Unit 1: Electric Charge and Coulomb’s Law
Lesson 1.2
Coulomb's Law
Contents
Introduction
1
Learning Objectives
2
Warm Up
2
Learn about It!
Coulomb’s Law
Mathematical Statement of Coulomb’s law
Free-Body Diagrams
Superposition of Forces
4
4
5
10
11
Key Points
17
Key Formula
18
Check Your Understanding
18
Challenge Yourself
20
Bibliography
21
Key to Try It!
22
Unit 1: Electric Charge and Coulomb’s Law
Lesson 1.2
Coulomb's Law
Introduction
From the spark that you feel as you walk through the mall and suddenly touched a cold
metal up to the laser printer that you can see in the photo above, all are applications of
electric charges and Coulomb’s law. The technology of laser printers is the same as when
you write a word in the air using a flashlight. When a document is sent to the printer, a
laser beam "draws" the document on a selenium-coated drum using electrical charges.
When the drum is charged, it is rolled in a toner, a dry powder type of ink. The toner
attaches to the charged image on the drum. The toner is transferred onto a piece of paper
and fused to the paper with heat and pressure. In this lesson, we will further discuss how
electric charges interact with each other.
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
Learning Objectives
DepEd Competencies
●
negative charges, and that charge
In this lesson, you should be able to do the
is measured in coulombs
following:
●
Predict the direction of the net
(STEM_GP12EMIIIa-4).
●
of point charges
charges.
(STEM_GP12EMIIIa-6).
●
State Coulomb’s law.
●
Calculate net electric force in
●
one-dimensional
and
two-dimensional problems.
Solve
Calculate the net electric force on a
point charge exerted by a system
force in a charge in a system of
●
State that there are positive and
problems
involving
Solve problems involving electric
charges, dipoles, forces, fields, and
flux in contexts such as, but not
limited to, systems of point
charges, classical models of the
atom, electrical breakdown of air,
charged pendulums, control of
electrostatic forces in the context
electron and proton beams,
of systems of point charges.
electrostatic ink-jet printers
(STEM_GP12EMIIIa-14).
Warm Up
Force Ring
10 minutes
The activity below allows you to play with forces using strings and binders.
Materials
●
metal ring binder
●
string
●
bottles
●
stopwatch
Procedure
1. Group yourselves with at most four members each.
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
2. Bring out the necessary materials and create the following set up:
Fig. 1.2.1. The strings are tied around the binder. The number of strings should correspond
to the number of members.
3. Place a bottle at the end side of the classroom (approximately 5 meters apart) and
another to the starting side.
4. Hold the strings at the chest level. Make sure that the ring should not go higher or
lower to the chest of your shortest groupmate. The ring should be taut at all times.
5. From the starting point, shoot the ring to the endpoint by gradually walking towards
it and following the rules, as stated in procedure number 4.
6. Once you shoot the ring to the bottle, let go of the string and rotate clockwise.
7. Grab the strings again and go back to the starting point while following the same
rules.
8. The group with the most number of rings shoot to the bottles within 5 minutes wins.
Guide Questions
1. What happens to the ring if it is not taut?
2. What happens to the ring when one of your group mates pulls it stronger than the
others?
3. How did you maintain the height of the ring?
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
Learn about It!
We know that if there are two oppositely charged spheres hanging from the ceiling by
insulating strings, they will have the tendency to attract each other. On the other hand, if we
charge the other sphere with the same charge as the other, the spheres will repel. We can
deduce from this that charged particles exert forces with each other. This force can be
described using Coulomb’s law.
What are the factors that affect the forces
between two charges?
Coulomb’s Law
Charles-Augustin de Coulomb (1736–1806) described the force between two charges using
a torsion balance. He discovered an inverse square relationship between the electrostatic
force (Fe) and the distance of separation (r) of the charges—that is
. He further
discovered that the force is dependent on the quantity of charge on each body, which is
denoted by q1 and q2. He found that the forces that two point charges q1 and q2 exert on
each other are proportional to each charge and therefore are proportional to the product
q1q2 of the two charges. Because of these findings, he was able to formulate the Coulomb’s
law.
Remember
Coulomb’s Law states that the magnitude of the electric force
between two point charges is directly proportional to the product of
the charges and inversely proportional to the square of the distance
between them.
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
Did You Know?
Charles Coulomb (1736–1806) measured the magnitudes of the
electric forces between charged objects using the torsion balance,
as shown in the figure below.
Fig. 1.2.2. An example of a torsion balance used by Coulomb to formulate
the Coulomb’s Law.
Mathematical Statement of Coulomb’s law
The Coulomb’s Law can be expressed mathematically through the following equation:
Equation
1.2.1
where Fe is the electrostatic force
q1 and q2 are the charges that are interacting
r is the distance separation between the two charges
is the permittivity of free space with a value of 8.8542 × 10−12 N-1 C2 m-2.
