14
DOC.
1
MANUSCRIPT ON SPECIAL RELATIVITY
E
=
J(pp
-
1/2Egrad2p)dT
also
represents
the
energy
of
the
system.
The latter
representation
has
a
property
that
makes
it
much easier
to construct
the variation that is
being sought. Namely,
if
one
varies
p
while the
values
for
p
and
e
are
kept
constant,
then the variation for
the
total
system
vanishes. For
we
have
Sȣ-/
psp
-
»•»•)
OX
ox
dr
=
f
pSp
+ +
•
+
=J[p
-
divi]S(pdr
=
0
Thus,
in order
to
find the variation of E that
corresponds
to
an
arbitrary
virtual
displacement,
one
need
only vary
p
and
E,
so
that
one
has
SE
=
j|p5p
-
^t2
8e\dr.
Because of the
indestructibility
of
the
conduction
electricity,
and because the latter
is
displaced together
with the
matter
in the static
problem
under
consideration,
we
have
dp
=
-
div
(p$),
where
3 denotes the
vector
of the
infinitely
small
spatial displacement
of the
system.
We determine the variation of
E on
the
assumption
that the dielectric
constant
e
of
a
material
particle
does
not
change during
the
displacement;
by making
this
assumption,
one
excludes the
phenomena
of
electrostriction from consideration. After
the
displacement
one
finds
at
the location of the radius
vector r
the material
particle
that
was
at
the location
r-s
before the
displacement. Therefore,
given
the indicated
assumption,
de
=
-3
grad
8,
so
that
we
have[25]
8E
=
J^~P
grad
(p3)
+
e2
grad
dr.
To
find the X-coordinate
fx
of
the
force
acting on
substance
per
unit
volume,
we
have
to
consider
only
that
part
6XE
of
öE
that
corresponds
to
the
component Sx
of the
elementary displacement.
After
having
transformed the first
term
of
the
integral by
means
of
integration by parts,
one gets[26]
ä,E
-
jt
C*P
+
lc2
2