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Objective: Writing about scientific principles and phenomena is an increasingly important skill in the 21st century. This assignment is designed to help you understand a major concept covered in this unit while also helping you develop your science writing skills. One key to effective science writing is explaining how science can be used in everyday life. You will do this in a 500-600 word assignment. The topic includes in the files.
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Introduction
Objective: Writing about scientific principles and phenomena is an increasingly
important skill in the 21st century. This assignment is designed to help you
understand a major concept covered in this unit while also helping you develop
your science writing skills. One key to effective science writing is explaining how
science can be used in everyday life. You will do this in a 500-600 word
assignment.
Instructions
You have two choices for this assignment:
Option 1: An essay or Option 2: A letter
Please choose one of these two options and follow instructions below.
Option 1: An essay
You should:
1) Formulate and answer a question
Select a concept from lecture content covered in the second midterm and
formulate a question. Use this question as the title of your essay.
Sample questions:
• How do electric field lines compare to magnetic field lines?
• How does a capacitor store energy?
2) Explain how this applies to your life
Write a 500-600 word essay answering this question, and discuss how the
information could be useful to you in your own life. Be sure to include some
concrete information that was covered in this unit, explaining why the information
is relevant to your life and useful for you. Be sure to explain how the information
applies to you personally and give examples.
Examples of applications:
• “Magnetic fields are not visible to the naked eye, but can be applied to Magnetic
Resonance Imaging (MRI) scans. MRIs use powerful magnetic fields to generate
signals from inside the body to create clear images of bones, organs, and tissues.
This is how the doctor was able to diagnose my injury when I tore my ligament in
a game.”
3) Structure your essay as suggested below
• State your question in the title.
• 1st section: Give an overview of the answer to your question.
• 2nd section: Provide the scientific details of the answer to your question. Be
sure to select the relevant information from class notes and the textbook.
• 3rd section: Make it personal. Explain why this information is relevant to your
life or useful for you and give examples.
Since you will be writing about science from a personal perspective, you can use
personal pronouns (I, we, you, etc.).
Option 2: A letter
You should:
1) Formulate and answer a question
Select a concept from lecture content covered in the second midterm and
formulate a question. Use this question as the title of your essay.
Sample questions:
• How do electric field lines compare to magnetic field lines?
• How does a capacitor store energy?
2) Explain how this applies to a friend or family member
Write a 500-600 word letter to a family member or close friend, answering this
question, and discuss how the information could be useful to this person
in their own life. Be sure to include some concrete information that was covered
in this unit, explaining why the information is relevant to this person’s life and
useful for this person. Be sure to explain how the information applies to this
person and give examples.
Examples of applications:
• “Brother, magnetic fields are not visible to the naked eye, but can be applied to
Magnetic Resonance Imaging (MRI) scans. MRIs use powerful magnetic fields to
generate signals from inside the body to create clear images of bones, organs,
and tissues. This is how the doctor was able to diagnose your injury when you
tore your ligament in last year’s game.”
3) Structure your letter as suggested below
• State your question in the title.
• Begin your letter by addressing your recipient: Dear _________,
• 1st section: Give an overview of the answer to your question.
• 2nd section: Provide the scientific details of the answer to your question. Be
sure to select the relevant information from class notes and the textbook.
• 3rd section: Make it personal. Explain why this information is relevant to this
person’s life or useful for them and give examples.
Since you will be writing about science from a personal perspective, you can use
personal pronouns (I, we, you, etc.).
This assignment requires that you formulate your own question and approach, so
do not use the sample questions or specific examples provided above for your
paper. Great ways to start questions that will help you think deeply in this class
include: How do/does ______? Why is/are ______? What is the difference
between ______?
For this assignment work independently and do the best you can; you’ll receive
feedback on your work after it’s turned in.
Formatting
Your paper should be 500-600 words. Save your paper in a Word document and
then copy and paste your paper into the text box below.