The absolute value symbol suggests that the force is always positive. The directions of the
forces the two charges exert on each other are always along the line joining them. When the
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
charges q1 and q2 have the same sign, either both positive or both negative, the forces are
repulsive; when the charges have opposite signs, the forces are attractive.
For brevity, we can also express the Coulomb’s Law using the constant k.
Equation
1.2.2
where the constant
is just noted by k with an equal value of 8.988 x 109 N m2 C-2. This
can be usually approximated to be 9 x 109 N m2 C-2.
It is important to note that given two charges q1 and q2 such that q1 has greater charge than
q2, q2 exerts the same force to q1 as q1 exerts to q2. Thus, two charges will exert an equal
force to each other regardless of their individual charge. This is a consequence of the third
law of motion.
Example 1
Two point charges, q1 = +5 C and q2 = -3 C, are separated by a distance r = 30 000 m. Find the
magnitude of the electric force that q1 exerts to q2. Predict whether it will be attractive or
repulsive.
Solution
Step 1:
Identify the given in the problem
The individual charges are given q1 = +5 C and q2 = -3 C. The distance separation is
also given r = 30000 m.
Step 2:
Identify what is asked in the problem.
You are asked to solve for the magnitude of the electric force that q1 exerts to q2.
Step 3:
1.2 Coulomb’s Law
Write the working equation.
6
Unit 1: Electric Charge and Coulomb’s Law
Step 4:
Substitute the given values.
Step 5:
Solve for the answer.
It is attractive since the two charges are opposite.
1 Try It!
Two charged spheres are hanging from two planes by an insulating string. One of the
spheres has a charge of 6 C, and the other has a charge of -12 C. When the two
planes are at a distance of 50 000 m, calculate the force that each sphere exerts on
the other. Assume that the only force that acts on the spheres is the electrostatic
force.
Example 2
Two equally charged spheres exert 12 N to each other. If they are separated by a distance of
12 m, calculate the charge on either sphere.
Solution
Step 1:
Identify the given in the problem
The force between the two spheres is given which is equal to 12 N. The distance
separation r = 12 m, is also given.
Step 2:
Identify what is asked in the problem.
You are asked to solve for the value of the charges.
Step 3:
1.2 Coulomb’s Law
Write the working equation.
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Unit 1: Electric Charge and Coulomb’s Law
Note that the spheres are equally charged. Thus, we can rewrite the
Coulomb’s law as
. Therefore,
.
Step 4:
Substitute the given values.
Step 5:
Solve for the answer.
The charge on either sphere is 4.38 ✕ 10‒4 C.
2 Try It!
Two spheres, one is charged twice as much as the other, are separated with a
distance of 6 m. At this distance, they exert 20 N to each other. Calculate the charge
of each sphere.
Example 3
Calculate the distance separation between charges q1 = 5 nC and q2 = 3 nC when they exert
a force equal to 3 N. Deduce what will happen to the electrostatic force when the original
distance separation is doubled.
Solution
Step 1:
Identify the given in the problem
The force between the two spheres is given, which is equal to 3 N. The charges are
also given q1 = 5 nC and q2 = 3 nC.
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
Step 2:
Identify what is asked in the problem.
You are asked to calculate for the distance separation.
Step 3:
Write the working equation.
Step 4:
Substitute the given values.
Step 5:
Solve for the answer.
The distance separation between the two charges is 2.12 ✕ 10‒4 m.
If the original distance is doubled, the electrostatic force will be reduced by
one-fourth since there is an inverse square relationship between the distance
separation and the force.
3 Try It!
How far will you separate two charges with an equal value of 3 mC in order for them
to exert 5 N to each other? Predict what will happen to the electrostatic force if the
distance separation was halved.
Tips
In solving problems involving Coulomb’s law, make sure that all of
the units are expressed in terms of SI.
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Unit 1: Electric Charge and Coulomb’s Law
Free-Body Diagrams
When dealing with forces, it is necessary to start the problem solving by drawing free-body
diagrams. Free-body diagrams are diagrams used to show the relative magnitude and
direction of all forces acting upon an object in a given situation.
An important concept that we have to remember in drawing free-body diagrams is to
determine the charges of the particles and relative to what particle we are considering the
force.
You can refer to these three interacting particles A, B, and C.
Fig. 1.2.2. An illustration showing the interaction of three charges.
Relative to A, the free body diagram shows
Fig. 1.2.3. A free-body diagram showing the forces at charge A.
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
Relative to B, the free-body diagram shows
Fig. 1.2.4. A free-body diagram showing the forces at charge B.
