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Eðlisfræði 2, vor 2007
23a. Electric Potential
Assignment is due at 2:00am on Wednesday, January 31, 2007
Credit for problems submitted late will decrease to 0% after the deadline has passed.
The wrong answer penalty is 2% per part. Multiple choice questions are penalized as described in the online help.
The unopened hint bonus is 2% per part.
You are allowed 4 attempts per answer.
Electric Potential and Properties
Electric Potential Energy versus Electric Potential
Learning Goal: To understand the relationship and differences between electric potential and electric potential
energy.
In this problem we will learn about the relationships between electric force , electric field , potential energy ,
and electric potential . To understand these concepts, we will first study a system with which you are already
familiar: the uniform gravitational field.
Gravitational Force and Potential Energy
First we review the force and potential energy of an object of mass in a uniform gravitational field that points
downward (in the
direction), like the gravitational field near the earth's surface.
Part A
Find the force
Express
ANSWER:
on an object of mass
in terms of
in the uniform gravitational field when it is at height
.
, , , and .
=
Because we are in a uniform field, the force does not depend on the object's location. Therefore, the variable
does not appear in the correct answer.
Part B
Now find the gravitational potential energy
potential to be at position
.
of the object when it is at an arbitrary height . Take zero
Express
in terms of , , and . Note that because potential energy is a scalar, and not a
vector, there will be no unit vector in the answer.
ANSWER:
=
Part C
In what direction does the object accelerate when released with initial velocity upward?
ANSWER:
upward
downward
upward or downward depending on its mass
downward only if the ratio of to initial velocity is large enough
Electric Force and Potential Energy
Now consider the analogous case of a particle with charge placed in a uniform electric field of strength ,
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pointing downward (in the
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direction)
Part D
Find
, the electric force on the charged particle at height .
Hint D.1 Relationship between force and electric field
Hint not displayed
Express
in terms of ,
ANSWER:
, , and .
=
Part E
Now find the potential energy
position
.
Express
of this charged particle when it is at height . Take zero potential to be at
(a scalar quantity) in terms of ,
ANSWER:
, and .
=
Part F
In what direction does the charged particle accelerate when released with upward initial velocity?
ANSWER:
upward
downward
upward or downward depending on its charge
downward only if the ratio of to initial velocity is large enough
Electric Field and Electric Potential
The electric potential
with charge .
is defined by the relationship
, where
is the electric potential energy of a particle
Part G
Find the electric potential
of the uniform electric field
. Note that this field is not pointing in the same
direction as the field in the previous section of this problem. Take zero potential to be at position
Express
in terms of ,
ANSWER:
.
, and .
=
Part H
The SI unit for electric potential is the volt. The volt is a derived unit, which means that it can be written in
terms of other SI units. What are the dimensions of the volt in terms of the fundamental SI units?
Express your answer in terms of the standard abbreviations for the fundamental SI units:
(meters),
(kilograms), (seconds), and
(coulombs)
ANSWER:
volts =
Part I
The electric field can be derived from the electric potential, just as the electrostatic force can be determined from
the electric potential energy. The relationship between electric field and electric potential is
, where is the
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gradient operator:
.
The partial derivative
means the derivative of
Consider again the electric potential
coordinate only, so
with respect to , holding all other variables constant.
corresponding to the field
and
.
Find an expression for the electric field
in terms of the derivative of .
Express your answer as a vector in terms of the unit vectors
derivative of
with respect to .
ANSWER:
. This potential depends on the z
, , and/or . Use
for the
=
Part J
A positive test charge will accelerate toward regions of ________ electric potential and ________ electric potential
energy.
Part J.1 Direction of the electric field
In what direction do electric fields point?
ANSWER:
from regions of higher electric potential to lower electric potential
from regions of lower electric potential to higher electric potential
Hint J.2 Formula for the force on a charge in an electric field
The force
on a charge in an electric field
is given by
.
This is similar to the equation
for the force on a mass in a uniform gravitational field.
Hint J.3 Formula for electric potential energy
The electric potential energy
of a charge at electric potential
is given by
.
This is similar to the equation
, for the gravitational potential energy of a particle with mass .
Choose the appropriate answer combination to fill in the blanks correctly.
ANSWER:
higher; higher
higher; lower
lower; higher
lower; lower
Part K
A negative test charge will accelerate toward regions of ________ electric potential and ________ electric
potential energy.
Part K.1 Direction of the electric field
Part not displayed
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Hint K.2 Formula for the force on a charge in an electric field
Hint not displayed
Hint K.3 Formula for electric potential energy
Hint not displayed
Choose the appropriate answer combination to fill in the blanks correctly.
ANSWER:
higher; higher
higher; lower
lower; higher
lower; lower
A charge in an electric field will experience a force in the direction of decreasing potential energy. Since the
electric potential energy of a negative charge is equal to the charge times the electric potential (
), the
direction of decreasing electric potential energy is the direction of increasing electric potential.
Electric Potential, Potential Energy, and Force
Learning Goal: To review relationships among electric potential, electric potential energy, and force on a test
charge
This problem is a review of the relationship between an electric field , its associated electric potential , the
electric potential energy , and the direction of force on a test charge.