Physics 7D:
Electricity and
Magnetism
Chapter #23
Electric Potential
1
Chapter 23
2
Introduction
In classical mechanics, the principle of energy conservation saves us an
enormous amount of time when solving certain types of problems
For example, I do not care what
shape the roller coaster has (if it
has loops, how many climbs and
descents, how long, … etc). If
there is no friction, all I need is the
difference in height (initial vs.
final) and I can predict the
difference in speed (initial vs. final)
Trying to determine the final speed from first principles using Newton’s laws would be
a very long task, even for a simple roller coaster like this one
Something similar happens in electromagnetism:
Just as the difference in height is all I need to know for a frictionless roller coaster, in
electromagnetism all we will care about most of the time is the difference in potential
energy. We won’t have to worry about all the fields created by individual atoms.
This is why we will introduce in this chapter concepts like voltage
Reminder: potential energy
Potential energy is the type of energy that an object has because of its
configuration and/or position in a force field
In other words, an object has potential energy if a force acting on it has the
ability (or potential) of doing work on it. You can think of it as “stored”
energy: the force field is ready to do work as soon as you let it
Examples:
Reminder: nothing is free in this life. Work was required to place these objects in
these configurations
4
Electric Potential Energy
Consider two point charges at a given distance. Does this configuration
have potential energy?
yes!
When you release the charges, they move!
(remember the principle of work and energy: Wnet=ΔK)
Another way of looking at this: in order for me to place the second
charge in this configuration I had to do work (or the electric force
did work, if the charges had opposite signs)
5
Work
Let’s first remember how to calculate the work done by a force field when an
object moves from point a to b:
This is a line integral. It means that in every little piece of line the
product between dl and F is taken, and all of the results summed
Clarification: θ in
this image is the
same as φ in the
equation above.
Same for dl and
ds
Work
Some questions about work:
1) Does it depend on the
initial and final positions?
In general yes. It’s not the same
to bring two charges to a
distance of 1cm as 1m
2) Can it be negative?
Yes. If the force and the
displacement occur in the same
direction work is positive, and
otherwise it is negative.
3) Does it depend on the path?
In the case of the electric force, no (see next slide)
7
Path Independence
Let’s consider a charge q0 moving
along an arbitrary path from a to b in
the presence of a field created by
another charge:
We can see clearly that, at any point:
The force only depends on r, which
means that this will be the only
variable in the integral. In other
words, we won’t need to worry how
θ and φ depend on r (i.e. the path),
and the work will only depend on
the initial and final values of r.
@ 2012 Pearson Education, Inc.
This means the electric force is conservative
(other examples are the gravitational force and the spring force)
8
Reminder: Work and Potential Energy
Since the electric force is conservative, there is a corresponding
potential energy:
To calculate the work done by a conservative force all we need are U and
the initial and final positions. No need to do a line integral!
A few comments:

Work and the change of potential energy have opposite signs. As the force
does positive work, potential energy decreases (e.g. think of when you drop
something). This is why U increases as the charge moves in the direction
opposite the electric force, and viceversa

If the electric force were not conservative we could not talk of electric
potential. With non-conservative forces mechanical energy is not conserved
(hence the name), as energy is dissipated (heat, sound… etc).
9
Electric Potential Energy (test charge q0)
To determine U let’s calculate the work
done when moving a charge q0 from a
radial distance ra to rb of another charge q.
To simplify our life, let’s use a radial path,
since we already established the path
doesn’t matter. We obtain:
We conclude that:
10
Electric Potential Energy (test charge q0)
Let’s clarify a few things about electric potential energy:
1) Units? energy (e.g. Joules)
2) Potential energy is always defined with respect to some
reference point where U=0. Which one is it here?
infinity (r→∞)
(notice that there are cases where it is better to choose a
different reference)
3) In words, what is this electric potential energy?
It’s the work that the electric force would do on q0 if it would move
from a distance r until infinity
It’s also the same amount of Joules as the work that an external agent
must do to bring the charge q0 from infinity to a distance r of q (at an
infinitesimally small speed, in such a way that the force done by the external agent is
the same as the electric force)
11
Experiment
Reminder: the molecule
of water is close to a
perfect dipole
12
Electric Potential Energy (test charge q0)
What happens now if we have several electric charges contributing
to the field that q0 feels?
The principle of superposition indicates:
The total work done on q0 is the sum of the
contributions from all the charges
Important: note how calculating the potential energy due to several
charges is much easier than calculating the electric field, since it is not a
vector sum but just a scalar one
If we have a charge continuously distributed in space then the sum becomes an
integral:
q0
dq
U=

4πε 0 r
A word of caution
Does the quantity in the previous slide correspond to the work
required to put this configuration together?
