Coulomb’s Law
1
Object
To determine the power law of the force between two “point charges” and the charge on each of the two
“point charges.”
2
Apparatus
Chamber with mirror, ruler, and glass window set on a rigid base with grid and guides for blocks and
topped with either a hardboard top or a plastic top; vinyl strip; felt square; two graphite coated spheres
mounted on guide blocks, one graphite coated sphere hanging from monofilament wire inside the chamber.
3
Theory
The electrical interaction between two charged particles is described in terms of the forces exerted between them. Augustin de Coulomb conducted the first quantitative investigation of these forces in 1784.
Coulomb used a very sensitive torsion balance to measure the forces between two “point charges,” that
is, charged bodies whose dimensions are small compared to the distance between them. Coulomb found
that the force grows weaker as the distance between the charges increases, and that it also depends on the
amount of charge on each body. More specifically, Coulomb’s force law states that:
The force of attraction or repulsion between two point charges is directly proportional to the product of the charges
and inversely proportional to the square of the distance between them.
The direction of the force on each particle is always along the line joining the two particles; pulling them
together when the two charges are opposite, and pushing them apart when the charges are the same.
The force on the “point charges” is measured in this experiment by balancing their electrostatic repulsion
against the force of gravity. By suspending a small charged sphere with an insulating thread, the electrostatic force can be found by measuring the deflection of the suspended sphere from vertical as a second
charged sphere is brought near. Thus the electric force can be determined from the sphere’s weight and
deflection.
The diagram in figure 1 shows a force diagram for the hanging sphere in equilibrium. When the sphere
is in
equilibrium, the sum of the tension T~ , gravitational force F~g , and electrostatic force F~E is zero. That
X
is,
F~ = T~ + F~g + F~E = 0. If we assume the electrostatic force is in the horizontal direction (x) with
no vertical (y) component (this is not strictly true, but it’s a reasonable approximation for small angular
displacements of the pendulum), then the component equations become
X
X
Fx ≈ −T sin θ + FE = 0 and
Fy ≈ +T cos θ − mg = 0
(1)
where m is the mass of the hanging sphere, g is the acceleration due to gravity, and θ is the angular
Figure 1: Illustration of the apparatus and the force diagram of the hanging mass.
displacement of the pendulum from vertical, as shown in figure 1. From these we can get
FE ≈ mg tan θ = mg
d
L
(2)
where d is the horizontal displacement of the pendulum and L is the length of the pendulum. Therefore,
if we know g and we measure m, L, and d, we can compute the electrostatic force FE .
We assume the force law takes the form FE = Crn , where we know C = kq1 q2 and n is the power we
wish to measure. From this, we can figure out how to plot our data so we can use linear regression to
determine the power n. We take the natural logarithm of both sides of the equation
ln (FE ) = ln (Crn ) = ln (C) + ln (rn ) = ln (C) + n ln (r)
(3)
and compare the result ln (FE ) = ln (C)+n ln (r) to the slope-intercept form of a straight line (y = b+mx).
We find that the power we are looking for is the slope of the graph of ln (FE ) versus ln (r). Further, since
we know the constant is C = kq1 q2 and k = 8.99 × 109 Nm2/C2 , we can also figure out what the charges on
our spheres are (assuming they have same the same charge or that one sphere has a known fraction of the
charge on the other sphere).
4
Procedure
In this experiment, we will use an average mass value of m = 0.066 g = 6.6×10−5 kg and measure both the
length L and the initial horizontal position of the hanging mass x0 once. We will then vary the position of
the the sliding mass xs while measuring the corresponding position of the hanging mass xh . From these
measurements we can determine the electrostatic force FE and the separation of the charges r to get an
experimental value for the power of the force law and to estimate the charge induced on the graphite
coated spheres.
1. If it has not already been done, lower the graphite coated sphere that is attached to the monofilament
wire into the center of the chamber. Pull the monofilament through the precut slits at the top of the
chamber. These slits are centered on the top edge of the front and back faces of the chamber. Once
the monofilament has been pulled into the slit, the height and position of the sphere can be adjusted
by pulling on the free ends of the monofilament. The suspended sphere’s final position should be
the same height as the sphere mounted on the guide block and should be centered (front to back) in
the chamber. Cover the top of the chamber with either the clear or hardboard top to help eliminate
the effects of air currents and breezes.
