# Full text of "Physics 12, Coulomb's Law"

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```Coulomb's Law

The torsion balance was invented by the Englishman John Mitchell in
the mid 1700's.

A torsion balance is a precision instrument which allows for the
measurement of a very small force. A wire is used to hang a lever
arm. The wire is uniform in diameter and composition.

An object at the end of the lever arm is what is experiencing the
force. The force experienced by the object causes the twisting of the
wire. As the wire twists it is possible to measure the angle through
which it twists.

The twisting force on a wire is related to the angle through which it
twists by a linear relationship, namely,

twist

= H6

where H is a constant of proportionality. By measuring the angle
through which the wire twists, you can measure the force that is being
experienced by the object.

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In some ways the relationship between force and angle for the torsion
balance is similar to the relationship between force and displacement
found in Hookes' Law F = kx, where x is the displacement. In the
case of the torsion balance, the angle 6 is the comparable to
displacement in Hookes' Law.

In 1785 Charles Coulomb borrowed the idea of the torsion balance
from the english to do an experiment to determine the relationship
between the electrostatic force on charged spheres, and the distance
between the spheres.

In his experiment Coulomb took two similarly charged spheres, one
on a fixed rod and a second on the torsion balance-lever arm.

By touching the two spheres, Coulomb was able to ensure the two
spheres were equally charged.

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Since like charges repel there is a force pushing the two spheres
apart from each other. It is this repulsive force between similarly
charged spheres which Coulomb measured using his torsion balance.

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By turning the knob at the top of the torsion arm he was able to
change the distance between the fixed charged spheres.

How the measurements were taken:

> An angular scale near the knob allowed Coulomb to
measure the angle through which the wire twisted.

> A scale near the spheres allowed Coulomb to measure the
distance between the spheres.

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Coulomb did two experiments: The first was to determine the
relationship between the force and the distance, and the second was
to determine how the charges on the two spheres affects the force.

Experiment 1 : The Relationship between Force and Distance

Consider a sample experiment with the following results (distance
and angle are in arbitrary measurements):

Distance

Angle (9)

1.6

0.39

1.4

0.51

1.2

0.69

1.0

1.0

0.8

1.6

0.6

2.8

0.4

6.3

Analysis:

1 ) Graph the data with the distance as the independent variable
and the angle as the dependent variable.

2) Use logarithms to find the functional dependence between
distance and angle.

3) If force is proportional to the angle, what is the relationship
between the force and the distance?

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Experiment 2: the Relationship between Charge and the Force

Coulomb was not interested in an absolute measure of the charge on
the spheres. By touching the two spheres to begin with, Coulomb
was able to ensure the two spheres were equally charged.

By grounding the sphere on the torsion arm from time to time he was
able to divide the charge several times over so that if he started with
a charge Q, on the first run he was comparing the force between two
spheres with charge Vz Q, and then two spheres with charge % Q,
and soon ....

Consider a sample experiment with the following results (charge and
angle are in arbitrary measurements):

Charge

Angle (9)

2.0

2.0

1.0

1.0

1/2

0.50

1/4

0.25

Analysis:

1 ) Graph the data with the charge as the independent variable and
the angle as the dependent variable.

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Coulomb's Law

In the his first experiment Coulomb determined that the electrostatic
force and the distance between the two charges are related by

Foe 1 /d 2

The electrostatic force relationship is an INVERSE SQUARE LAW .

In his second experiment Coulomb determined that electrostatic force
and the charge is related by

F <k Q! Q 2

where Qi is the charge on the first sphere and Q 2 is the charge on
the second sphere.

Combining these two

relationships

and using a constant

of

proportionality Coulomb arrived at his Law of Electrostatic Force

F = K Q 1 Q 2

/d 2

In MKS units K = 8.998 x

10 9 Nm 2 /C 2 ,

where the unit of charge (C)

is known as the Coulomb

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If two charges each of 1 C are one metre apart they would
experience a repulsive force of 8.998 x 10 9 N. A force of 9 billion
Newtons is an enormous force, equal to about 1 million tonnes.

Charges we typically experience are around a microcoulomb (1 uC =
10~ 6 C) and so we experience electrostatic forces on the order of 1
Newton or less.

The charge on an individual electron (e) is - 1 .602 x 10~ 19 C. The
charge of a proton is equal in magnitude but opposite in sign.

It is a yet to be explained fact that the charge of the electron and
proton are equal yet opposite.

1 C of charge is equal to 1 19 electrons. In one mole of matter there
are therefore on the order of 10 5 Coulomb of charge!

If the charge in matter were not neutral the matter would fly apart with
a considerable force.

For instance, when a uranium atom splits during nuclear fission (the
nucleus is no longer able to stick together) most of the energy that is
released from the nuclei is a result of the electrostatic repulsion of the
92 protons of positive charge so close together within the nuclei.

' PSCB/physics 12/2004

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```