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Physics 210 Q1 2012
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Week 4 - Electric Potential
Due: 10:00pm on Monday, October 22, 2012
Note: You will receive no credit for late submissions. To learn more, read your instructor's Grading Policy
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
charge
. Charges A, B, and C have charge
, and D has
.
Throughout this problem, use
in place of
.
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
factor, in
. In the space provided, enter the numeric value that multiplies the above
.
Hint 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 2. Electric potential and potential energy
Recall that the electric potential at a point at distance
defines the electric potential to be zero at
from a charge
is
, where
. The electric potential energy of a charge
. Note that this equation implicitly
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.
Hint 3. Work required to place charge A
What is
, the work required to assemble the charge distribution shown in the figure?
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Express your answer in terms of some or all of the variables , , and
.
Hint 1. Find the potential at the location of charge A
What is
, the electric potential at the upper left corner of the square before charge A is placed there?
Express your answer in terms of some or all of the variables , , and
.
ANSWER:
= 0
ANSWER:
= 0
Hint 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?
Express your answer in terms of some or all of the variables , , and
.
Hint 1. Find the potential at the location of charge B
What is
, the potential at the upper right corner due to charge A, before charge B is placed there?
Express your answer in terms of some or all of the variables , , and
.
ANSWER:
=
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ANSWER:
=
Hint 5. Work required to place charge C
What is
, the amount of work required to add charge C to the configuration, as shown in the figure?
Express your answer in terms of some or all of the variables , , and
.
Hint 1. Find the potential at the location of charge C
What is
, the potential at the lower right corner of the square before charge C is placed there?
Hint 1. How to approach this part
The potential at C is the sum of the individual potentials due to the charges at B and A.
Hint 2. Find the potential at C due to the charge at B
What is the potential at C due to the charge at B?
Express your answer in terms of some or all of the variables , , and
.
ANSWER:
=
Hint 3. Find the potential at C due to the charge at A
What is the potential at C due to the charge at A?
Express your answer in terms of some or all of the variables , , and
.
ANSWER:
=
.
ANSWER:
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=
ANSWER:
=
Hint 6. Find the work required to place charge D
What is
, the amount of work required to add charge D to the configuration?
Express your answer in terms of some or all of the variables , , and
.
Hint 1. Find the potential at the position of charge D
What is
, the potential at the lower left corner of the square before charge D is placed there?
Express your answer in terms of some or all of the variables , , and
.
ANSWER:
=
ANSWER:
=
Because D has a negative charge, it is attracted to charges A, B, and C. This accounts for the negative sign on the work. We
would have to do positive work to remove charge D from the configuration.
ANSWER:
= 0
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The hints led you through the problem by adding one charge at a time. A little thought shows that this is equivalent to simply adding the
energies of all possible pairs:
.
Note that this is not equivalent to adding the potential energies of each charge. Adding the potential energies will give you double the
correct answer because you will be counting each charge twice.
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:
figure a
figure b
figure c
Moving a Charge
Part A
A point charge with charge
( 0.170
= 2.10
, 0) to the point ( 0.255
is held stationary at the origin. A second point charge with charge
, 0.275
Express your answer in joules. Use
). How much work
= 8.99×10
9
= -4.10
moves from the point
is done by the electric force on the moving point charge?
for Coulomb's constant:
.
Hint 1. How to approach the problem
Use the equation for the electric potential energy between two point charges to calculate the work done by the electric force. Recall
that the work done by a conservative force is
, the difference between the initial and final potential energies. A
conservative force is one for which the work done on a particle by the force is independent of the path taken and depends only on the
initial and final points. The electric force is a conservative force. Gravity is another example of a conservative force.
Hint 2. Calculate the initial electric potential energy
Calculate the initial electric potential energy
Express your answer in joules. Use
when the moving point charge is at the point ( 0.170
9
= 8.99×10
for Coulomb's constant:
, 0).
.
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Hint 1. Derivation of electric potential energy
The force between two point charges
the charges and
point
and
is given by Coulomb's law as
, where
is the separation between
. The work done by the electric force between the charges as one charge moves from point
and the other is held fixed is calculated using
to
. Since the force depends only on the distance between
the charges, it follows that
,
where
and
are the distances between the fixed charge and points
and , respectively. Since the work done is equal to
the change in potential energy, this equation is consistent with defining the electric potential energy between two point charges a
distance
apart by
.
ANSWER:
= -0.455
=
Hint 3. Calculate the final electric potential energy
Calculate the final electric potential energy
Express your answer in joules. Use
when the moving charge is at the point ( 0.255
9
= 8.99×10
for Coulomb's constant:
, 0.275
).
.
Hint 1. Derivation of electric potential energy
The force between two point charges
the charges and
point
and
is given by Coulomb's law as
, where
is the separation between
. The work done by the electric force between the charges as one charge moves from point
and the other is held fixed is calculated using
to
. Since the force depends only on the distance between
the charges, it follows that
,
where
and
are the distances between the fixed charge and points
and , respectively. Since the work done is equal to
the change in potential energy, this equation is consistent with defining the electric potential energy between two point charges a
distance
apart by
.
