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Unit 2: Students Absolutely Must Learn…
Weekly Activity 3: Electric Field Mapping
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The relationship between potential energy and force, visually and mathematically.
The relationship between electric potential and electric field, visually and
mathematically.
How to calculate partial derivatives.
The difference between vector fields and scalar fields.
How to visualize a vector field with arrow graphs.
How to visualize a vector field with field lines.
How electric fields cause electric forces on charged particles.
How to draw equipotential lines and electric field lines.
How to estimate electric fields from electric potential measurements.
Weekly Activity 4: Cathode Ray Tube
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How charged particles behave in a scalar electric potential field.
How charged particles behave in an electric vector field.
How to relate kinematics/mechanics concepts to electricity concepts.
How a CRT works, how parallel plates of different voltages affect electrons.
How to complete and explain a long physics derivation.
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Unit 2 Grading Guidelines
Staple the lab report, then graphs, and finally worksheets together. Please put
your worksheets in order. Turn in your work to your TA at the beginning of the
next lab meeting following the completion of the unit.
Unit Lab Report [50%, graded out of 25 points]
Write a separate section using the section titles below (be sure to label these sections in your
report). In order to save time, you may add diagrams and equations by hand to your final
printout. However, images, text or equations plagiarized from the internet are not allowed!
Remember to write your report alone as collaborating with a lab partner may make you both
guilty of plagiarism. Pay close attention to your teacher for any changes to these guidelines.

Title [0 points] – A catchy title worth zero points so make it fun.

Concepts & Equations [9 points] – {One small paragraph for each important concept, as
many paragraphs as it takes, 2+ pages.} Go over the lab activities and make a list of all
the different concepts and equations that were covered. Then simply one at a time
write a short paragraph explaining them. You must write using sentences & paragraphs;
bulleted lists are unacceptable.
Some example concepts for this unit report include (but are not limited to):
 How the concept of electric potential is related to topographical maps.
 How the steepness of the equipotential landscape is related to the electric field.
 How to relate E and V mathematically. Describe in words the meaning of the
equations.
 Discuss how to relate F and U mathematically. Compare the math of E &V to
the math of F &U.
 Describe (without math) using text and diagrams how a cathode ray tube works.
 Examine the vertical CRT deflection equation. Explain how the deflection
equation
works (how to think about this equation). What happens to the

deflection distance when Vd,y is increased and WHY? What happens to the
deflection distance when Va is increased and WHY?
 Derive the deflection equation. Explain every step of the derivation. This is the
largest part of this section of the report and will be worth a large percentage of
the grade. It should take at least a full page. (Note: see the post-lab, you may
simply insert your work into your lab report if you leave space.)
 Any other equations that were used in the activities will need explained.
 Any other specific TA requests:
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Selected In-Class Section [6 points]: {3-5 paragraphs, ~1 page}
This week's selection is: Weekly Activity 4, In-Lab Section 2, Subsection A
Write a "mini-report" for this section of the lab manual. Describe what you did
succinctly, and then what you found accurately. Then explain what the result means
and how it relates to some of the concepts in the previous section. You must write using
sentences & paragraphs; bulleted lists are unacceptable.
o Procedure: Do not provide a lot of specific details, but rather you should
summarize the procedure so that a student who took the course a few years ago
would understand what you did.
o Results: Do not bother to rewrite tables of data, but rather refer to the page
number on which it is found. State any measured values, slopes of ilnes-of-bestfit, etc. Do not interpret your results, save any interpretation for the discussion.
o Discussion: Analyze and interpret the results you observed/measured in terms of
some of the concepts and equations of this unit. It is all right to sound repetitive
with other parts of the report.

