8
DOC.
1
MANUSCRIPT ON SPECIAL RELATIVITY
and
ffdx =--^{|J-SJT}
...(6b)
It follows from
(5b)
that
the
work
Jafdr
that
is
performed by
the field
per
unit time
is
always
associated with
an
equally large
decrease of
the
quantity
jwdr
of
the field;
hence
we
have
to
view the latter
as
the
energy
of
the total
field,
and
we
remain in
accord with the
energy
law if
we
view
w as
the
density
of the
energy
of the
electromagnetic
field. It
follows from
(6b)
that the volume
integral
of the
ponderomotive
force exerted
by
the total field
is
equal
to
the decrease of the
vector
integral
J1/c2sdr;
hence,
we
have
to
view the latter
as
the
expression
for the
momentum
of the total
field,
and
we
remain in
accord
with the
momentum
law
if
we
view
s
as
the
(vectorial) momentum
density
of the field.
From
the formal
point
of
view,
we
note
that the momentum and
energy
laws
are
satisfied in Lorentz's
theory
due
to
the
existence of
equations
of the
following
form
for the
ponderomotive
force
f per
unit volume:
dP.xx
dPxv dPxz
1
3sr
r
dx dv dz
c2
dt
dsx
dsy
ds.
dw
^
dx
dv
dz
dt
(7)
It should be
emphasized
that
H. A.
Lorentz's
expression
for the force
density f
is
one
of the
empirically
best
supported
results in
electrodynamics.
It
yields
immediately
the
electromotive forces that
are
induced in conductors
moving
in
stationary magnetic
fields,[16]
the
simplest
case
of
the Zeemann
effect,[17]
the
way
magnetic
fields influence the cathode
rays.[18]
But
we
will
not
get
into
this here.
§3. Completion
of Lorentz's
Theory for
the
Case
Where
Electrically
and
Magnetically
[p.
6]
Polarizable Media Are
Present
(Bodies
at
Rest)
The
brilliant idea
through
which
H. A.
Lorentz advanced
electrodynamics
and
optics
in such
an
extraordinary
way
can
be
expressed
as
follows.
Every
influence of
matter
on
the
electromagnetic
field
and,
conversely, every
influence of the
electromagnetic
field
on
matter is
based
on
the fact that
matter
contains movable
electrical
masses
that interact with
the
electromagnetic
field
according
to
equations
(I).
In this