No!
U is the work required to bring q0 from infinity to that particular
location.
In order to calculate the energy required to assemble the
configuration we need to sum the work needed to bring all charges
from infinity one by one:
sometimes referred to as
“total potential energy”
(should really be using a
letter other than “U”)
14
Example: a system of point charges
(Example 23.1 in Y&F)
Two point charges are located on the x-axis, q1=-e at x=0, q2=+e at x=a.
(a) Find the work that must be done by an external force to bring a third
point charge q3=+e from infinity to x=2a. (b) Find the total potential energy
of the system of 3 charges.
2
Answer:
e
(a) W = U =
8πε 0 a
−e
(b) U =
8πε 0 a
2
15
The answer to (b) is negative
because the configuration has
less potential energy than if the
charges were at infinity (i.e. the
electric force “helped” build the
configuration). An external agent
would have to do positive work to
get the charges back to infinity
Reminder so far
The potential energy of charge q0 from being in the presence of another point
charge q is given by:
If charge q0 is in the presence of several charges, then we just sum the
contributions from all of them:
If the charges are continuosly distributed, the above sum becomes an integral:
q0
dq
U=
4πε 0 ∫ r
The quantities above are *not* the same as the energy required to assemble a
configuration of charges, even though they are also called “U” (!!)
Sum of the work needed
to bring all charges from
infinity one by one
Experiment
(with iClickers)
Rice krispies (or something similar) are
brought near a Van der Graaff generator.
Rice Krispies have the property that some
of their electrons are very loose. What
happens to them?
A) Nothing
B) They fly away one by
one
C) They fly away at the
same time ✔
D) They stick to the Van
der Graaff
E) They start rotating
17
Experiment
(with iClickers)
Now 5 cup cake holders (metallic) are
put on top of the van der Graaff. What
happens to them?
A) Nothing
B) They fly away one by
one ✔
C) They fly away at the
same time
D) They stick to the Van
der Graaff
E) They start rotating
18
Electric Potential
Now we are ready to make one of the most important definitions in
this course: electric potential
When we looked at Coulomb’s law we saw that it would be useful to
express it per unit charge
In this way we can calculate this quantity (electric field) and then apply it
to any test charge to obtain the force
Same here. Let’s define an “electric potential energy per unit
charge”
Caution! “electric
potential” is per unit
charge (V), while
“electric potential
energy” is what we
saw in the previous
slides (U)
We call this “electric potential”
It has units of energy/charge (e.g. J/C), and is a scalar. This makes
it a lot easier to use and handle than electric field
(Note: 1 J/C is the same as 1 V, where V = “volt”)
Voltage
Is electric potential the same as voltage?
Almost. A voltage is a
difference of electric potential
For example, if I put a +1C charge in the
positive pole of a 1.5 V battery, the electric
force will do 1.5 J of work on it as it moves
from the positive to the negative pole
(just as with gravity a massive object wants to go from high potential to lower
potential, a positive charge wants to go from high potential to low potential; for a
negative charge it’s the contrary)
20
Advantages of Electric Potential
As mentioned in the introduction, the concept of voltage is
extremely useful and is thus widely used
All I need to know to
predict the impact a
roller coaster is going to
have on the car is “the
difference in height”
between the initial and
final points. Same in
electricity, where all I
need is “the difference in
electric potential”
Comment: In a frictionless roller coaster the loss of gravitational potential
energy goes entirely into increasing the speed of the car. However, in electric
circuits the average speed of the electrons does not change. We will see why
in two more chapters
21
Calculating Electric Potential #1
Now, how can we calculate the electric potential V?
If I know the charge configuration throughout space, I can calculate the
potential energy of a charge q0 due to the field produced by that configuration.
Then I just have to take those equations and divide by q0:
(Remember that r is the distance between dq and the point where I want to
know the electric potential; it does not necessary coincide with the r of
cylindrical and spherical coordinates)
22
Calculating Electric Potential #2
But it’s not the only way. It can also be determined from the E field
Remember the definition of work:
We divide both sides by q0 and we are left with:
Note that you can use any path you like between a and b to do this integration.
Needless to say, you should simplify your life and choose the simplest one!