Open-Ended / Creative Design [6 points] – {3-5 paragraph, ~1 page} Choose one of the
open-ended experiments from the two weekly activities to write about. Describe your
experimental goal and the question you were trying to answer. Explain the ideas you
came up with and what you tried. If your attempts were successful, explain your results.
If your attempts resulted in failure, explain what went wrong and what you would do
differently in the future. You must write using sentences & paragraphs; bulleted lists are
unacceptable.
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Graphs [4 points] - {attach to typed report} Graphs must be neatly hand-drawn during
lab and placed directly after your typed discussion (before your quizzes and selected
worksheets). Your graphs must fill the entire page (requires planning ahead) and must
include: a descriptive title, labeled axes, numeric tic marks on the axes, unit labels on the
axes, and if the graph is linear, the line of best fit written directly onto the graph.
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Thoroughly Completed Activity Worksheets [30%, graded out of 15 points]
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Week 3 In-Class [7 points]: Pages assigned to turn in:
_TA signature page, Post-lab pages, ____________________________________
___________________________________________________________________
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Week 4 In-Class [8 points]: Pages assigned to turn in:
_TA signature page, _____________ ____________________________________
___________________________________________________________________
The above lab report and worksheets account for 80% of your unit grade. The
other 20% comes from your weekly quizzes, each worth 10%. These will be
entered into D2L separately.
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Weekly Activity 3: Electric Field Mapping
Pre-Lab
!
You must complete this pre-lab section before you attend your lab to prepare
for a short quiz. Be sure to complete all pages of the pre-lab.
Continue until you see the stop pre-lab picture:
Subsection 0-A
One view of particle dynamics is to think about how a particle responds to the forces exerted
on it (the vector force field). Another equally valid point of view is to may think of how a
particle responds to the potential energy landscape that surrounds it (the scalar energy field).
An example of how particles move in one dimension is shown below. Imagine that some
external force creates the “hilly” potential energy graph shown, and that four particles are
placed at locations A through D.
The particle at A is at a minimum of the potential energy curve so that if it is moved, the
particle will rise in potential energy. Therefore, the particle at A remains at rest because it is in
its "lowest potential energy state". Even if it is perturbed a little, it will simply oscillate back
and forth around A. The particle at B sits at a part of the potential energy curve that has a
positive slope. If it moves to the left, it can lower its potential energy. Therefore, the particle
at B feels a force pushing it to the left.
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The particle at C sits at a precarious position. The slope of the potential energy curve at C is
zero so that the particle feels no force. However, any small perturbation in the particles
position will cause it to “tumble” from its unstable equilibrium. The particle at D has been
placed where the slope of the potential energy curve is negative. If the particle moves to the
right, it will lower its potential energy. Therefore, the particle at D feels a force pushing it to
the right. Notice that the magnitude of the slope at B is greater than at D. Thus the particle at
B does not have to move as far to lower its energy as the particle at D. This corresponds to the
particle at B feeling a stronger force than the particle at D.
The big idea here is that given a graph of potential energy, you can find the
corresponding force at each location by realizing that force equals the negative slope of the
 
dU(x)
potential energy, F (x)  
xˆ . Remember that acceleration is proportional to a  F
dx
and that the acceleration vector points in the same direction as the force, but the velocity
vector usually does not point in the same direction as the force (only for acceleration in one
direction).

¿
0-A-1
1
2
If a potential energy is given as U(x)  k x  22 where k  3 [J/m 2 ], what is the
force from this potential energy on a particle placed at x = 1 m (always use SI
units)?


¿
0-A-2
For the system described in the previous question, at which positions is the
force zero?
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Subsection 0-B
The previous technique may be applied to motion in more than one direction:
In this situation, the particle can move in the x and y directions. The potential energy landscape
shows a single unstable equilibrium (“mountaintop”). Dotted lines have been drawn to show
where the potential energy is constant. These are called equipotential lines. If this were a
topographical map, these lines would represent constant height (called contours), and
mountaineers maneuvering around the mountain at a constant height would be “contouring”.
The force on the particle “down the mountain” is always perpendicular to the equipotential
lines.
Again the force will equal the negative slope of the potential energy graph, but how do you
describe a two-dimensional slope? The answer is that the slope will be a two-dimensional
vector, one component to describe the slope in the x-direction and another component to
describe the slope in the y-direction. These two slopes can be found using partial derivatives to
U(x, y)
U(x, y)
xˆ 
yˆ . (For those students
yield a two dimensional force vector: F (x, y) 
x
y
who have not yet used partial derivatives, they are very simple and will be explained next.) This
equation makes intuitive sense: if you want the component of the force in the x-direction, find
out how much the potential energy is changing in the x-direction using the partial x-derivative.

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Partial derivatives are incredibly easy; all you do is take the regular derivative of the given
 (3x 2 y 3 )
variable while treating all other variables as constants. For example,
 6 xy 3 . Here
x
you see that taking the derivative of the x squared brings down a 2, which multiplies the 3 to
give 6. Meanwhile, the y cubed is treated like a constant and so does nothing. Also,
 (5 y 2 )
 (3x 2 y 3 )
0
 9 x 2 y 2 (notice that x squared did not change). Be careful because
x
y
since 5y 2 has no x dependence.
Because derivatives point in the upward direction of increasing slope while forces push objects
downward on the potential energy "hills", the negative signs in the force equation are needed
U(x, y)
U(x, y)
xˆ 
yˆ . One could
to give the correct directions for the force: F (x, y) 
x
y
imagine a two-dimensional potential energy "hill" with a maximum at x  0 and y  0 with
U ( x, y )  10  x 2  y 4 (notice that any non-zero value of x or y only decreases the value of U).

)  2 xxˆ  4 yyˆ (notice that the force grows faster in the yThis would give a force field of F ( x, y
direction because the quartic function is steeper than the quadratic function). The force at the

point (2,3) would be given by the vector F ( 2,3)  4 xˆ  12 yˆ . (Note that you must evaluate the
force at the point after taking the partial derivatives; never plug values into U before taking
derivatives.)
¿
0-B-1
If a potential energy is given as U(x, y)  k2x 2  xy  3xy2  where k  1 [N/m] , what
is the force in the x-direction from this potential energy on a particle placed at
x  2 m and y  0 m (SI units)?




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Subsection 0-C
A charge creates an electric (vector) field around it. This electric field can be described by
drawing vector arrows around the charge with the size of the vectors representing the strength
of the field at that point in space. Note that the strength of the electric field decreases as you
move away from the charge.


