Download 17 27 Line Charge - nchsdduncanapphysics

LivePhoto Physics Activity 27
Name: _____________________________________
Page 1 of 4
Electric Field Due to a Line of Charge
Consider a thin insulated rod that carries a known negative
charge Qrod that is uniformly distributed. It is possible to
determine the electric field along a line perpendicular to the rod
that passes through its center using the following equation –
often derived in introductory texts:
 theory
Erod

kQrod 
r r 2  ( L / 2) 2
iˆ
x
r
L
[Eq. 1]
where L is the length of the charged part of the rod, r is the
distance from the test charge to the center of the charged part
of the rod, and Qrod is its total charge. The constant k is the
well-known Coulomb constant.
y
Qrod
Qball
Test
Charge
Figure 1: A uniformly charged rod
exerts a force on a test charge
In this activity you will examine a digital movie of a charged rod exerting a force on a hanging “test charge” along
with a Logger Pro analysis to determine if the theoretical equation [Eq. 1] describes the relationship between r, L and

meas at the location of the test charge.
the measured electric field, Erod

R
x
How a Horizontal force on a Hanging Ball Displaces It:
Before doing this activity you should have completed
Activity 25 entitled Coulomb’s Law for Two Charged
Spheres. There you found that if the center of a small ball of
mass m is hanging vertically a distance R from a point of
support and then becomes displaced by a distance x <<R
from a vertical axis (as shown in Fig. 2), it must be
experiencing a horizontal force given by
Fig. 2: A force acting in the horizontal
direction on a hanging ball can cause it to
be displaced by a horizontal distance x.
1. Preliminary Questions
Fx 
mgx ˆ
i
R
[Eq. 2]

Note: You will receive full credit for each prediction made in this preliminary section whether or not it matches
conclusions you reach in the next section. As part of the learning process it is important to compare your predictions
with your results. Do not change your predictions!
As you proceed with this assignment, you’ll be working with a short video clip entitled < V-Rod_Ball1.mov> in which
a uniformly charged rod is brought closer and closer to a hanging ball that is also charged. The insulated plastic rod
has been rubbed with cat’s fur so that it is negatively charged. Before answering the questions in this section view the
QuickTime movie by using the arrow keys on the keyboard to view it frame by frame.
(a) Is the hanging ball in the movie positively or negatively charged? Explain your answer. Hint: What happens when
a rubber rod that has been rubbed with cat’s fur is brought in contact with a small ping pong ball covered with
conducting paint?
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Name: _______________________________________
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(b) Sketch electric field vectors at the location of a positive test charge for the three small segments (labeled 1, 2, and
3) based on its interaction with the negatively charged rod shown in the Figure 3 below. Please indicate both the
direction and relative magnitude of each of your vectors. Base your scale on the vector associated with segment
2 and assign it a length of 2 cm. Hint: Assume each segment behaves like a point charge so you can use
Coulomb’s Law to determine the relative magnitudes of the vectors.
1
2 cm
y
L 2
x
3
Figure 3: A uniformly charged rod exerts a force on a test
charge
(c) Find the direction of the E-field at the location of the test charge. Turn your thoughts back to the movie.
Assume that the center of the charged part of the rod shown in the movie (i.e., between its bottom end and the
bottom of the hand) lies on the x-axis. What will the direction of the net electric field be at the negatively charged
hanging ball’s location if the charged rod, like that shown in Figure 3, acts on it? Use the x-y coordinate system
shown in Figures 1 and 3 to describe the direction of this E-field. Note: For now, ignore the fact that center of the
rod in the movie is not quite on the x-axis.
(d) Equation 1 predicts that the E-field lies along the x-axis only. Does this agree with the conclusion you reached?
Why or why not?
(e) Assume the hanging ball acts as a “test charge” by carrying an excess charge, Qball. If its center is displaced from
its original vertical axis by a distance, x, write down the equation need to calculate the electric field at the ball’s
center based on measurements of the ball’s horizontal displacement, x, its mass, m, charge, Qball, the gravitational
constant, g, and the distance, R, from the ceiling to the center of the ball (with x << R).
m eas
Erod,
x 
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[Eq. 3]
Physics with Video Analysis
2. Activity-Based Questions
In this section, you will be working with a Logger Pro file entitled <LineCharge.cmbl> in which data for the position
of the center of the rod and the center of the ball have been collected in each frame of the movie. To check this out
open the file <LineCharge.cmbl> and click on the Start button in the Replay box to see the movie play. At the same
time you will see a graph emerge of the measured x-component of the electric field produced by the rod, Erod, x (as
determined by Eq. 3) at the location of the hanging ball vs. the measured distance, r, between the ball and the center of
the rod.
Note: The origin of the video analysis coordinate system was placed at the location of the center of the ball when it
was hanging vertically.
(a) Determine the length of the charged part of the rod: Use the Photo Distance tool ( ) to find the length L
of the rod (between the rod’s bottom and the bottom edge of the hand). Enter your measured value in the
space below.
L=
m
(b) Determine whether the calculated values of the electric field based on measurements are consistent with
m eas
theory: The key to this determination is to see if the Logger Pro graph of Erod,
x vs. r (based on Equation 3) can
be fit by the function described in Equation 1.
To do this fit:
(1) Choose Curve Fit in the Analyze menu.
(2) Scroll down through the equations and
select the one that looks like Equation 1
(V-rod_Pt Charge Theory).
(3) Do the fit and draw the fit line in the
graph. Enter the values of A (to
four significant figures) and L to
two significant figures) on the graph to
the right.
Best Fit:
A = ________N•m2/C
L = ________ m
(c) Comment on whether you think the fit is good enough to verify that the theoretical equation adequately describes
the electric field due at various distances, r, along a perpendicular axis passing through the center of the rod.
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Name: _______________________________________
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(d) Note that, to within a sign, the A in your fit equation corresponds to the product of the Coulomb constant
( k  8.99109 N m2 /C2 ) and the charge on the rod. Use these facts to calculate the charge on the rod, Qrod in nC
(nanocoulombs). Show your calculations and express your answer using three significant figures.

3. Reflections on Your Findings
(a) Given the methods used to charge the rod and the hanging ball, do you expect the charge on the rod to be greater
than the charge on the ball or less than the charge on the ball? Why? (The 35 nC listed for the ball is based on
measurements made with the charge sensor distributed by Vernier Software & Technology.)
(b) In light of your answer in part (a), comment on how charge you calculated for the rod compares with the 35 nC
charge measured for the hanging ball.
(a) You should have noticed that the “fit value” of the rod length L is almost 10% lower than the directly measured
value. Hint: What impact might the rod being a bit off center have of the experimentally determined electric field
value?
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Physics with Video Analysis