By the way, you can also see from this equation that:
V/m is the most common unit for electric field
23
Summary: calculating electric potential
There are two ways of calculating electric potential
1) From the charge configuration:
2) From the electric field:
24
Example: charged conducting sphere
(Example 23.8 in Y&F)
A solid conducting sphere with radius R has a total charge q. Find the
electric potential everywhere, both inside and outside the sphere
Answer:
Important Lessons
There are three important lessons that we can derive from the last
exercise:
1) The electric potential is always continuous
In other words, even if you go from a conductor to vacuum, the electric potential will
not suffer a discontinuity. Otherwise this would imply an infinite electric field. Note that
the electric field can be discontinuous.
2) There is a maximum voltage that a conducting sphere can
hold
Any isolating material becomes a conductor in the presence of a strong enough
electric field. We call this “electrical breakdown”. For air, this typically occurs around
3×106 V/m
In the past example we saw that:
Vsurface = Esurface R
This means that a 1cm sphere in air can hold a
maximum voltage of:
(otherwise air becomes a conductor and the charge “flees”)
Important Lessons
3) In electrostatics, a conductor is always equipotential
In other words, the electric potential in the surface and everywhere within a
conductor that has reached equilibrium is the same
This is because in a conductor in equilibrium the electric field is zero. Therefore,
there are no potential differences in it:
0
This is a very important result. It is the foundation for
electrical circuits and cables
If I have a certain electric
potential, I can transmit it
anywhere I want by means of a
good conductor (e.g. a cable)
Needless to say, you cannot do the
same with gravitational potential
energy
27
Experiment
Is it possible to turn on a light-bulb “wirelessly”?
28
Example: infinite line of charge
(example 23.10 in Y&F)
Find the electric potential at a distance r from a very long wire carrying a
linear charge density λ
Note that we get the same
result as in the problem
done in the discussion
session, where the
potential energy was
obtained by integrating the
contributions from every
infinitesimal piece of
charge
Answer:
29
Watch out! Here we could not
use r=∞ as a reference and
we had to use an arbitrary
point r0 such that V(r0)=0. We
are allowed to do this, as
there is nothing special about
the place where the potential
is set to 0. It’s arbitrary.
Example: two shells and one sphere
(not in Y&F)
A solid conducting sphere with radius r1 is inside a spherical conducting
shell with internal radius r2 and external radius r3, which is inside another
spherical conducting shell of internal radius r4 and external radius r5. As
expected, r1 < r2 < r3 < r4 < r5. The internal sphere has a total charge +2q, while the outermost shell has a total charge -q, where q>0. The shell in the
middle is neutral.
Determine:
(a) The superficial charge density σind
induced at the exterior surface of
the interior shell (i.e. at r3)
(b) The magnitude of the electric field in
all space (i.e. for all values of r)
(c) The potential difference between the
sphere and the outermost shell
(See next slide for answers)
30
Example: two shells and one sphere
(not in Y&F)
(a) σ ind
Answers:
q
=
2π r32
⎧0, 0 < r < r1 ⎪ q ⎪ , r1 < r < r2 2 ⎪ 2πε 0 r ⎪0, r < r < r 4 ⎪⎪ 3 (b) ⎨ q ⎪ 2πε r 2 , r3 < r < r4 0 ⎪ ⎪0, r3 < r < r4 ⎪ q ⎪ , r5 < r 2 ⎪⎩ 4πε 0 r q ⎛1 1 1 1⎞ (c) V (r1 ) − V (r4 ) = − + − ⎟ ⎜ 2πε 0 ⎝ r1 r2 r3 r4 ⎠ 31 Extra Material (the material in the next few slides is just for your information and will not be evaluated in this course) 32 Electric Potential and Electric Field We saw that the electric potential can be obtained from the electric field by doing an integral This suggests that you should be able to go the other way as well, that is to obtain the electric field from the potential We can find out how without making a formal demonstration: The equation above can be rewritten as: If we remove the integrals on both sides and develop the right side we have: This suggests that: 33 Electric Potential and Electric Field In other words: ! ! E = −∇V Reminder: (in cartesian coordinates) In cylindrical and spherical coordinates, If V depends only on r then we have: ! dV E=− r̂ dr 34 Basic Example: Field of a Point Charge The ... Purchase answer to see full attachment

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