If a test charge q is placed in this electric field, the test charge feels an electric force of F  qE
where the direction of the force is given by the direction of the electric field. (If q is a negative
charge, then the force is in the opposite direction of the electric field direction.) Note that in


the equation F  qE , all the charge does is multiply the electric field components by a number,
so finding the force on a charge q is very easy if you know the electric field. This is why you will
spend much of your time in this course calculating electric fields, because they provide you
with so much information about the system.
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Another equally valid point of view is that the charge creates an electric potential landscape (a
scalar field). This electric potential is a new concept that often confuses students, but it is very
similar to potential energy, U. V is just the potential energy given by the electric field for each
unit of charge that interacts with the electric field. The SI unit of the electric potential is the
volt [V]. The SI unit of the electric field is volts per meter [V/m].

Just as the force was the slope of the potential energy graph, the electric field is the slope of
V (x, y)
V (x, y)
xˆ 
yˆ which may be abbreviated to
the electric potential graph: E (x, y) 
x
y
V V 
E  
xˆ,
yˆ .
y 
 x

These ideas are easily extended to three dimensions though you will only need to work in two
dimensions for this week's activities:
U(x, y,z)
U(x, y,z)
U(x, y,z)
F (x, y,z) 
xˆ 
yˆ 
zˆ
x
y
z
(mathematically identical)
E (x, y,z) 
V (x, y,z)
V (x, y,z)
V (x, y,z)
xˆ 
yˆ 
zˆ
x
y
z
10

Notice that compared to a positive point charge, a negative point charge has an electric vector
field that point radially inward.
This makes sense because if you bring a positive test charge nearby, it will feel an attractive
force toward the negative point charge. Thus the electric field shows the direction of force a
positive test charge would feel.
Another way to think of the fields produced by a negative charge is to think of an hole in the
ground or valley that tend to cause positive test charges to feel a force downward and inward
toward the negative point charge.
It is important to remember that a negative charge will always feel a force from an electric field


in the opposite direction that a positive charge would simply due to the equation F  qE since
making q negative simply changes the sign of the force vector F.
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¿
0-C-1
If an electric potential is given as V ( x, y )  k 2 x 2  xy  3xy 2  where k  1 [V] , what
is the force in the x-direction from this electric potential on a charge of 2
coulombs placed at x  2 m and y  0 m (SI units)?


¿
0-C-2
Imagine that the long bar shown below has positive charge uniformly
distributed over its surface. Sketch the electric field vectors (arrows) that you
would expect above and below the bar (direction and size). Pretend the bar is
much wider than the page so that you do not need to worry about the electric
field around the bar edges.
¿
0-C-3
Briefly explain how your answer would change if the bar had uniform negative
charge.
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Subsection 0-D
There are four fields described: two vector fields, the force vector field F and the electric vector
field E, and two scalar fields, the potential energy scalar field U and the electric potential scalar
field V. These fields are intimately connected and much of your coursework will involve using
one field to find another. Some students find the following picture helpful though it should be
noted that the formulas for the fields only apply to simple charges and not more complicated
2
9  Nm 
systems. In SI units, the Colomb constant k is given by 8.99 10  2  .
 C 
¿
0-D-1
Most often you will be asked to find the electric field for a variety of systems.
The above equation for the electric field is only useful for which
physical system? (Many introductory students often only know one
method/formula for finding E and get stumped on their exams!)
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In-Lab Section 1: sketching electric fields
If you examine a two dimensional electric potential landscape (first picture) from above you will
see an 'aerial view' (second picture) that obscures the third dimension (height or voltage):
Nevertheless, if you know that the middle is supposed to have a higher electric potential
(voltage), then you can imagine that each sequentially outward dotted ring represents a lower
step down the 'voltage mountain'. The solid lines show the direction of electric field, but
without arrowheads so that you must figure out the direction yourself. That is not hard
because electric fields always point in the direction of lower electric potential ("down the
voltage hill"). Also note that electric field lines are always perpendicular to equipotential lines
where they intersect.
What is not obvious is where the electric field and thus forces on test charges would be the
strongest. A good rule of thumb is that the strength of the electric field is largest where the
density of electric field lines is greatest. This would imply that the electric field magnitude is
greater nearer to the center of the system.
Also, just as with magnetic field lines, electric field lines
cannot intersect. If they did, it would be ambiguous in which
direction a test charge would feel a force if it were at that
point. But the force direction caused by an electric field is
always unique so such crossings cannot happen. If you draw
electric field lines crossing, it makes it easy for your TA to
mark your answer wrong!
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¿
1-1
In the following picture depicting equipotential lines (dashed lines of constant
voltage), sketch the corresponding electric field lines (draw the electric field
lines with solid lines). Be sure to label the direction of the electric field lines
using arrows using the assumption that VOUT < VIN. Note that Vout gives the value
of the voltage for the entire dotted line it touches (similarly for Vin). Think
topographical maps! Draw on the picture below.
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¿
1-2
In the following picture assume VOUT < VIN. Describe how each of the two test
charges shown would move if placed where shown. Draw on the picture below
and write an explanation below the picture.
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¿
1-3
In the following picture depicting equipotential lines (dashed lines of constant
voltage), label the two regions that have the strongest electric field and the two
regions that have the weakest electric field. Next to the picture, explain how
you made your decision. Draw on the picture below and write an explanation
below the picture.
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In the laboratory, we find is easiest to use a voltmeter to make measurements of the electric
potential (using a DMM). However, this gives us values and not formulas for V so that we are
not able to find E by taking partial derivatives of V. We must have a way of estimating E in the
lab from voltage measurements. This technique will be illustrated using the following problem.
The following picture shows that a student has used a DMM to construct two equipotential
voltage lines and wishes to find the electric field at some point between them.
First, we draw an estimate of the best electric field line component that lies between the two
equipotential lines (and measure its distance d):
Next we approximate the magnitude of E using differences in place of derivatives:

 V
2
V
E 
  1   . This gives the magnitude (the size of the hypotenuse), and now we
d
2
m
must break it up into components to get the electric field as a vector.
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Now notice that for this particular coordinate system, the x component points in the negative
direction along the x-axis and the y component points in the negative y-direction so that the

V
final electric field vector is given by E  0.82 xˆ  0.57 yˆ   .
m
¿
1-4
In the following picture depicting labeled equipotential lines, calculate the
electric field at the marked point. Be sure to find EX and EY, and write your final
answer in vector notation. Show your calculations in the space next to the
picture (a protractor is provided above).
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In-Lab Section 2: 2-D electric fields on conductive paper
!
Do not write on the conductive paper, especially with pencil because the
graphite conducts as well as the paper (and thus ruins it). Erase any pencil
markings you see.
Subsection 2-A
The above picture shows two point charges with 10 volts of electric potential difference
between them. The dotted lines represent lines of constant voltage. For instance, all points on
the dotted line to the left might be at 7 volts while all points on the dotted line on the right
might be at 3 volts (as 10 volts decreases to 0 volts). A charge would not raise or lower its
potential energy if it moved along a given dotted line because there would be no change in its
potential energy. This means that there would be no force on the charge in the direction
parallel to the dotted line. However, the charge would definitely change its electric potential
(and thus its potential energy) if it moved off a dotted line. This means it feels an electric force
pushing it away from (and perpendicular to) the dotted equipotential lines.
The solid lines represent electric field lines. They also show the direction of electric force that a


charge would feel since F  qE . Note they are perpendicular to the equipotential lines where
they intersect.
21
¿
2-A-1
Electrify the dipole conductive paper (two dots of silver paint) with V  10 volts .
Use a DMM to measure the locations of at least 7 dashed equipotential lines
and sketch them on graph paper. Be sure to use the physical symmetry of the
system to reduce your work by 75%. Make a copy for both lab partners.
(answer on separate graph paper)
¿
2-A-2
Sketch at least 8 electric field lines using solid lines on top of the dashed
equipotential lines you just graphed. Make a copy for both lab partners.
(answer on separate graph paper)
¿
2-A-3
Label the part(s) of your sketch where the electric field appears to be the
strongest. Make a copy for both lab partners.
(answer on separate graph paper)
¿
2-A-4
Calculate the electric field at the point that is 2 cm to the left and 3 cm above
the center of the dipoles using the derivative approximation. Be sure to express
your answer for the electric field in vector notation.
22
¿
2-A-5
Draw both electric field components for the point in the previous problem as
arrows on your field mapping paper and label the strength of the field in each
direction. At this point your field mapping paper should look something like this
only with many more lines:
(answer on separate graph paper)
¿
2-A-6
Discuss where the electric field should be the strongest for the dipole system
based upon the spacing of the equipotential lines.
23
¿
2-A-7
Place an x-y coordinate system on your graph paper as shown above. On
another sheet of regular graph paper, graph voltage versus the x-coordinate
(V(x,0) vs. x) along the line of symmetry that goes through both poles of the
dipole conducting paper using your previous results. Your voltage axis and
distance axis should be in SI units. An example is shown below. (answers on
separate graph papers)
24
¿
2-A-8

