Maxwells Equations

Maxwells Equations

Between 1785 and 1787, the French physicist Charles Augustine de
Coulomb performed a series of experiments involving electric charges,
and eventually established what is nowadays known as Coulomb’s law.
According to this law, the force acting between two static electric charges
is central, inverse-square, and proportional to the product of the charges.
Two like charges repel one another, whereas two unlike charges attract.
Suppose that two charges, q1 and q2, are located at position vectors r1
and r2, respectively. The electrical force acting on the second charge is
written

in vector notation—see Figure 3.1. An equal and opposite force acts on
the first charge, in accordance with Newton’s third law of motion. The SI
unit of electric charge is the coulomb (C). The magnitude of the charge
on an electron is 1.6022 × 10?19 C. Finally, the universal constant 0 is
called the permittivity of free space, and takes the value
0 = 8.8542 × 10?12 C2 N?1m?2. (3.2)
Suppose that two masses, m1 and m2, are located at position vectors
r1 and r2, respectively. According to Newton’s law of gravity, the
gravitational force acting on the second mass is written

50 MAXWELL’S EQUATIONS AND THE PRINCIPLES OF ELECTROMAGNETISM

in vector notation. The gravitational constant G takes the value
G = 6.6726 × 10?11 Nm2 kg?2. (3.4)
Note that Coulomb’s law has the same mathematical form as Newton’s
law of gravity. In particular, they are both inverse-square force laws: i.e.,

However, these laws differ in two crucial respects. Firstly, the force due
to gravity is always attractive (there is no such thing as a negative mass).
Secondly, the magnitudes of the two forces are vastly different. Consider
the ratio of the electrical and gravitational forces acting on two particles.
This ratio is a constant, independent of the relative positions of the
particles, and is given by
|felectrical|
|fgravitational|

For electrons, the charge-to-mass ratio is |q|/m = 1.759 × 1011 Ckg?1, so
|felectrical|
|fgravitational|
= 4.17 × 1042.