Coulomb's law
Experiments on electrically charged bodies yield the following observations:
1. Like charges repel and opposite charges attract each other.
2. The force between charges is:
(a) inversely proportional to the square of the distance between them;
(b) dependent on the medium in which they are embedded;
(c) acts along the line joining the charges;
(d) is proportional to the product of the charge magnitudes.
These facts are summarized in Coulomb's law of force which gives the
force, F, on a charge q2 due to a charge q1 like sign as
v
qq )
F = 1 22 a newtons…………………………(1D)
4πεr
Where q1 and q 2 are the charges in coulombs, r is the distance between
them in meters, and ε is called the permittivity of the medium in which they are
)
embedded. The vector a is a unit vector from q 2 , pointing away from q1 in the
direction of the line joining the charges and reflects the fact hat when the charges
have the same sign the force is one of repulsion. The units of permittivity may be
deduce from eq.(1D) thus:
(coulombs ) 2
ε=
newtons (meters ) 2
The properties of a material as a dielectric enter into Coulomb's law only
through this permittivity, which is also a measure of its ability to store charge.
This follows from Coulomb's law of capacitance which states that the capacitance
of a body to store charge is defined by the equation
Q = CV …………………………………………..(2D)
Where + Q is the charge on one surface of the body and − Q is the charge
on the opposite surface, V is the potential drop between the surfaces, and C is the
capacitance of the body. The dimensions of C are given by eq.(2D) as
coulombs
Capaci tan ce =
volts
And the unit of the capacitance is called the farad. The capacitance of a
body is found experimentally to be an intrinsic property of the material which
forms the body and of its geometry, that is, its physical shape. It is most
conveniently defined in terms of a parallel plate capacitor. If we have two metal
electrodes, each of area a square meters, separated by a distances, d meters and
parallel to each other, filled with a material of permittivity, ε , (Fig.@) the
capacitance of the system is given by
εa
C=
………………………………………………(3D)
d
Fig @ : Parallel –plate capacitor .
It can be shown that the potential energy stored by a capacitor is given by
1 Q2 1
E=
= CV 2 joules …………………………..(4D)
2 C
2
When it has Q coulombs of charge stored. From this we see that
(coulombs ) 2
joules = newtonmeters =
farads
Combining this with eq.(1D) shows that the dimensions of permittivity
reduce to farads per meter. When the medium in the capacitor is empty space, the
1
× 10 −9 farads per meter.
permittivity is written as ε o and has the value
36π
The permittivity, ε , of a dielectric material may be related to the
permittivity of vacuum by writing it as
ε = ε r ε o …………………………………………..(5D)
Where ε r is the relative permittivity and is simply a number. It has exactly
the same value as the dielectric constant, k, defined in the CGS system of units.
Values of relative permittivity for a range of a materials are given in table 1D. it
should be noted that dry air has a value approximately equal to one, the relative
permittivity of vacuum.
We may now return to the question of the potential of a friction – charged
surface. We may take the surface as one plate of a capacitor with the other being
the nearest earthed surface. That is, the nearest surface at zero potential.
Let us take the example of the three–inch spark which may happen. The
cross-sectional area of the arc is small, say one square millimeter, and so only that
tiny area of the nylon surface is discharged by the spark. We can calculate the
charge stored on that area, knowing that the potential across the three-inch
capacitor is 400,000 volts, and from the capacity, C, which is given by
ε a
C= O
d
Where a = 1 mm2 and d = 0.076 m. Thus
C = 10 −9 × 10 −6 / 36π × 0.076 = 1.2 × 10 −16 farads .
The charge stored is then
Q = CV = 1.2 × 10 −16 × 4 × 10 5 = 4.6 × 10 −11 coulombs
This charge would correspond to Q / e single electronic charges, that is to
(4.6×10-11)/(1.6×10-19)=2.9×108 electronic charge in an area of one square
millimeter. This is an electron charge density of 2.9×1010 per square centimeter.
Now, in a Nylon fabric there are about 1015 dipolar side-chains per square
centimeter of surface so that this rough calculation suggest that on average one
side- chain in 105 acquires a single electronic charge as a result of friction.