On another sheet of regular graph paper, use differences E x 
 V
x
to
approximate and graph EX(x,0) vs. x. Use your previous voltage graph along the
line of symmetry (y=0). This requires calculating the electric field in the xdirection only for points along the line of symmetry, but this is alright since Ey =
0 along this line due to the symmetry of the system. Make a copy for both lab
partners. An example is shown below.
(answer on separate graph paper)
25
Subsection 2-B
¿
2-B-1
Find and draw the equipotential lines for the parallel plate conductive paper
(two long bars of silver paint) and sketch them on graph paper using dashed
lines. Be sure to find the potential in each region outside the parallel plates.
(This should take only a few minutes!) Make a copy for both lab partners.
(answer on separate graph paper)
¿
2-B-2
Discuss the strength of the electric field everywhere between the parallel plates
as well as outside the parallel plates based on your measurements.
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Subsection 2-C
¿
2-C-1
Find a novel conducting paper of silver paint that looks interesting to you. Find
and sketch its equipotential lines on graph paper using dashed lines. Make a
copy for both lab partners.
(answer on separate graph paper)
¿
2-C-2
Sketch the electric field lines on your previous equipotential graph using solid
lines.
(answer on separate graph paper)
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In-Lab Section 3: authentic assessment
(Students must be supplied with a strip of conductive paper 1 [cm] by 10 [cm] with alligator
clips attached at either end.)
Find the average electric field strength and direction between two points of conductive paper
10 [cm] apart electrified by a 10 [volt] electric potential difference.
Hint: You may solve this problem experimentally by making measurements or theoretically by
making a calculation. One method is much easier.
¿ 3-1
Show a student in a different group that you can successfully calculate the
average electric field magnitude and direction between 10 [cm] of conducting
paper electrified by 10 [volt]. Once you are successful, have them sign below.
Note: if someone is stuck, please give them advice!
"Yes, I have seen this student theoretically calculate the electric field in a 10
[cm] strip of conductive paper with 10 [volt] across it. They understand how
important it is to think about a physical situation before manipulating it!"
Student Signature:___________________________________________________
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In-Lab Section 4: open-ended / creative design
Use the cathode ray tube conductive paper (shown below) to find out what would happen to a
positive test charge of +3.0 [nC] placed at three unique locations on the conductive paper
(points A, B, and C shown in the picture below). You must use the equation FE  qE and
Newton's second law of motion to calculate the magnitude of the acceleration of the test
charge if its mass was 10 [g].

You are allowed to "cheat" by talking to other groups for ideas, but are not allowed to "cheat"
by just stating an answer you may already know, looking it up online or asking your TA.
Below you are given three prompts:
hypothesizing/planning, observations/data,
calculations/conclusion. Your job is to figure out the answer using these prompts as your
problem-solving model. In the event that you should run out of time, you may not discover the
correct answer, but you should make an attempt at each prompt. Grades are based on honest
effort.
Your open-ended solution should probably include some of the following items: sketches of
circuit diagrams, tables of data, calculations, recorded observations, random ideas, etc.
Write at the prompts on the next page.
30
¿
4-1
hypothesizing/planning:
¿
4-2
observations/data:
¿
4-3
calculations/conclusion
I, the physics 241 laboratory TA, have examined this student's Weekly Activity pages and found
them to be thoroughly completed.
!
TA signature: _______________________________________________________________
31
Post-Lab: electric field mapping
!
You must complete this post-lab section after you attend your lab. You may
work on this post-lab during lab if you have time and have finished all the other
lab sections.
¿
X-1
Below is a sketch of some equipotential lines (dashed) and some field lines
(solid). The points where these intersect are labeled. If an electron is released
from rest at point e, which intersection will it reach some time later? Explain
your reasoning. Now find the velocity of that electron when it reaches that
point. Hint: use energy conservation with K  U  qV   eV  where
1
K  mv 2 .
2
Note: me = 9.1x10-31 kg. Write your explanations and work around
the picture:


32
¿
X-2
Imagine a long vertical wire held at 4 V. Two points of interest A and B are
delineated. Also, imagine that you have a positive test charge q that may move
along the wire.
a. At which point would the charge q have a higher electrical potential energy,
A or B? Explain.

b. What is the magnitude of E y at point A? And point B?

c. What is the magnitude of E y directly between A and B?