Part A
Electric field lines always begin at _______ charges (or at infinity) and end at _______ charges (or at infinity).
One could also say that the lines we use to represent an electric field indicate the direction in which a _______ test
charge would initially move when released from rest. Which of the following fills in the three missing words
correctly?
ANSWER: Answer not displayed
Part B
Part not displayed
Part C
Part not displayed
Charges moving in Electric Potential
The Fate of an Electron in a Uniform Electric Field
In this problem we will study the behavior of an electron in a uniform electric field. Consider a uniform electric
field (magnitude ) as shown in the figure within a parallel plate capacitor in vacuum.
First, let us review the relationship between an electric field and its associated
electric potential . For now, ignore the electron located between the plates.
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Part A
Calculate the electric potential
plate to be zero.
inside the capacitor as a function of height . Take the potential at the bottom
Hint A.1 Relationship of field and potential
Hint not displayed
Hint A.2 Limits of integration
Hint not displayed
Express
in terms of
ANSWER:
and .
=
Now an electron of mass
the figure) at height .
and charge
(where
is a positive quantity) is placed within the electric field (see
Part B
Calculate the electon's potential energy
, neglecting gravitational potential energy.
Hint B.1 Definition of electric potential energy
Hint not displayed
Express your answer in terms of
ANSWER:
,
, and
.
=
Part C
The electron, having been held at height , is now released from rest. Calculate its speed (i.e.,
reaches the top plate.
) when it
Part C.1 How to approach the problem
Part not displayed
Part C.2 Find the initial energy
Part not displayed
Part C.3 Find the final energy
Part not displayed
Express
in terms of
ANSWER:
,
,
,
,
, and other given quantities and constants.
=
Now we consider the effect of changing either the charge or the mass of the charged particle that is released from
rest at height .
Part D
Imagine a particle that has three times the mass of the electron. All other quantities given above remain the same.
What is the ratio of the kinetic energy
that this heavy particle would have when it reaches the upper plate to
the kinetic energy that the electron would have? That is, what is
?
Hint D.1 Energy conservation
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In the motion of both particles, total energy is conserved. Does the potential energy of the particles depend on
their mass?
ANSWER:
3
1
none of the above: the heavy particle will never reach the upper plate
Part E
Imagine a third particle, which we will call a cyberon. It has three times the mass of an electron ( ). It has a
positive charge that is three times the magnitude ( ) of the charge on an electron. What is the ratio of the speed
that the cyberon would have when it reaches the upper plate after being released from rest at position to the
speed that the electron would have? That is, what is ?
ANSWER:
3
1
none of the above: the cyberon will never reach the upper plate
Because it has positive instead of negative charge, the cyberon will accelerate downward, toward the lower plate.
Bouncing Electrons
Two electrons, each with mass and charge , are released from positions very far from each other. With respect to
a certain reference frame, electron A has initial nonzero speed toward electron B in the positive x direction, and
electron B has initial speed toward electron A in the negative x direction. The electrons move directly toward each
other along the x axis (very hard to do with real electrons). As the electrons approach each other, they slow due to
their electric repulsion. This repulsion eventually pushes them away from each other.
Part A
Which of the following statements about the motion of the electrons in the given reference frame will be true at
the instant the two electrons reach their minimum separation?
ANSWER: Answer not displayed
Part B
Part not displayed
Energy in Configuration of Charges
Energy Stored in a Charge Configuration
Four point charges, A, B, C, and D, are placed at the corners of a square with side length . Charges A, B, and C
have charge
, and D has charge .
Throughout this problem, use in place of
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Part A
If you calculate , the amount of work it took to assemble this charge configuration if the point charges were
initially infinitely far apart, you will find that the contribution for each charge is proportional to . In the space
provided, enter the numeric value that multiplies the above factor, in
.
Hint A.1 How to approach the problem
The Coulomb force is conservative. If we define the potential energy of the system to be zero when the charges
are infinitely far apart, the amount of work needed to place any one charge in a configuration is equal to its
electric potential energy. Imagine moving charge A, then B, then C, and finally D into place. Find the work
required to add each charge to the configuration by calculating the potential energy of each just after it is added.
Add the work required for each charge to find the total work required.
Hint A.2 Electric potential and potential energy
Recall that the electric potential at a point at distance from a charge is
, where
. Note that this
equation implicitly defines the electric potential to be zero at
. The electric potential energy of a charge is
equal to , where is the electric potential at the position of the charge before the charge is placed there. To find
the potential at a point due to multiple charges, sum the potentials at that point due to each charge.
Part A.3 Work required to place charge A
What is
, the work required to assemble the charge distribution shown in the figure?
Part A.3.a Find the potential at the location of charge A
Part not displayed
Express your answer in terms of some or all of the variables , , and .
ANSWER:
= 0
Part A.4 Work required to place charge B
What is , the amount of work required to add charge B to the configuration, as shown in the figure?
Part A.4.a Find the potential at the location of charge B
Part not displayed
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Part B
Which of the following figures depicts a charge configuration that requires less work to assemble than the
configuration in the problem introduction? Assume that all charges have the same magnitude .
ANSWER:
Summary
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figure a
figure b
figure c
3 of 5 problems complete (56.77% avg. score)
14.19 of 15 points
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