Mathematics of Physics and Engineering

(^164) Maxwell's Equations
divE =
divB =
curl E =
P_
£o
0;
= -
1
dB
dt
Gauss and Stokes from vector analysis.
In the International System of Units (SI), Maxwell's equations are
(3.3.2)
(3.3.3)
(3.3.4)

dE

cmlB = /j, 0 J + fJ.oSo-Kr- (3.3.5)

The positive numbers Mo = 47T • 10"^7 N/A^2 , s 0 = 8.85 • 10 -12 C/(N-m^2 ) are
called the magnetic permeability and electrical permittivity of the
free space, respectively; the relation

c^2 = — (3.3.6)

Mo£o

holds, where c is the speed of light in vacuum.

The starting point in the derivation of (3.3.2) is Coulomb's Law, discov-

ered experimentally in 1785 by the French physicist CHARLES AUGUSTIN

DE COULOMB (1736-1806). Originally stated as the inverse-square law for

the force between two charges, the result also provides the electric field E

produced by a point charge q: if q is located at the point O, then, for every

point P ^ O,

E{p)

= ^\oPJ

=

^^-

(3

3

7)

Let S be closed piece-wise smooth surface enclosing the point O. Then

E(P) is denned at all points P on S. According to (3.2.5), page 152, // E •

s

da = q/eo. By linearity, if there are finitely many point charges qi,-.-,qn

inside S, and E is the total electric field produced by these charges, then

f f E • dtr = (l/e 0 )f29k- (3-3.8)

s fc=1

Now assume that there is a continuum of charges in the domain G enclosed

by S, and a small region GB around a point B G G has the approximate

charge /j(B)m(Gs), where p is a continuous density function and m(Gg) is

the volume of GB ', the smaller the region GB , the better this approximation