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Physics is Life
Coulomb’s Law Review Guide
Charge Interactions Revisited
In the first few pages of Chapter 16 in your textbook, the interactions between charged objects of like
charge and opposite charge were discussed. At that time, the two fundamental charge interactions were stated:
oppositely charged objects attract and like charged objects repel. These mutual interactions resulted in an electric
force between the two charged objects. This force is commonly observed in physics lab activities and classroom
demonstrations. The next step will be to explore these charge interactions in more detail and begin to quantify it. We
will introduce Coulomb's law of electric force which is a mathematical expression of these charge interactions.
Ultimately, we will combine Coulomb's law of electric force with Newton's laws of motion to attempt to explain the
variety of charge interactions which we have observed throughout the unit.
Charge Interactions are Forces
It is possible that you might have watched two balloons repel each other a dozen or more times and never
even thought of the balloon interaction as being a force. Or perhaps you have used a plastic golf tube or other object
to raise small paper bits off the lab table and never thought of Newton's laws of motion. Perhaps even now you're
thinking "Why should I? That was the Newton's Laws unit and this is the Static Electricity unit." True. However, the
physical world which we study does not separate itself into separate topics as we teachers and students are prone to
do. Physics has an amazing way of fitting together in a seamless fashion. The information which you have forgotten
about from the Newton's laws unit has a mischievous way of creeping up on you in other units. That forgetfulness
(or negligence or mere ignorance) will haunt you as you try to learn new physics. The more physics which you learn
(as in really learn), the more that you come to recognize that the pieces of the physics puzzle fit together to form a
unified picture of the world of sight, sound, touch and feel. Here we will explore how Newton's laws of motion fit
together with the interaction of charged objects.
Suppose that you hold a charged plastic golf tube above a handful of paper bits at rest on the table. The
presence of the charged tube is likely to polarize a few bits of paper and then begin to exert an upward pull upon
them. The attraction between a charged tube and a polarized (yet neutral) paper bit is an electrical force - Felect. Like
all the forces studied so far, the electrical force is a push or pull exerted upon an object as a result of an interaction
with another object. The interaction is the result of electrical charges and thus it is called an electrical force.
Unlike many forces which we study, the electrical force is a non-contact force - it exists despite the fact
that the interacting objects are not in physical contact with each other. The two objects can act over a separation
distance and exert an influence upon each other. In this case, the plastic golf tube pulls upward upon the paper bit
and a paper bit pulls downward upon the golf tube. In this case, the force is significantly small. If you were holding
the golf tube, you would not likely sense the downward pull exerted upon it by the paper bit. On the other hand, the
force is often large enough to either balance or even overwhelm the downward pull of gravity (F grav) upon the paper
bit and cause it to be elevated or even accelerated off the table. Of course the actual result of the force upon the
paper bit is related to Newton's laws and a free-body analysis. If at any moment, the electrical force is greater in
magnitude than the gravitational force, the paper bit would be accelerated upward. And if at any moment, the
electrical force is equal in magnitude to the gravitational force, the paper bit will be suspended (or levitated) in
midair. The paper bit would be said to be at equilibrium.
Now consider the case of the rubber balloons hanging by light threads from the ceiling. If each balloon is
rubbed in the same manner (with animal fur), they each become negatively charged and exert a repulsive affect upon
each other. This charge interaction results in a force upon each balloon which is directed away from the balloon with
which it interacts. Once more, we can identify this repulsive affect as an electrical force. This electrical force joins
two other forces which act upon the balloon - the tension force and a force of gravity. Since the balloons are at rest,
the three forces must balance each other such that the net force is zero.
Both of these examples illustrate how the interaction between two charges results in a mutual force acting
upon the charged objects. An electrical interaction is a force which, like any force, can be analyzed using a freebody diagram and Newton's laws. But what factors affect the magnitude of this force? Is there an equation which
can be used to quantify it the same manner as was done for the force of gravity (F grav = m•g) and the force of friction
(Ffrict = mu•Fnorm)? The answer is Yes! Coulomb's law holds the key to understanding the answer to these questions.
