Mathematics of Physics and Engineering

Equations in Vacuum 165

of the charge. Then (3.3.8) becomes

S G

and equation (3.3.2) follows from Gauss's Theorem (see Theorem 3.2.2).

EXERCISE 3.3.l.c Verify that (3.3.9) indeed implies (3.3.2).
Equation (3.3.9) is important in its own right and is known as Gauss's
Law of Electric Flux.

Equation (3.3.3) is the analog of (3.3.2) for the magnetic field and re-
flects the experimental fact that there are no single magnetic charges, or
monopoles.

The starting point in the derivation of (3.3.4) is Faraday' s Law, discov-
ered experimentally in 1831 by the English physicist and chemist MICHAEL
FARADAY (1791-1867): a change in time of a magnetic field B induces
an electric field E. This law is the underlying principle of electricity gen-
eration: the electric field E produces an electric current in a conductor.
Mathematically, if S is a piece-wise smooth orientable surface with a closed
piece-wise smooth boundary dS so that the orientations of S and dS agree,
then

/„•—//£ •"• <3-3io>
s
Equation (3.3.4) now follows from Stake's Theorem (Theorem 3.2.3). Note
that §QS E • dr = V is the voltage drop along dS. For a coil without a
ferromagnetic core, electric current I through the coil produces magnetic
field B whose flux is proportional to /: — JJ B • da = LI, where L is
s
the inductance of the coil. Then (3.3.10) becomes V = Ldl/dt, which is a
fundamental relation in the theory of electrical circuits.

EXERCISE 3.3.2.c Verify that (3.3.10) indeed implies (3.3.4).
The starting point in the derivation of (3.3.5) is an experimental fact
that a steady electric current produces a magnetic field. In 1820, the Danish
physicist HANS CHRISTIAN 0RSTED (1777-1851) observed that a steady
electric current deflects a compass needle. The French scientist ANDRE-
MARIE AMPERE (1775-1836) learned about this discovery on September
11, 1820; one week later, he presented a paper to the French Academy with