Finally, relative to C, the free-body diagram would look like:
Fig. 1.2.5. A free-body diagram showing the forces at charge C.
Note that the arrows represent the direction of the force. The fundamental rule for charges
apply in this situation: like charges repel and unlike charges attract.
How can we solve the net force in a system of point
charges?
Superposition of Forces
Deducing from the equation provided by the Coulomb’s law—only the interaction of two
charges are considered. When more than two charges are present, the resultant force on
any one of them equals the vector sum of the forces exerted by the various individual
charges. For example, if four charges are present, then the resultant force exerted by
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
particles 2, 3, and 4 on particle 1 is F1 = F2 to 1 + F3 to 1 + F4 to 1. This important property, called
the principle of superposition of forces, holds for any number of charges. By using this
principle, we can apply Coulomb’s law to any collection of charges to get the net force acting
on a specific charge.
Example 4
Two point charges are located on the x-axis of a coordinate system: q1 = 3.0 C is at x = +2.0
m, and q2 = -5.0 C is at x = +4.0 m. What is the total electric force exerted by q1 and q2 on a
charge q3 = 5.0 C at x = 0? Where is the net force directed?
Solution
Step 1:
Identify the given in the problem
The individual charges are given and their positions in the cartesian plane (x-axis).
Step 2:
Identify what is asked in the problem.
You are asked to solve the magnitude of the net electric force at q3.
Step 3:
1.2 Coulomb’s Law
Draw the free-body diagram of the problem.
12
Unit 1: Electric Charge and Coulomb’s Law
Step 4:
Write the working equation.
Note that the first term is negative since it is directed to the -x direction (going to
the left).
Step 5:
Substitute the given values.
Step 6:
Solve for the answer.
The net force on q3 is ‒1.97 ✕ 1010 N. It is directed to the left because of the
negative sign.
4 Try It!
Three charges q1 = 3 C, q2 = -4 C, and q3 = 5 C are placed along the y-axis with the
positions y1 = 0, y2 = 5 m and y3 = 20 m, respectively. Calculate the net force at q2.
Example 5
Two equal positive charges q1 = q2 = 10.0 mC are located at x = 0, y =
0.70 m and x = 0, y = -0.70 m, respectively. Calculate for the net force
at q3 = 10.0 mC at x = 0.80 m, y = 0.
Solution
Step 1:
Identify the given in the problem
The individual charges and their position are given.
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
Step 2:
Identify what is asked in the problem.
You are asked to solve the net force at q3.
Step 3:
Draw the free-body diagram of the problem.
Step 4:
Write the working equation.
By symmetry, we can see that the y-components of the net force is just equal to
zero. Therefore,
Step 4:
.
Substitute the given values.
Since the y-components will just be cancelled, we are only after the x-component.
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
Step 5:
Solve for the answer.
The net force at q3 at x = 0.80 m, y = 0 is 1.04 ✕ 106 N.
5 Try It!
Two point charges are arranged in the Cartesian plane as follows: charge q1 = -1.50
nC at y = -0.600 m, and charge q2 =+3.20 nC at the origin (y = 0). What is the total force
(magnitude and direction) exerted by these two charges on a third charge q3 = +5.00
nC located at y = -0.400 m?
Example 6
Three charges lie along the x-axis as shown in the figure below. A positive charge q1 = 15.0
nC is at x = 2.00 m and another positive charge q2 = 6.0 nC is at the origin, and the resultant
force acting on q3 is zero. What is the x coordinate of q3?
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
Solution
Step 1:
Identify the given in the problem
The value of the charges are given q1 = 15 nC, q2 = 6 nC. Further information are
given in the diagram above.
Step 2:
Identify what is asked in the problem.
You are asked to solve for the position of q3 where its net force is equal to zero.
Step 3:
Write the working equation.
Since we are looking for the position where there is zero net force for q3, the force
caused by q1 should be equal to the force caused by q2. Thus,
Further simplification will give us
Recalling about the quadratic equation will also be handy for this problem:
Step 4:
Substitute the given values.
Simplifying the equation will give us:
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
Step 5:
Solve for the answer using the quadratic equation.
The charge q3 is located at x = 0.775 m.
6 Try It!
Three charges are along the x-axis. Charge q1 is at x = -2.00 cm, q2 = -3 nC and is
located at x = 4 cm and q3 is at the origin with a charge of 5 nC. Calculate for the
charge of q1 if the net force at q3 is zero.
How are we going to solve the net force on a charge
in a two-dimensional system?
Key Points
___________________________________________________________________________________________
●
Coulomb’s law quantifies the amount of force between two stationary charged
particles.
●
The electrostatic force is directly dependent on the amount of charge of the
interacting particles.
