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Week 1 Coulomb's Law and Electric Fields
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Week 1 Coulomb's Law and Electric Fields
Due: 10:00pm on Monday, October 1, 2012
Note: You will receive no credit for late submissions. To learn more, read your instructor's Grading Policy
Placing Charges Conceptual Question
Below are free-body diagrams for three electric charges that lie in the same plane. Their relative positions are unknown.
Part A
Along which of the lines (A to H) in the figure should charge 2 be placed so that the
free-body diagrams of charge 1 and charge 2 are consistent?
Hint 1. How to approach the problem
Newton’s 3rd law states that the forces exerted by a pair of objects on each other are always equal in magnitude and opposite in
direction. Identifying the forces that correspond to 3rd-law pairs in the free-body diagrams will enable you to place the particles in their
proper relative position.
Hint 2. Placing charge 2
The two forces acting on charge 2 correspond to the forces exerted on it by charge 1 and charge 3. This means that one of these
forces must pair with a force on charge 1 of equal magnitude and opposite direction and the other must pair with a force on charge 3 of
equal magnitude and opposite direction. Also note that charge 2 should be repelled by charge 1, since both are negative. Therefore,
the vector that represents the force of charge 1 on charge 2 must point away from charge 1. This information is all you need to place
charge 2 in its correct position.
ANSWER:
Part B
Along which of the lines (A to H) in the figure should charge 3 be placed so that the
free-body diagrams of charge 1, charge 2, and charge 3 are consistent?
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ANSWER:
Part C
Along which of the lines (A to H) in the figure should charge 2 be placed so that the
free-body diagrams of charge 1 and charge 2 are consistent?
ANSWER:
Part D
Along which lines (A to H) in the figure should charge 3 be placed so that the
free-body diagrams of charge 1, charge 2, and charge 3 are consistent?
ANSWER:
± PSS 21.1 Coulomb's Law
Learning Goal:
To practice Problem-Solving Strategy 21.1 Coulomb's Law.
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Three charged particles are placed at each of three corners of an equilateral triangle whose sides are of length 4.0
negative charge:
= -8.9
and
= -17.8
. The remaining particle has a positive charge,
= 8.0
. Two of the particles have a
. What is the net electric force acting on
particle 3 due to particle 1 and particle 2?
Problem-Solving Strategy: Coulomb's law
IDENTIFY the relevant concepts:
Coulomb’s law comes into play whenever you need to know the electric force acting between charged particles.
SET UP the problem using the following steps:
1. Make a drawing showing the locations of the charged particles, and label each particle with its charge.
2. If three or more particles are present and they do not all lie on the same line, set up an xy coordinate system.
3. Often you will need to find the electric force on just one particle. If so, identify that particle.
EXECUTE the solution as follows:
1. For each particle that exerts a force on the particle of interest, calculate the magnitude of that force using
.
2. Sketch a free-body diagram showing the electric force vectors acting on the particle(s) of interest due to each of the other particles.
Recall that the force exerted by particle 1 on particle 2 points from particle 2 toward particle 1 if the two charges have opposite signs,
but points from particle 2 directly away from particle 1 if the charges have the same sign.
3. Calculate the total electric force on the particle(s) of interest. Recall that the electric force, like any force, is a vector.
4. As always, using consistent units is essential. If you are given non-SI units, don’t forget to convert!
5. If there is a continuous distribution of charge along a line or over a surface, divide the total charge distribution into infinitesimal
pieces, use Coulomb’s law for each piece, and then integrate to find the vector sum.
6. In many situations, the charge distribution will be symmetrical. Whenever possible, exploit any symmetries to simplify the problemsolving process.
EVALUATE your answer:
Check whether your numerical results are reasonable, and confirm that the direction of the net electric force agrees with the principle that like
charges repel and opposite charges attract.
IDENTIFY the relevant concepts
To determine the angle of the force vector on a single charged particle, you will need to calculate the vector sum of all the forces on that particle
due to the presence of other charged particles. To do this, you will need to use Coulomb's law.
SET UP the problem using the following steps
Part A
Identify the most appropriate xy coordinate system.
ANSWER:
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You are asked to find the net force acting on particle 3. Centering the xy coordinate system on particle 3 will make this easier.
EXECUTE the solution as follows
Part B
Find the net force
direction
acting on particle 3 due to the presence of the other two particles. Report you answer as a magnitude
and a
measured from the positive x axis.
Express the magnitude in newtons and the direction in degrees to three significant figures.
Hint 1. How to approach the problem
To calculate the electric force acting on particle 3, you should begin by drawing a free-body diagram indicating the forces acting on
particle 3 due to particle 1 and particle 2. You know that
.
Use Coulomb's law to calculate the magnitude of each of these forces. Apply vector algebra to find the component forces in the
the
and
directions. Then, sum the component forces for each direction:
.
From
and
you can find the magnitude and direction of the resulting electric force vector.
Hint 2. Draw a free-body diagram
Identify the forces on the positively charged particle 3.
Draw your vectors starting at the origin. The orientation of your vectors will be graded but their precise length will not.
ANSWER:
Hint 3. Calculate the force on particle 3 due to particle 1
Using the equation for Coulomb's law, calculate the magnitude of the force on particle 3 due to particle 1. Keep in mind that
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.
Express your answer in newtons using three significant figures.