ANSWER:
=
= -0.206
ANSWER:
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= -0.249
=
Also accepted:
= -0.249
Exercise 23.5
A small metal sphere, carrying a net charge of
with a net charge of
speed 22.0
= -7.20
= -2.50
, is held in a stationary position by insulating supports. A second small metal sphere,
and mass 1.80 , is projected toward
. When the two spheres are 0.800
apart,
is moving toward
with
. Assume that the two spheres can be treated as point charges. You
can ignore the force of gravity.
Part A
What is the speed of
when the spheres are 0.400
apart?
ANSWER:
= 16.1
=
Part B
How close does
get to
?
ANSWER:
=
= 0.254
Exercise 23.13
A small particle has charge -3.30
where the electric potential
particle has a speed of 3.40
and mass 2.00×10
= 520
−4
. It moves from point
is greater than the potential at point
at point
, where the electric potential is
= 250
, to point
,
. The electric force is the only force acting on the particle. The
.
Part A
What is its speed at point
?
ANSWER:
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= 4.52
=
Part B
Is it moving faster or slower at
than at
?
ANSWER:
Faster
Slower
Exercise 23.19
Two point charges
from
= 2.10
and
= -6.90
are 0.100
apart. Point
is midway between them; point
is 0.080
from
and 0.060
(the figure ). Take the electric potential to be zero at infinity.
Part A
Find the potential at point
.
ANSWER:
=
= -863
Part B
Find the potential at point
.
ANSWER:
=
= -798
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Part C
Find the work done by the electric field on a charge of 2.80
that travels from point
to point
.
Express your answer using two significant figures.
ANSWER:
= 1.8×10−7
=
Exercise 23.24
At a certain distance from a point charge, the potential and electric field magnitude due to that charge are 4.98
and 12.0
, respectively.
(Take the potential to be zero at infinity.)
Part A
What is the distance to the point charge?
ANSWER:
= 0.415
Part B
What is the magnitude of the charge?
ANSWER:
= 2.30×10−10
Part C
Is the electric field directed toward or away from the point charge?
ANSWER:
Toward
Away
Exercise 23.25
A uniform electric field has magnitude
point
(at
0.80
) is 350
and is directed in the negative
direction. The potential difference between point
(at
0.70
) and
.
Part A
Which point,
or , is at the higher potential?
ANSWER:
a
b
Part B
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Calculate the value of
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.
Express your answer using two significant figures.
ANSWER:
=
= 3500
Part C
A negative point charge
is moved from
to . Calculate the work done on the point charge by the electric field.
Express your answer using two significant figures.
ANSWER:
= −7.0×10−5
=
Exercise 23.45
In a certain region of space, the electric potential is
, where
,
, and
are positive constants.
Part A
Calculate the
-component of the electric field.
Express your answer in terms of the given quantities.
ANSWER:
=
Part B
Calculate the -component of the electric field.
Express your answer in terms of the given quantities.
ANSWER:
=
Part C
Calculate the -component of the electric field.
Express your answer in terms of the given quantities.
ANSWER:
= 0
Part D
At which point is the electric field equal to zero?
ANSWER:
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Problem 23.62
A small sphere with mass 1.60 hangs by a thread between two large parallel vertical plates 5.00
uniform surface charge densities
9.80×10
−6
and
. The charge on the sphere is
apart . The plates are insulating and have
=
.
Part A
What potential difference between the plates will cause the thread to assume an angle of 30.0 with the vertical?
ANSWER:
=
= 46.2
Problem 23.83
A metal sphere with radius
has a charge
. Take the electric potential to be zero at an infinite distance from the sphere.
Part A
What is the electric field at the surface of the sphere?
Express your answer in terms of the given quantities and appropriate constants.
ANSWER:
=
Also accepted:
,
Part B
What is the electric potential at the surface of the sphere?
Express your answer in terms of the given quantities and appropriate constants.
ANSWER:
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=
Also accepted:
,
Part C
This sphere is now connected by a long, thin conducting wire to another sphere of radius
that is several meters from the first sphere.
Before the connection is made, this second sphere is uncharged. After electrostatic equilibrium has been reached, what is the total charge on
each sphere? Assume that the amount of charge on the wire is much less than the charge on each sphere.
Express your answer in terms of the given quantities and appropriate constants.
ANSWER:
=
Part D
Express your answer in terms of the given quantities and appropriate constants.
ANSWER:
=
Part E
What is the electric potential at the surface of each sphere? Assume that the amount of charge on the wire is much less than the charge on
each sphere.
Express your answer in terms of the variables
,
,
and appropriate constants.
,
,
and appropriate constants.
ANSWER:
=
Also accepted:
,
Part F
Express your answer in terms of the variables
ANSWER:
=
Also accepted:
,
Part G
What is the electric field at the surface of each sphere? Assume that the amount of charge on the wire is much less than the charge on each
sphere.
Express your answer in terms of the variables
,
,
and appropriate constants.
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ANSWER:
=
Also accepted:
,
Part H
Express your answer in terms of the variables
,
,
and appropriate constants.
ANSWER:
Also accepted:
,
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