d. What is the magnitude of F y at point A? And point B?
c. Explain how much work would be required to move charge q from point A to
point B.
33
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34
Weekly Activity 4: Cathode Ray Tube
Name:___________________________________________________ Date:______________
Pre-Lab
!
You must complete this pre-lab section before you attend your lab to prepare
for a short quiz. Be sure to complete all pages of the pre-lab.
Continue until you see the stop pre-lab picture:
A cathode ray tube works by ‘boiling’ electrons off a cathode heating element and accelerating
them with a large voltage difference. Then the high-speed electrons pass between a pair of
charged deflection plates so that the path of the electron is altered. Finally, the electrons strike
a screen coated with a fluorescent material and you see a scintillation take place (i.e. you see
light emitted). All of this is done in a vacuum so that the electron can travel through the CRT
unhindered by collisions with air molecules. Actually, there are two pairs of deflection plates: a
pair of charged deflection plates for vertical deflection and another pair of charged deflection
plates for horizontal deflection. (See figure.)
Three-dimensional figure showing the operation of the CRT.
The dotted line shows the path traversed by an example electron.
35
¿
0-1
In the cathode ray tube, an electron is initially at rest (approximately) and is
accelerated by a force produced by an electric field. However, in a lab setting
you usually only know/measure voltage differences. If an electron is placed
between two conducting plates with different voltages, explain which plate will
repel the electron and which plate will attract the electron. (Remember that
the electron is negatively charged.) Draw a picture to elucidate your answer.
36
¿
0-2
Imagine an electron lies between two charged plates of different voltages. The
following picture shows how this physical system may be described with a
'voltage landscape' graph. Explain whether electrons move downhill or uphill
on an electric potential hill. Be sure to discuss how a negative sign on a charge
can affect the way we think about potential energy.
37
¿
0-3
The electric field is given by the negative slope of the electric potential. For this
system, determine whether the electric field points to the left or right. Then
explain whether electrons move in the direction of the electric field or against
the electric field.
38
¿
0-4
There are three electric fields affecting the trajectory of the electron in the
cathode ray tube. Ea accelerates the electron initially to a high speed. Ed,v
causes a deflection in the vertical direction and Ed,h causes a deflection in the
horizontal direction. In the CRT figure below, draw arrows correctly depicting
the direction and magnitudes of these fields. Do you best to draw Ed,v in the
three-dimensional picture.
39
¿
0-5
What is the shape of trajectory of a ball launched into the air at an angle and
experiencing constant gravitational acceleration?
¿
0-6
The electric field between two charged plates is approximately constant. This
means that a charged particle between two charged plates feels a constant
force (and thus undergoes constant acceleration). If an electron is traveling at a
constant speed and enters the electric field between two charged plates of
different voltage, what is the shape of the trajectory it makes while in between
the two plates?
¿
0-7
In the picture below, sketch the trajectory of the electron entering the region
between the two charged conducting plates. Be sure to show what happens to
its trajectory when it leaves the area between the two plates. (Assume the
electron has a high enough initial velocity to not collide with the plates.)
40
In-Lab Section 1: cathode ray tube derivations
Subsection A
The operation of a cathode ray tube is described by just a few simple physics concepts.
However, the chain of logic/math that relates the plate voltages to where the electron strikes
the scintillation screen is quite cumbersome. Many student become confused in the derivation
that calculates this deflection distance of the electron. Much of the theoretical work that
follows is designed to help the students understand each part of this long derivation.
Three-dimensional figure showing the operation of the CRT.
The dotted line shows the path traversed by an example electron.
¿
1-A-1
In the cathode ray tube, an electron is initially at rest (approximately) and is
accelerated by a force produced by an electric field. However, in lab you will
only know the positive change in voltage Va (really “Va”) of the plates through
which the electron is accelerated. What simple formula using Va, q and W can
you write to relate the work done on the electron to the change in voltage of
the apparatus? Be careful, the charge of an electron is –e where e=1.60×10−19 C.
Also, me=9.11×10−31 kg (Check your answer with other students or your TA!)
Your formula:
Now plug in q = -e into your formula (for an electron):
41
¿
1-A-2
Now slightly extend this formula using the work-energy theorem. The workenergy theorem states that the change in kinetic energy is equal to the work
done on the object. Using this concept, write a formula relating the change in
the electron’s kinetic energy to the accelerating voltage Va of the apparatus (use
Va, e and K). (Be sure to check your answer with other students or your TA!)
Your formula:
¿
1-A-3
Write a formula that describes the final velocity of the electron if it starts from
rest and you know the work done on the electron (use vf, me, Va, and e). (Be sure
to check your answer with other students or your TA!)
Your formula:
¿
1-A-4
Explain what the sign of the accelerating voltage difference Va must be in order
for your formula in 1-A-2 to make sense? Reconcile this with your knowledge of
how negatively charged particles respond to an electric field. Your explanation:
42
¿
1-A-5
Now try out your formula using some numerical values. If the electron starts at
the position x=0 on the following graph, find out the speed of the electron once
it has reached the area of constant electric potential. (Remember that electrons
flow upward in the voltage landscape.) You should use the electron charge, -e =
-1.60x10-19 C, and the electron mass, me = 9.11x10-31 kg. Your calculations and
answer in SI units:
43
Subsection B
Now you know how to find speed of the electrons after they are initially accelerated, you will
study how the electrons are deflected by the charged deflecting plates.
¿
1-B-1
There are three electric fields affecting the trajectory of the electron. Ea
accelerates the electron initially to a high speed. Ed,v causes a deflection in the
vertical direction and Ed,h causes a deflection in the horizontal direction. In the
CRT figure below, draw arrows correctly depicting the direction and
magnitudes of these fields. Do you best to draw Ed,v in the threedimensional picture.
¿
1-B-2
Now examine a pair of charged deflection plates (see figure). The voltage
difference between the plates is Vd,y. Assume the plates are separated by a
distance d. Assume Vd,y is positive
44
¿
1-B-3
Find the magnitude and direction of the electric field between the plates Ey in
terms of d and Vd,y (ignore any edge effects). Your answer:
¿
1-B-4
Determine the acceleration ay felt by an electron inside the space between the
plates using e, me and Ey. Your answer:
¿
1-B-5
Now examine what happens when an electron enters the space between the
vertical deflection plates (see figure).
If the electron enters with a velocity in the x-direction of vo,x and travels the
length of the plates w, how long does it take for the electron to reach the other
side, t? Write your answer for t using w and vo,x. Your work and answer:
45
¿
1-B-6
Explain why this time t is not affected by the acceleration in the y-direction
caused by the deflection plates? Your explanation:
¿
1-B-7
As the electron traverses the space between the deflection plates, it is
accelerated in the y-direction.
Using the kinematics equation y = ½ ay(t)2 to find vertical displacement y of
the electron once it has reached the other side of the deflection plates. Write
your answer for y using w, vo,x, e, me and Ey. Your work and answer in SI units:
¿
1-B-8
Use the kinematics equation vf,y = ay(t) to determine the final y-velocity vf,y
when the electron has reached the other side of the deflection plates. Write
your answer for vf,y using w, vo,x, e, me and Ey. Your work and answer:
46
Subsection C
This section is not a problem, but you must read through it in lab. This is where all the analysis
you have done from the previous section is synthesized together to derive the cathode ray tube
equation. You will need to understand the following work in order to write your lab report.
The derivation will use the following figure to find Dy.
¿
1-C-1
READ THE FOLLOWING SENTENCE TWICE! Our goal here is to derive a final
equation that relates Dy to the only things we can control in the lab Va and Vd,y
(as well as the things we can’t control: the geometric parameters of the CRT d,
w and L). Note that capital V will always represent a voltage while a lower case
v will always represent a velocity. In the derivation ignore directional negative
signs for simplicity, and also we need to use
1
2
mev o,x
 e  Va (usually to substitute
2
for vo,x). Check here if you read that sentence twice:_____