It is the topic of Chapter 22.
Coulomb's Law
The interaction between charged objects is a non-contact force which acts over some distance of separation.
Charge, charge and distance. Every electrical interaction involves a force which highlights the importance of these
three variables. Whether it is a plastic golf tube attracting paper bits, two like-charged balloons repelling or a
charged Styrofoam plate interacting with electrons in a piece of aluminum, there is always two charges and a
distance between them as the three critical variables which influence the strength of the interaction
Force as a Vector Quantity
The electrical force, like all forces, is typically expressed in units of Newtons. Being a force, the strength of
the electrical interaction is a vector quantity which has both magnitude and direction. The direction of the electrical
force is dependent upon whether the charged objects are charged with like charge or opposite charge and upon their
spatial orientation. By knowing the type of charge on the two objects, the direction of the force on either one of them
can be determined with a little reasoning. In the diagram below, objects A and B have like charge causing them to
repel each other. Thus, the force on object A is directed leftward (away from B) and the force on object B is directed
rightward (away from A). On the other hand, objects C and D have opposite charge causing them to attract each
other. Thus, the force on object C is directed rightward (toward object D) and the force on object D is directed
leftward (toward object C). When it comes to the electrical force vector, perhaps the best way to determine the
direction of it is to apply the fundamental rules of charge interaction (opposites attract and likes repel) using a little
reasoning.
Electrical force also has a magnitude or strength. Like most types of forces, there are a variety of factors
which influence the magnitude of the electrical force. Two like-charged balloons will repel each other and the
strength of their repulsive force can be altered by changing three variables. First, the quantity of charge on one of the
balloons will affect the strength of the repulsive force. The more charged a balloon is, the greater the repulsive force.
Second, the quantity of charge on the second balloon will affect the strength of the repulsive force. Gently rub two
balloons with animal fur and they repel a little. Rub the two balloons vigorously to impart more charge to both of
them, and they repel a lot. Finally, the distance between the two balloons will have a significant and noticeable
affect upon the repulsive force. The electrical force is strongest when the balloons are closest together. Decreasing
the separation distance increases the force. The magnitude of the force and the distance between the two balloons is
said to be inversely related.
Coulomb's Law Equation
The quantitative expression for the affect of these three variables on electric force is known as Coulomb's law.
Coulomb's law states that the electrical force between two charged objects is directly proportional to the product of
the quantity of charge on the objects and inversely proportional to the square of the separation distance between the
two objects. In equation form, Coulomb's law can be stated as
where Q1 represents the quantity of charge on object 1 (in Coulombs), Q2 represents the quantity of charge on object
2 (in Coulombs), and d represents the distance of separation between the two objects (in meters). The symbol k is a
proportionality constant known as the Coulomb's law constant. The value of this constant is dependent upon the
medium that the charged objects are immersed in. In the case of air, the value is approximately 9.0 x 109 N • m2 / C2.
If the charged objects are present in water, the value of k can be reduced by as much as a factor of 80. It is
worthwhile to point out that the units on k are such that when substituted into the equation the units on charge
(Coulombs) and the units on distance (meters) will be canceled, leaving a Newton as the unit of force.
The Coulomb's law equation provides an accurate description of the force between two objects whenever
the objects act as point charges. A charged conducting sphere interacts with other charged objects as though all of
its charge were located at its center. While the charge is uniformly spread across the surface of the sphere, the center
of charge can be considered to be the center of the sphere. The sphere acts as a point charge with its excess charge
located at its center. Since Coulomb's law applies to point charges, the distance d in the equation is the distance
between the centers of charge for both objects (not the distance between their nearest surfaces).
