What would be the force on an antiproton (whose charge is –e and whose mass is the same as the
mass of a proton) if it were located at ⟨ – 0.05, 0.1, 0⟩ ?
<6.1, 14, 0> 10e-16 N
Problem 4 (18 pts)
A small, very lightweight hollow aluminum ball is suspended from a cotton thread. The events depicted in frames 1-6
then occur, in sequence. All diagrams show cross-sectional views of the objects.
For each frame (1-6), write the letter of the corresponding diagram (A-P) that best depicts the distribution of charge
in and/or on the aluminum ball, following the conventions for diagrams discussed in the textbook and in class.
Some letters may be used more than once; others may not be used at all.
1. You touch the ball
briefly with your fingers,
then release it.
2. Holding only the thread,
you bring the ball near a
charged metal block.
Ball diagram:
Ball diagram:
K
3. The ball swings toward
the block.
Ball diagram:
D
D
Ne
pla gativ
stic ely
pe cha
n
rg
ed
4. The ball briefly touches
the block.
5. The ball swings away
from the block.
Ball diagram:
Ball diagram:
B
6. The ball is repelled by a
negtively charged plastic pen.
Ball diagram:
H
Diagrams showing distribution of charge in and/or on the aluminum ball. Only patterns are shown, not magnitudes: both distributions at right would be represented by
diagram A.
or
A
B
C
F
G
H
I
J
L
M
O
P
K
D
H
E
4A
Problem 3 (15 pts)
The arrangement of charged particles shown in the diagram is called an electric quadrupole. At locations on the x axis far
from the quadrupole, the electric field due to the quadrupole has the form:
E =
1 h
0, 0⟩
4πε0 x 4
⟨ ------------ -----,
( h is a constant with units C ⋅ m 2 , combining the magnitudes of the charges, and their separations).
The diagram below is not drawn to scale; both locations A and B are far from the quadrupole. The origin is at the center of
the quadrupole.
+
–
–
< 0, 0, 0>
+
A
< a, 0, 0>
B
<b, 0, 0>
(a: 2 pts) At two locations between A and B, draw arrows showing the electric field at that location. The relative magnitudes
of the arrows should be correct (longer arrow means larger magnitude).
(b: 3 pts) Is V B – V A positive, negative, or zero? Briefly but clearly explain your reasoning:
Negative, because the electric field is in the the direction of the path.
(c: 10 pts) What is V B – V A ? Your answer should be a symbolic (algebraic) expression, which may include the constants h, a,
b, and 1 ⁄ ( 4πε 0 ) . Show all steps in your work.
4
Problem 8 (Bonus: 5 pts)
This bonus problem is worth only 5 points, so don’t work on it until you have completed and checked your work
on all of the other problems in the test.
A long rod of length L carries a uniform negative charge –Q. In the diagram, A, B, and C are locations. a, b, and c are distances. Calculate the
potential difference VA – VC. Your answer should be a symbolic (algebraic) expression. All of the distances are small compared to L. Explain
your work carefully.
Uniform
charge –Q
b
c
L
C
B
a
A
6