Verification of Coulomb’s Law using a Coulomb Balance
Raghuveer Dodda, Physics 425
Coulomb’s Law states that the force acting on a charge due to another charge is directly
proportional to the product of the these charges and to the inverse of the square of the
distance separating these charges. There is also a constant of proportionality called the
Coulomb’s constant. The Coulomb Balance is an apparatus that allows the experimenter
to alter the charge present on two conducting spheres that behave as point charges. The
experimenter can also change the separation between them. This paper presents the
results of an experiment to verify Coulomb’s law using this apparatus.
INTRODUCTION
Coulomb’s Law is one of the basic
experimental laws of physics. Though it
describes the interaction between two
point charges, it is extremely powerful in
its application because one can
understand all charges as collections of
point charges. It is therefore important to
verify that the force between two
charges is proportional to each of the
individual charges, and that the force
obeys the inverse square law for
distance. The constant of proportionality,
called the Coulomb constant, has the
value 8.9876 x 109 N m2 C-2.
MATERIALS AND METHODS
A Coulomb Balance consists of a torsion
balance that has a conducting sphere
attached to its wire. The sphere is
perpendicular to the axis of the wire and
a small shaft supports it. This sphere is
free to move about the wire, and it twists
the wire when it moves. The angle
through which the wire twists is
proportional to the force the wire exerts
on the sphere (i.e., on the shaft and
hence on the sphere).
The other sphere moves freely along a
straight path, which has a scale attached
to it. The scale shows the distance
between the two spheres at any given
time. A power supply enables the
experimenter to impart a desired
potential to the sphere, and hence a
desired charge (because the capacitance
of the sphere is only a property of the
sphere and is the proportionality
constant between the potential and the
charge, it will suffice to use the potential
instead of the charge for this
experiment).
The experiment had the following parts:
1. Verifying the dependence of
Theta on R where Theta is the
angle through which the torsion
wire rotates and the R is distance
of separation between the two
charged spheres.
2. Verifying the dependence of
Theta on Q where Q is the charge
on a sphere (which is
proportional to V, the potential of
the sphere).
3. Calculating
the
Coulomb’s
constant
The charges on the two spheres were
fixed, and the distance between the
spheres was changed to observe how
Theta changes with R. To see how Theta
depended on Q, the distance between the
charges was kept constant and the charge
on the spheres was altered (in our
experiment, only the charge on one
sphere was changed). The torsion
constant of the wire was calculated in
the final part of the experiment so that
the actual force acting on the spheres
could be calculated; this allowed for the
calculation of the Coulomb constant
because all the other parameters were
known.
then the inverse square dependence of R
stands verified. The next two columns in
Table1 – Part A contain LN [THETA]
and LN[R]. A linear regression yields
the following value :
n = (-2.36800 ± 0.212661)
Hence, inverse square dependence is
verified.
Part B
RESULTS AND DISCUSSION
Table 1 contains the data for all the parts
of the experiment.
Part A
The spheres were maintained at a fixed
potential, and torsion was observed in
the wire for varying separation between
the spheres. The results are tabulated in
Table1 – Part A in the two columns
labeled THETA and R. The third column
B is the correction factor that is
necessary to validate the assumption that
the spheres are point charges (The closer
the charges are to one another, the more
they affect the distribution of charge on
each other’s surfaces. The correction
factor considers the fact that these
objects are spheres and not just point
charges per se).
Now, assume that:
THETA = b R^n , where b and n are
constants. Then,
LN[ THETA] = n LN[R] + LN[b]
This is the equation of a straight line in
the form y = nx + c where x = LN[R]
and y = LN[THETA]. If we are able to
show that the slope of this line n is -2,
The spheres were separated by the
distance of 0.1 m, and one of the spheres
was charged to 6000 V. The other sphere
was successively given different
potentials varying between 6000V and
2000V. The Table1 – Part B contains
two columns labeled THETA and V2
where V is the potential that is being
changed on one sphere. The correction
factor B serves the same purpose as
before.
Now, assume that
THETA = b V^ n , where b and n are
constants.Then,
LN[ THETA] = n LN[V] + LN[b]
This is the equation of a straight line in
the form y = nx + c where x = LN[V]
and y = LN[THETA]. If we are able to
show that the slope of this line n = 1,
then we will have proven that THETA is
proportional to the V and hence to Q, the
charge on the sphere. The next two
columns in Table1 – Part B contain LN
[THETA] and LN[V]. A linear
regression yields the following value :
n = 1.41378 ± 0.154996
The error 0.154996 is the standard
deviation of the error, which means that
there is about 68% chance that the value
on n lies in that interval. If we allow the
error to be up to three standard
deviations, i.e., ± 0.464898, then we
have a 99% chance that the value lies
within this interval. So,
n = 1.41378 ± 0.464898
contains the value of n = 1, which is the
value Coulomb’s law expects for n.
Part C
This part focuses on finding coulomb’s
constant. This is very straight forward
because the data required to compute
this value is already present in Part B.
The following formula is used :
F = k Q1 Q2 / R2 where Q1 = CV1 ,
Q1 = CV1 and C = a / k for a conducting
sphere.
Therefore, k = V1 V2 a2 / ( R2 F).
We know all the parameters in the above
equation except F where F = K x
THETA. Once the torsion constant, K, is
known, the Coulomb constant, k, is also
known.
Table 1 – Part C contains a table to
determine K. Since K = F / THETA, the
value of K is found for three different
sets of Forces and torsion displacements.
The average value of K is 1.0975E-06 ±
1.35982E-09 Newtons/degree.
The average value of k is, then,
1.60632E+11 ± 1.35982E-09 N m2 C-2.
This value is at least two orders of
magnitude larger than the actual value.
As explained in Table1 – Part C, the
source of this error is unkown.