The symbols Q1 and Q2 in the Coulomb's law equation represent the quantities of charge on the two interacting
objects. Since an object can be charged positively or negatively, these quantities are often expressed as "+" or "-"
values. The sign on the charge is simply representative of whether the object has an excess of electrons (a negatively
charged object) or a shortage of electrons (a positively charged object). It might be tempting to utilize the "+" and "" signs in the calculations of force. While the practice is not recommended, there is certainly no harm in doing so.
When using the "+" and "-" signs in the calculation of force, the result will be that a "-" value for force is a sign of
an attractive force and a "+" value for force signifies a repulsive force. Mathematically, the force value would be
found to be positive when Q1 and Q2 are of like charge - either both "+" or both "-". And the force value would be
found to be negative when Q1 and Q2 are of opposite charge - one is "+" and the other is "-". This is consistent with
the concept that oppositely charged objects have an attractive interaction and like charged objects have a repulsive
interaction. In the end, if you're thinking conceptually (and not merely mathematically), you would be very able to
determine the nature of the force - attractive or repulsive - without the use of "+" and "-" signs in the equation.
Calculations Using Coulomb's Law
In physics courses, Coulomb's law is often used as a type of algebraic recipe to solve physics word problems. Three
such examples are shown here.
Example A
Suppose that two point charges, each with a charge of +1.00 Coulomb are separated by a distance of 1.00 meter.
Determine the magnitude of the electrical force of repulsion between them.
This is not the most difficult mathematical problem which could be selected. It certainly was not chosen for its
mathematical rigor. The problem-solving strategy utilized here may seem unnecessary given the simplicity of the
given values. Nonetheless, the strategy will be used to illustrate its usefulness to any Coulomb's law problem.
The first step of the strategy is the identification and listing of known information in variable form. Here we know
the charges of the two objects (Q1 and Q2) and the separation distance between them (d). The next step of the
strategy involves the listing of the unknown (or desired) information in variable form. In this case, the problem
requests information about the force. So Felect is the unknown quantity. The results of the first two steps are shown in
the table below.
Given:
Q1 = 1.00 C
Q2 = 1.00 C
d = 1.00 m
Find:
Felect = ???
The next and final step of the strategy involves substituting known values into the Coulomb's law equation and using
proper algebraic steps to solve for the unknown information. This step is shown below.
Felect = k • Q1 • Q2 / d2
9
Felect = (9.0 x 10 N•m2/C2) • (1.00 C) • (1.00 C) / (1.00 m) 2
Felect = 9.0 x 109 N
The force of repulsion of two +1.00 Coulomb charges held 1.00 meter apart is 9 billion Newtons. This is an
incredibly large force which compares in magnitude to the weight of more than 2000 jetliners.
This problem was chosen primarily for its conceptual message. Objects simply do not acquire charges on the order
of 1.00 Coulomb. In fact, more likely Q values are on the order of 10-9 or possibly 10-6 Coulombs. For this reason, a
Greek prefix is often used in front of the Coulomb as a unit of charge. Charge is often expressed in units of
microCoulomb (µC) and nanoCoulomb (nC). If a problem states the charge in these units, it is advisable to first
convert to Coulombs prior to substitution into the Coulomb's law equation. The following unit equivalencies will
assist in such conversions.
1 Coulomb = 106 microCoulomb
1 Coulomb = 109 nanoCoulomb
1.
2.
3.
The problem-solving strategy used in Example A included three steps:
Identify and list known information in variable form.
List the unknown (or desired) information in variable form.
Substitute known values into the Coulomb's law equation and using proper algebraic steps to solve for the
unknown information. (In some cases and for some students, it might be easier to first do the algebra using
the variables and then perform the substitution as the last step.)
This same problem-solving strategy is demonstrated in Example B below.
Example B
Two balloons are charged with an identical quantity and type of charge: -6.25 nC. They are held apart at a separation
distance of 61.7 cm. Determine the magnitude of the electrical force of repulsion between them.