ANSWER:
= 4.00×10−4
=
Use the same method to calculate the force on particle 3 due to particle 2.
Hint 4. Calculate the component forces on particle 3 due to particle 1
Calculate the x component and the y component forces acting on particle 3 due to particle 1, using simple trigonometry. The angle
between particle 1 and particle 3 is 60 :
Enter the components of the force in newtons separated by a comma.
ANSWER:
,
=
,
= 2.00×10−4, 3.46×10−4
,
Use the same method to calculate the component force on particle 3 due to particle 2. The sum of all the components in each
direction
will provide you with the information needed to calculate the magnitude and direction of the net force on particle 3.
Hint 5. How to calculate the component forces on particle 3 due to particle 2
Because particles 2 and 3 both lie on the x axis, there will be no y component to calculate. The x component of force will therefore be
equal to the value calculated from Coulomb's law, and the y component will be zero.
Hint 6. How to determine the magnitude and direction of a vector from its components
If a vector
has components
and
, the magnitude
and direction
are given by
,
where
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ANSWER:
,
=
,
= 1.06×10−3, 19.1
,
EVALUATE your answer
Part C
Assume that particle 3 is no longer fixed to a corner of the triangle and is now allowed to move. In what direction would particle 3 move the
instant after being released?
Draw the velocity vector for particle 3 below. The orientation of your vector will be graded, but not its length.
ANSWER:
Specifically, from Newton's 2nd law,
, you know that a mass accelerates in the same direction as the net force acting upon it.
Therefore, at the instant after being released, particle 3 accelerates in the same direction as
rest, its velocity at that instant will be
. Moreover, since particle 3 starts from
. In other words, the initial direction of particle 3 is the same direction as its acceleration, and
therefore the same direction as the applied net force.
Let us interpret this result in terms of electric forces. In general, like charges repel and unlike charges attract. If particle 3 were free to
move, it would move toward the negative charges
and . If
and
were the same size, particle 3 would start to move toward them
along a direction equidistant from each charge, that is, at an angle of
from the positive x axis. Instead,
, so particle 3 will be
more strongly attracted toward particle 2 and will move off in a direction less than
.
Exercise 21.8
Two small aluminum spheres, each having mass
, are separated by
.
Part A
How many electrons does each sphere contain? (The atomic mass of aluminum is
, and its atomic number is
.)
ANSWER:
= 7.25×1024
Part B
How many electrons would have to be removed from one sphere and added to the other to cause an attractive force between the spheres of
magnitude
(roughly one ton)? Assume that the spheres may be treated as point charges.
ANSWER:
=
15
5.27×10
Part C
What fraction of all the electrons in each sphere does this represent?
ANSWER:
=
−10
7.27×10
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Exercise 21.7
An average human weighs about 700
.
Part A
If two such generic humans each carried 3.0 coulomb of excess charge, one positive and one negative, how far apart would they have to be
for the electric attraction between them to equal their 700 weight?
Express your answer using two significant figures.
ANSWER:
= 11
=
Exercise 21.27
6
A proton is traveling horizontally to the right at 5.00×10
.
Part A
Find (a)the magnitude and (b) direction of the weakest electric field that can bring the proton uniformly to rest over a distance of 3.00
.
ANSWER:
= 4.35×106
=
Part B
ANSWER:
= 0
counterclockwise from the left direction
Part C
How much time does it take the proton to stop after entering the field?
ANSWER:
=
= 1.20×10−8
Part D
What minimum field ((a)magnitude and (b)direction) would be needed to stop an electron under the conditions of part (a)?
ANSWER:
=
= 2370
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Part E
ANSWER:
= 180
counterclockwise from the left direction
Exercise 21.30
A point charge is placed at each corner of a square with side length . The charges all have the same magnitude . Two of the charges are
positive and two are negative, as shown in the following figure.
Part A
What is the direction of the net electric field at the center of the square?
ANSWER:
rightward direction
leftward direction
upward direction
downward direction
Part B
What is the magnitude of the net electric field at the center of the square due to the four charges in terms of
and ?
Express your answer in terms of the variables , , and appropriate constants.
ANSWER:
=
Also accepted:
Exercise 21.47
Three negative point charges lie along a line as shown in the figure .
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Part A
Find the magnitude of the electric field this combination of charges produces at point
, which lies
from the
charge
measured perpendicular to the line connecting the three charges.
ANSWER:
= 1.04×107
Part B
Find the direction of the electric field this combination of charges produces at point
, which lies
from the
charge
measured perpendicular to the line connecting the three charges.
ANSWER:
outward
toward
Problem 21.64
Two charges, one of 2.50
and the other of -3.50
, are placed on the x-axis, one at the origin and the other at
= 0.600
, as shown in the
figure .
Part A
Find the position on the
-axis where the net force on a small charge
would be zero.
ANSWER:
= -3.27
Problem 21.73
A small 12.2 plastic ball is tied to a very light 27.0
string that is attached to the vertical wall of a room. (See the figure .) A uniform horizontal
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electric field exists in this room. When the ball has been given an excess charge of -1.50
, you observe that it remains suspended, with the string making an angle of 17.4
with the wall.
Part A
Find the magnitude of the electric field in the room.
ANSWER:
=
= 2.50×104
Part B
Find the direction of the electric field in the room.
ANSWER:
to the right
to the left
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