●
There is an inverse-square relationship between the electrostatic force and the
distance separation of the charges.
●
In getting the net force on a point charge in a system of charges, superposition of
forces should be applied.
___________________________________________________________________________________________
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
Key Formula
___________________________________________________________________________________________
Concept
Formula
Coulomb’s Law
where:
● Fe is the electrostatic force
●
●
●
is the electric
constant which can also be
noted as k with a value of
8.988 x 109 N m2 C-2
q1 and q2 are charges
expressed in terms of
Coulomb (C).
r is the distance separation
of q1 and q2.
Description
Use this formula in solving
for the electrostatic force
between two stationary
charges. Note that this
shows an inverse-square
relationship between the
force and the distance
separation.
___________________________________________________________________________________________
Check Your Understanding
A. Fill in the blanks.
In order to quantify the forces between two charges, ________________ law is used. This
law states that the magnitude of the electric force between two point charges is
________________ proportional to the product of the ________________ and inversely
proportional to the ________________ of the distance between them. Thus, if you
________________ the distance separation, the electrostatic force will decrease by
one-fourth.
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
B. Write T if the statement is true. Otherwise, change the underlined
word(s) to make the statement correct.
____________________
1. There
is
an
inverse-square
relationship
between
electrostatic force and distance separation of forces.
____________________
2. Two positive charges will have an attractive force.
____________________
3. If two charges are along the x-axis, it is possible for a third
charge placed along their line of connection a net force
along the y-axis.
____________________
4. The direction of the electrostatic force is determined by
the charges of the particles.
____________________
5. Coulomb’s law assumes that charges which are interacting
are stationary.
C. Solve the following problems.
1. Three particles are along the x-axis.
q1 = q3 = 5 nC and q2 = –5 nC.
Calculate the:
a. force that q1 exerts to q2
b. force that q2 exerts to q3
c. force that q3 exerts to q1
d. net force (magnitude and direction) at q1
e. net force (magnitude and direction) at q2
f.
net force (magnitude and direction) at q3
2. How far should you separate two particles which have 5 nC and 16 nC in order to
have a force of 5.0 ✕ 10-4 N?
3. Two equal charges are interacting in a vacuum. What is the charge of either particle
when they exert 2 N when they are 5 km away?
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
4. Charges q1 and q2 are placed at (0, 3) and (3, 0). At which position should you place a
third charge (q3) in order for q2 to have a zero net force along the y-axis? Note that q1
= q2 = q3.
5. If q1 and q2 are of opposite charges, what would be the value of the electrostatic
force if the absolute value sign is removed?
Challenge Yourself
A. Briefly answer the question in two to three sentences only.
1. Three point charges q1, q2, and q3 are placed in the Cartesian plane at positions (5,
0), (0, 6) and (0, -6). Explain why q1 will have a net force which is purely along the
x-axis.
2. What can you say about the charge of two particles whose acceleration is increasing
each time? Assume that only electrostatic force exerts on them. Defend your
answer.
3. There are two charges in a vacuum. Is it possible for either two charges to be in
equilibrium? Why or why not?
B. Answer the following question.
4. Five charged particles are suspended from the ceiling using insulating strings.
Observations show that ball A attracts B and A repels C, ball D attracts B and D has
no effect on E, and a negatively charged rod attracts both A and E. Determine
whether the particle is charged negatively, positively or just neutral.
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
C. Four charges are placed at the corner of a square. Draw the free-body
diagram at q3 and determine at which quadrant would the net force be
directed to.
Bibliography
Freedman, Roger A. and Young, Hugh D. Sears and Zemansky's University Physics with Modern
Physics (13th ed). USA: Pearson Education, 2012.
Homer, David and Bowen-Jones, Michael. Physics Oxford IB Diploma Programme. UK: Oxford
University Press, 2014.
Hewitt, Paul G. Conceptual Physics (11th ed). New York: Pearson Education, 2010.
Sang, David, Graham, Jones, et.al. Cambridge International AS and A Level Physics Coursebook.
UK: University Printing House, 2014.
1.2 Coulomb’s Law
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Unit 1: Electric Charge and Coulomb’s Law
Young, Hugh D., Roger A. Freedman, and A. Lewis Ford. Sears and Zemansky’s University
Physics with Modern Physics (13th ed). USA: Pearson Education, 2012.
Key to Try It!
1. 259.2 N, it is attractive since the charges are opposite.
2. q1 = 2 ✕ 10-4 C and q2 = 4 ✕ 10-4 C
3. 127 m, the electrostatic force will be doubled.
4. F2 = 3.52 ✕ 109 N, directed downwards.
5. 2.58 x 10-6 towards the y-direction
6. -0.750 nC
1.2 Coulomb’s Law
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