2. Record the initial position of the hanging sphere x0 . In practice, this is best done by recording the
position of either the left or right edge of the sphere (be sure that you consistently measure the
same edge – left or right – of both spheres). A mirror is fastened to the back surface of the chamber
to help eliminate measurement errors due to parallax. When making measurements, always make
sure the sphere’s image in the mirror is completely covered by the sphere itself. This ensures that
your measurements will be consistent.
3. Begin the experiment by inductively charging the sphere fastened to the guide block. To do this,
follow these steps:
(a) Vigourously rub a strip of vinyl (white) with a bit of felt (black) to charge it.
(b) Hold the charged vinyl strip close to (but NOT touching!) the graphite coated sphere that is
attached to the sliding block.
(c) Place a finger from your other hand onto the sphere, thereby grounding it to allow electrons to
flow.
(d) Remove your finger.
(e) Remove the charged vinyl strip
The sphere on your guide block should now be charged.
4. From now on work quickly but carefully. If the air is humid, the charged placed on the coated
spheres will eventually “leak off.” If you touch the charged sphere with anything at this point, it
will immediately discharge and you will have to charge it inductively again.
5. Insert the charged sphere into the chamber through one of the holes in the base. Gently slide the
charged sphere up to the suspended sphere and bring them into contact. When they touch, the
charge will be equally distributed between the two spheres and each sphere will have the same
amount of charge. Note: It may be necessary to charge the two spheres several times initially to
get a sufficient repulsive force. To do this, repeatedly charge the sphere on the guide block by
induction, and then bring it into contact with the suspended sphere. The more charge you put on
the suspended sphere, the farther it will be displaced. It may take a little practice to get the right
amount of charge for your situation.
6. For a given “charged situation,” change the distance between the spheres r and measure the corresponding displacement of the hanging sphere d. In practice, you’ll want to set the position of the
guided sphere and measure the corresponding position of the hanging sphere. Later, you’ll calculate
r and d from these measurements and the initial position of the hanging sphere x0 .
7. Repeat parts 3 through 6 twice so you have three sets of data.
5
Calculations
1. For each set of position measurements, compute the separation of the spheres r by taking the absolute value of the difference between their positions.
2. For each measurement, compute the displacement of the hanging sphere d by taking the absolute
value of the difference between the position of the hanging sphere and its initial position x0 .
3. For each measurement, compute the electrostatic force FE on the hanging sphere using equation 2 .
4. For each data set, plot ln(FE ) versus ln(r).
5. From the slope and intercept of each graph, determine the power of the force law and the magnitude
of the charge on your spheres (recall k = 8.99 × 109 N·m2/C2 ). Compare the power of the force law to
the theoretical value in your summary of results.
6
Questions
1. Why would the charge “leak off” your spheres more quickly in humid weather than in dry?
2. When you initially brought the charged guided sphere into contact with the uncharged hanging
sphere, what happened just before the spheres touched? Did they attract or repel? What is a possible
explanation for this behavior?
3. After the spheres touched, they repelled. Why did they repel and not attract?
4. Do your measured value of the power of the force law agree with the theoretical value? If not, what
are some possible reasons why?
5. What is the charge on your spheres? Does this seem like a reasonable number to you? Explain.
6. Based on the measurements you’ve made, can you determine the signs of the charge on your
spheres?
7. Based on the magnitude of your charge, how many extra electrons were placed on each of the
spheres (or by how many electrons are each of your spheres deficient)?
Coulomb’s Law Data Sheet
m (g)
L (cm)
Data Set 1
x0 (cm)
Data Set 2
Data Set 3
Sliding
Hanging
Sliding
Hanging
Sliding
Hanging
xs (cm)
xh (cm)
xs (cm)
xh (cm)
xs (cm)
xh (cm)