The problem states the value of Q1 and Q2. Since these values are expressed in units of nanoCoulombs (nC), the
conversion to Coulombs must be made. The problem also states the separation distance (d). Since distance is given
in units of centimeters (cm), the conversion to meters must also be made. These conversions are required since the
units of charge and distance in the Coulomb's constant are Coulombs and meters. The unknown quantity is the
electrical force (F). The results of the first two steps are shown in the table below.
Given:
Q1 = -6.25 nC = -6.25 x 10-9 C
Q2 = -6.25 nC = -6.25 x 10-9 C
d = 61.7 cm = 0.617 m
Find:
Felect = ???
The final step of the strategy involves substituting known values into the Coulomb's law equation and using proper
algebraic steps to solve for the unknown information. This substitution and algebra is shown below.
Felect = k • Q1 • Q2 / d2
9
2 2
Felect = (9.0 x 10 N•m /C ) • (6.25 x 10-9 C) • (6.25 x 10-9 C) / (0.617 m)2
Felect = 9.23 x 10-7 N
Note that the "-" sign was dropped from the Q1 and Q2 values prior to substitution into the Coulomb's law equation.
As mentioned above, the use of "+" and "-" signs in the equation would result in a positive force value if Q 1 and Q2
are like charged and a negative force value if Q1 and Q2 are oppositely charged. The resulting "+" and "-" signs on F
signifies whether the force is attractive (a "-" F value) or repulsive (a "+" F value).
Example C
Two balloons with charges of +3.37 µC and -8.21 µC attract each other with a force of 0.0626 Newtons. Determine
the separation distance between the two balloons.
The problem states the value of Q1 and Q2. Since these values are in units of microCoulombs (µC), the conversion to
Coulombs will be made. The problem also states the electrical force (F). The unknown quantity is the separation
distance (d). The results of the first two steps are shown in the table below.
Given:
Q1 = +3.37 µC = +3.37 x 10-6 C
Q2 = -8.21 µC = -8.21 x 10-6 C
Felect = -0.0626 N (use a - force value since it is repulsive)
Find:
d = ???
As mentioned above, the use of the "+" and "-" signs is optional. However, if they are used, then they have to be
used consistently for the Q values and the F values. Their use in the equation is illustrated in this problem.
The final step of the strategy involves substituting known values into the Coulomb's law equation and using proper
algebraic steps to solve for the unknown information. In this case, the algebra is done first and the substitution is
performed last. This algebra and substitution is shown below.
Felect = k • Q1 • Q2 / d2
d2 • Felect = k • Q1 • Q2
d2 = k • Q1 • Q2 / Felect
d = SQRT(k • Q1 • Q2) / Felect
d = [(9.0 x 109 N•m2/C2) • (-8.21 x 10-6 C) • (+3.37 x 10-6 C) / (-0.0626 N)]1/2
d = Sqrt [ +3.98 m2 ]
d = +1.99 m
Comparing Electrical and Gravitational Forces
Electrical force and gravitational force are the two non-contact forces discussed in The Physics Classroom tutorial.
Coulomb's law equation for electrical force bears a strong resemblance to Newton's equation for universal
gravitation.
The two equations have a very similar form. Both equations show an inverse square relationship between force and
separation distance. And both equations show that the force is proportional to the product of the quantity that causes
the force - charge in the case of electrical force and mass in the case of gravitational force. Yet there are some
striking differences between these two forces. First, a comparison of the proportionality constants - k versus G reveals that the Coulomb's law constant (k) is significantly greater than Newton's universal gravitation constant (G).
Subsequently a unit of charge will attract a unit of charge with significantly more force than a unit of mass will
attract a unit of mass. Second, gravitational forces are only attractive; electrical forces can be either attractive or
repulsive.
The inverse square relationship between force and distance which is woven into the equation is common to both
non-contact forces. This relationship highlights the importance of separation distance when it comes to the electrical
force between charged objects.