4 DOC.
1
MANUSCRIPT ON SPECIAL RELATIVITY
electrical
quantities
that in unit time
pass through
surfaces of
magnitude
1
that
are
normal
to
the
coordinate
axes.
Thus,
with
an
appropriate
choice
of
the orientation
of
the surface
normal,[5] we
have the
equation
ff)d$
=
-f
indo
...(1)
Since, according
to Stokes's theorem
Jff
d$
=
J
(curl
ff)ndo,
and
since the above
equation
should be valid for
arbitrary curves
and thus also for
plane
curves
of
infinitesimally
small
dimensions,
there follows from it
i
curl
h
=
1/ci.
...(1a)
[p. 2]
But
equations (1)
and
(1a)
can
claim
general validity only
in the
case
where the
current
is
stationary.
For if
one
takes the
divergence on
both sides
of
(1a),
one
obtains div
i
=
0;
this
equation
cannot
be
generally
valid
since there
are
also
currents
that
are
not
closed. Maxwell
got
rid of this contradiction
by introducing
the
hypothesis
that,
besides the conduction
current
i,
"the electrical
displacement
current
e
also
participates
in the
production
of the
magnetic
field.[6]
The
equation
thus
completed
reads[7]
curl
f)
=
-(C
+
i).
...(1b)
Taking
the
divergence again
on
both
sides,
one
obtains
0
=
4^-(div
e)
+
div i.
at
The
following
should be noted about this
equation.
In the definition of
current
density
the law of the
conservation of the
quantity
of
electricity
was
already implicity
assumed. For the
quantity
of
electricity traversing a
cross
section cannot
be
measured
directly,
but
only
the
change
that the electrical
charge
on a
body undergoes
with time.
Implicitly,
we
set
this
equal
to
the
quantity
of
electricity
that flows from the
body,
or
flows
to
it;
i.e.,
in order
to
invest
our
definition of
i
with
physical meaning,
we
already
had
to
assume
the
indestructibility
of
the
electrical
quantities.
The last-derived
equation shows, therefore,
that div e is
nothing
other but the
density
p
of
the
electrical
charge.
Hence
we
can
set
div e
=
p
...(2)
It
can
also
easily
be shown that this
equation agrees
with the definition
of
the unit
of
electrical
quantity given
earlier.
Equation (1b)
is
exactly
correct-at
least
as
far
as our
current
knowledge
goes-as long
as
the material carriers of the
currents
and
charges
are
at rest.
But if
moving, electrically
charged
bodies
are present,
curl
h
is different from
zero even
if