Page 45 - Engineering Electromagnetics, 8th Edition

CHAPTER 2 Coulomb’s Law and Electric Field Intensity 27

Coulomb stated that the force between two very small objects separated in a vacuum
or free space by a distance, which is large compared to their size, is proportional to
the charge on each and inversely proportional to the square of the distance between
them, or
Q 1 Q 2
F = k
R 2
where Q 1 and Q 2 are the positive or negative quantities of charge, R is the separation,
1
and k is a proportionality constant. If the International System of Units (SI) is used,
Q is measured in coulombs (C), R is in meters (m), and the force should be newtons
(N). This will be achieved if the constant of proportionality k is written as
1
k =
4π 0
The new constant 0 is called the permittivity of free space and has magnitude, mea-
sured in farads per meter (F/m),

1
0 = 8.854 × 10 −12 ˙ = 10 −9 F/m (1)
36π
The quantity 0 is not dimensionless, for Coulomb’s law shows that it has the
label C /N · m .We will later define the farad and show that it has the dimensions
2
2
2
C /N · m; we have anticipated this definition by using the unit F/m in equation (1).
Coulomb’s law is now
Q 1 Q 2
F = (2)
4π 0 R 2
The coulomb is an extremely large unit of charge, for the smallest known quantity
of charge is that of the electron (negative) or proton (positive), given in SI units as
1.602 × 10 −19 C; hence a negative charge of one coulomb represents about 6 × 10 18
2
electrons. Coulomb’s law shows that the force between two charges of one coulomb
9
each, separated by one meter, is 9 × 10 N, or about one million tons. The electron
has a rest mass of 9.109 × 10 −31 kg and has a radius of the order of magnitude of
3.8 × 10 −15 m. This does not mean that the electron is spherical in shape, but merely
describes the size of the region in which a slowly moving electron has the greatest
probability of being found. All other known charged particles, including the proton,
have larger masses and larger radii, and occupy a probabilistic volume larger than
does the electron.
In order to write the vector form of (2), we need the additional fact (furnished
also by Colonel Coulomb) that the force acts along the line joining the two charges

1 The International System of Units (an mks system) is described in Appendix B. Abbreviations for the
units are given in Table B.1. Conversions to other systems of units are given in Table B.2, while the
prefixes designating powers of ten in SI appear in Table B.3.
2 The charge and mass of an electron and other physical constants are tabulated in Table C.4 of
Appendix C.