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Eðlisfræði 2, vor 2007
23b. Electric Potential
Assignment is due at 2:00am on Wednesday, February 7, 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.
Potential of Conductors that Touch and Separate
Conducting Tetrahedra
Two conductors, A and B, are each in the shape of a tetrahedron, but of different sizes. They are charged in the
following manner:
1. Tetrahedron A is charged from an electrostatic generator to charge .
2. Tetrahedron A is briefly touched to tetrahedron B.
3. Steps 1 and 2 are repeated until the charge on tetrahedron B reaches a maximum value.
Part A
If the charge on tetrahedron B was
tetrahdedron B?
after the first time it touched tetrahedron A, what is the final charge
on
Hint A.1 How to approach the problem
Hint not displayed
Part A.2 Find the ratio of the conductors' charges
Part not displayed
Part A.3 Find the maximum charge on A
Part not displayed
Express your answer in terms of .
ANSWER:
= Answer not displayed
Charged Mercury Droplets
A uniformly charged spherical droplet of mercury with electric potential
breaks into identical spherical
droplets, each with electric potential
. The small droplets are far enough apart form one another that they do
not interact significantly.
Part A
Find
, the ratio of
, the electric potential of the initial drop, to
, the electric potential of one of the
smaller drops.
Hint A.1 How to approach the problem
Hint not displayed
Part A.2 Find the charge on the small droplets
Keeping
conservation
chargeof
in mind,
finddroplet
the charge
on each small droplet.
Part A.3
Find the of
radius
a small
Express
answer
in terms
of
, the charge on the big droplet, and
What is theyour
radius
of each
small droplet?
.
ANSWER:
=
Hint A.3.a Consider
volume
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Hint not displayed
Express your answer in terms of
ANSWER:
, the radius of the big droplet, and
.
=
The ratio should be dimensionless and should depend only on
ANSWER:
=
Potential of Various Charge Configurations
Electric Force and Potential: Spherical Symmetry
Learning Goal: To understand the electric potential and electric field of a point charge in three dimensions
Consider a positive point charge , located at the origin of three-dimensional space.
Throughout this problem, use in place of
.
Part A
Due to symmetry, the electric field of a point charge at the origin must point _____ from the origin.
Answer in one word
ANSWER: Answer not displayed
Part B
Find
, the magnitude of the electric field at distance from the point charge .
Express your answer in terms of , , and .
ANSWER:
= Answer not displayed
Part C
Find
, the electric potential at distance from the point charge .
Express your answer in terms of , , and .
ANSWER:
= Answer not displayed
Part D
Which of the following is the correct relationship between the magnitude of a radial electric field
and its
associated electric potential
? More than one answer may be correct for the particular case of a point charge at
the origin, but you should choose the correct general relationship.
ANSWER: Answer not displayed
Now consider the figure, which shows several functions of the variable .
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Part E
Which curve could indicate the magnitude of the electric field due to a charge located at the origin (
)?
Hint E.1 How to approach the problem
Hint not displayed
ANSWER: Answer not displayed
Part F
Which curve could indicate the electric potential due to a positive charge located at the origin (
)?
Hint F.1 How to approach the problem
Hint not displayed
ANSWER: Answer not displayed
Part G
Which curve could indicate the electric potential due to a negative charge located at the origin (
)?
ANSWER: Answer not displayed
Part H
For either a positive or a negative charge, the electric field points from regions of ______ electric potential.
ANSWER: Answer not displayed
Potential of a Charged Ring
A ring with radius
and a uniformly distributed total charge
lies in the xy plane, centered at the origin.
Part A
What is the potential
due to the ring on the z axis as a function of ?
Hint A.1 How to approach the problem
Hint not displayed
Hint A.2 The potential due to a point charge
Hint not displayed
Express your answer in terms of
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, ,
, and
or
.
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Part B
What is the magnitude of the electric field
on the z axis as a function of , for
?
Part B.1 Determine the direction of the field
Part not displayed
Hint B.2 The relationship between electric field and potential
Hint not displayed
Express your answer in terms of some or all of the quantities
ANSWER:
|
, ,
, and
or
.
| = Answer not displayed
Potential of a Finite Rod
A finite rod of length has total charge , distributed uniformly along its length. The rod lies on the x -axis and is
centered at the origin. Thus one endpoint is located at
, and the other is located at
. Define the electric
potential to be zero at an infinite distance away from the rod. Throughout this problem, you may use the constant
in place of the expression
.
Part A
What is , the electric potential at point A (see the figure), located a distance
above the midpoint of the rod on the y axis?
Hint A.1 How to approach the problem
Divide the rod into infintesimal pieces of length . Find an expression for the potential due to one of these
segments, and then integrate to find the potential due to the entire rod.
Part A.2 Find the electric potential of a section of the rod
What is the electric potential
position
?
at point A due to an arbitrary section of the rod with length
centered at
Hint A.2.a A general formula for electric potential
The electric potential
at a distance from a point charge
is given by
, where
.
Part A.2.b Find the distance from a section to point A
What is , the distance from point A to an arbitrary section of the rod with length
Express your answser in terms of
ANSWER:
centered at position
?
and .
=
Part A.2.c Find the charge on a section of the rod
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What is , the amount of charge on a section of the rod with length ?
Express your answer in terms of
ANSWER:
,
=
Express your answer in terms of ,
ANSWER:
, and .
,
, ,
, and .
=
Now integrate. Be sure to choose the right limits of integration: You want to integrate over the whole length
of the rod. You can use the symmetry of the rod to simplify your integral.
Hint A.3 A helpful integral
Express your answer in terms of
ANSWER:
If
,
, , and .
=
, this answer can be approximated as
.
For
,
. For this problem, this means that the logarithm can be further approximated as
, and the expression for potential reduces to
. This is what we expect, because it means that
from far away, the potential due to the charged rod looks like that due to a point charge.
Part B
What is
, the electric potential at point , located at distance from one end of the rod (on the x axis)?
Hint B.1 How to approach the problem
As in Part A, divide the rod into infintesimal pieces of length . Find an expression for the potential due to one
of these segments, and then integrate to find the potential due to the entire rod.
Part B.2 Find the distance from point B to a segment of the rod
What is , the distance from point B to an arbitrary section of the rod with length
centered at position
?
Part B.2.a Find the x coordinate of B
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What is
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, the x coordinate of B?
Express your answer in terms of some or all of the variables
ANSWER:
=
Express your answer in terms of
ANSWER:
,
, and .
=
Give your answer in terms of ,
ANSWER:
and .
,
, and .
= Answer not displayed
Potential of a Charged Disk
A disk of radius has a total charge uniformly distributed over its surface. The disk has negligible thickness and
lies in the xy plane. Throughout this problem, you may use the variable in place of
.
Part A
What is the electric potential
on the z axis as a function of , for
?
Hint A.1 How to approach the problem
Hint not displayed
Part A.2 Find the potential due to a ring
Part not displayed
Hint A.3 A useful antiderivative
Hint not displayed
Express your answer in terms of
ANSWER:
, , and . You may use
instead of
.
= Answer not displayed
Part B
What is the magnitude
of the electric field on the axis, as a function of , for
?
Part B.1 Direction of the electric field
Part not displayed
Hint B.2 Electric field from potential
Hint not displayed
Express your answer in terms of some or all of the variables
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, , and . You may use
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instead of
ANSWER:
Summary
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.
= Answer not displayed
1 of 6 problems complete (16.33% avg. score)
4.9 of 10 points
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