47
First we need to find y: (You will need to justify each step below in your lab report.)
1
2
y  ay t 
2
2
1 e  E y  w 
 
 
2  me v o,x 

 Vd ,y 
e


  
1   d  w 2 

2  me 2e  Va 




 me 
2
1 e  Vd ,y  w  me

4 d  me  e  Va

w 2 Vd ,y
4d Va
Next we need to find y’: (You will need to justify each step below in your lab report.)
y' v f ,y t'

 L 
 ay t  
v o,x 
e  E w  L 
 

 
 me v o,x v o,x 
 Vd ,y 

e 

 d  w  L 



 
 me
v o,x v o,x 




w  L  e  Vd ,y

2
d  me v o,x

w  L  e  Vd ,y
2d  e  Va
w  L Vd ,y

2d Va
Finally this gives our equation for the total deflection on the oscilloscope screen Dy:
w 2 w  L  Vd ,y
. However, we cannot open up the CRT to measure d, w or L so we
Dy   

2d  Va
4d

might as well replace all these geometric factors with a single unknown geometric constant kg,y:
V
Dy  kg,y d ,y . THIS IS OUR FINAL CRT DEFLECTION EQUATION. Note that by symmetry we get
Va


48

Vd ,z
.
Va
THEREFORE, THERE IS A CRT DEFLECTION EQUATION FOR EACH DEFLECTION DIRECTION EACH
WITH ITS OWN GEOMETRIC CONSTANT.
the same derivation for the total deflection in the horizontal z-direction: Dz  kg,z
¿
1-C-2

Explain how we have we achieved the goal set out in 1-C-1?
Subsection D
Now you need to answer some questions about the CRT deflection equations.
¿
1-D-1
Given the CRT deflection equation in the vertical direction Dy  kg,y
Vd ,y
Va
, what you
would see on the CRT screen if the deflection voltage was increased. Explain
why this would happen using a physical argument (i.e. not using math). Your
answer and explanation:

¿
1-D-2
Given the CRT deflection equation in the vertical direction Dy  kg,y
Vd ,y
Va
, what you
would see on the CRT screen if the accelerating voltage was increased. Explain
why this would happen using a physical argument (i.e. not using math). Your
answer and explanation:

49
In-Lab Section 2: testing the cathode ray tube equations
Subsection A
Vd ,z
by
Va
adjusting Vd,z and observing Dz. If all goes well, you will be able to find an experimental value
for the z-geometrical constant without breaking the CRT open to measure it by hand.
You will now experimentally test the horizontal CRT deflection equation Dz  kg,z
¿
2-A-1

Use tape on screen to mark position of electron beam when there is NO
DEFLECTION (Vd,z set to zero to find the ‘origin’ of the CRT). Be sure to record Va
and keep this value constant for the rest of this section. (Va is the sum of VB and
VC on the CRT power module and should be set as high as possible while the
scintillation dot is still in focus). Record your constant accelerating voltage Va:
¿
2-A-2
Adjust Vd,z on the horizontal plates and mark Dz on the tape for several values of
Vd,z (make a data table with at least 5 data points). Record your data table of
Vd,z and Dz:
¿
2-A-3
Create graph of Dz vs Vd,z by hand. Graph Dz vs Vd,z on graph paper. Your data
should give you a straight line.
¿
2-A-4
Measure the slope of the line of best fit. Since Dz  kg,z
k g,z
Va
Vd ,z
, the slope will equal
Va
so multiply by Va to obtain kg,z. Record your result for kg,z here in SI units:


50
Subsection B
You will now experimentally test the vertical CRT deflection equation Dy  kg,y
Vd ,y
that you
Va
have derived by adjusting Vd,y and observing Dy. in order to estimate the y-geometrical factor.
The y-geometrical constant in this other direction is different from z because the geometry of
the deflecting plates is different in the z-direction from the y-direction, i.e. one set of plates is

closer to the screen than the other.
¿
2-B-1
Use tape on screen to mark position of electron beam when there is NO
DEFLECTION (Vd,y set to zero to find the ‘origin’ of the CRT). Be sure to record Va
and keep this value constant for the rest of this section. (Va is the sum of VB and
VC on the CRT power module and should be set as high as possible while the
scintillation dot is still in focus). Record your constant accelerating voltage:
¿
2-B-2
Adjust Vd,y on the horizontal plates and mark Dy on the tape for several values of
Vd,y (make a data table with at least 5 data points). Record your data table of
Vd,y and Dy:
¿
2-B-3
Create graph of Dy vs Vd,y by hand. Graph Dy vs Vd,y on graph paper. Your data
should give you a straight line.
¿
2-B-4
Measure the slope of the line of best fit. Since Dy  kg,y
k g,y
Va
Vd ,y
Va
, the slope will equal
so multiply by Va to obtain kg,y. Record your result for kg,y here in SI units:


51
Subsection C
You will now experimentally test in another way the horizontal CRT deflection equation
V
Dz  kg,z d ,z by adjusting Va and observing Dz.
Va
¿

2-C-1
Set Va to about ½ to ¾ its maximum value and adjust Vd,z to the largest value
possible that still enables you to see the scintillation dot (it may be fuzzy, but
you should measure deflections using the center of the dot). Be sure to record
Vd,z and to keep this value constant for the remainder of this section. Record
your constant deflecting voltage Vd,z:
¿
2-C-2
Adjust Va to larger and larger values and record the corresponding horizontal
screen displacement Dz for several values of Va (make a data table with at least
5 data points). Record your data table of Va and Dz:
¿
2-C-3
Linearize your data by making a graph of Dz vs 1/Va by hand. Graph Dz vs 1/Va
on graph paper. Your data should give you a straight line.
¿
2-C-4
Measure the slope of the line of best fit. Since Dz  kg,z
Vd ,z
, the slope will equal
Va
kg,zVd ,z so divide by Vd,z to obtain kg,z. Record your result for kg,z here in SI units.


52
In-Lab Section 3: authentic assessment
Have you ever seen how a charged particle interacts with a magnetic field?
¿
3-1
The magnetic force is given by FM  qv  B , but usually the speed of the charged
particle is too slow to be able to visually see the effects of the magnetic force.
However, electrons in CRTs move so fast, you can actually see them being
deflected by a magneticfield. In fact, this is how old-fashioned “television” first
worked. You must correctly predict whether the scintillation dot will be
deflected horizontally or vertically when a magnetic field is created nearby the
CRT. Then check your prediction. For this you will need to remember what the
cross product means in the Lorenz force equation (better ask around if you
don’t ), FM  qv  B and how to use the right-hand rule.
Circle your predictions then check them experimentally while showing a
student in a different group:

"Yes, I have seen this student determine how magnetic fields affect moving
electrons. They will be able to protect themselves from dangerous ions on their
trip to Mars (if they take a magnet)!"
Student Signature:___________________________________________________
53
In-Lab Section 4: open-ended / creative design
If you were able to see inside a CRT, you would see that one set of deflection plates is closer to
the scintillation screen than the other set of deflection plates. The deflection plates closer to
the screen have a smaller L and therefore a smaller geometric constant kg. Your job is to
determine which is set of deflection plates inside your CRT are closer to the scintillation screen,
the vertical plates or the horizontal plates. This is equivalent to finding which geometric
constant is larger, kg,z or kg,y.
You might say that you already know the answer to this based on your previous work.
However, do not use these results to answer the question, but instead think of an easier way to
determine which geometric constant is larger. Here is a hint: given a certain deflection voltage,
which direction seems to “use” that deflection voltage more. You should need to collect very
little data to answer which geometric constant is larger.
You are allowed to "cheat" by talking to other groups for ideas, but are not allowed to "cheat"
by just stating an answer you may already know, looking it up online or asking your TA.
Below you are given three prompts:
hypothesizing/planning, observations/data,
calculations/conclusion. Your job is to figure out the answer using these prompts as your
problem-solving model. In the event that you should run out of time, you may not discover the
correct answer, but you should make an attempt at each prompt. Grades are based on honest
effort.
Your open-ended solution should probably include some of the following items: sketches of
circuit diagrams, tables of data, calculations, recorded observations, random ideas, etc.
Write at the prompts on the next page.
54
¿
4-1
hypothesizing/planning:
¿
4-2
observations/data:
¿
4-3
calculations/conclusion
I, the physics 241 laboratory TA, have examined this student's Weekly Activity pages and found
them to be thoroughly completed.
!
TA signature: _______________________________________________________________
55
Post-Lab: cathode ray tube
!
You must complete this post-lab section after you attend your lab. You may
work on this post-lab during lab if you have time and have finished all the other
lab sections.
¿
X-1
Derive the CRT deflection equation. Clearly explain every step of the derivation
in words (i.e. explain why one step of the equation is equal to the next which
might be mathematical or based on physics concepts). You will not get credit
for simply copying equations without demonstrating you understand the math
and physics that connect all the steps. (You may want to simply insert this postlab into your lab report!)
56
(keep going…)
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