Volume 4: The Swiss Years: Writings 1912-1914 (English translation supplement) Page 18 (30 of 326)

18
DOC.
1
MANUSCRIPT
ON
SPECIAL RELATIVITY
[p. 14]
normal
n,
then,
according
to
the definition of
a,
the
product ano
is
equal
to
the
quantity
of
electricity traversing
o
per
unit time. If
we
succeed in
finding
this for
an
arbitrarily
oriented surface
element,
then
we
also have
along
with it the
expression
for
a.
Now
we inquire
about
the
quantity
of
electricity
that, per
unit
time,
traverses
that
particular
surface element
moving
with
the
matter
that
coincides with
o
at
the
beginning
of the unit of
time under
consideration,
but takes
up
the
position
o'
at
the time
t+dt.
The total
amount
of conduction
electricity passing through
this
element
is
inodt.
But the total
amount
of
polarization electricity passing through
the
comoving
element
is
equal
to
the
change
that the
expression
pno
undergoes during
the unit time
considered,
and thus
equal
to
pn'o'-pno.
This
change according
to
the calculation
rule
derived in the
appendix
()
is
equal to[31]
(Jl
+
q
div
p
-
curl
[Qp])n
odt,
where the index
at the bracket
means
that
one
is
to
take the
component along
the
normal of
o
of
the
vector
given
in the
bracket.
The
sum
of these
two terms is equal
to the total electrical
current
flowing through
the
comoving
surface element. The total
current
through
the
element
at rest
o
exceeds the former
by
the total
quantity
of
electricity
present
in the
space nno
that is
swept by
the
moving
element
o
in unit
time. It is
given by
div
eo.
Thus,
the total
current
passing through
the element
at rest
o
is
equal
to
i
+
p +
t
div
-
curl
[q,$J]
+
cj
div
C)no
or,
since div e
+
div
p
=
div
b
=
p,
it
is
equal
to
(i +
p
-
curl
[np] + np)no.
Thus,
the
expression
within the bracket
is
equal
to
the
vector
of
the total
current
that
was previously designated,
for the sake of
brevity, by
a.
Thus in
our
case,
the first
of
equations
(I)
assumes
the form
curl
h
=
-
(i
+ p
c
curl
[i\,p]
+
i
+
jp).
For the second
of
equations
(I) one
obtains
again
div
e
=
-div
p
+ p,
but
here,
of
course,
p
does
not
denote the total
density
of
electricity as
in
(I)
but
only
the
density
of the conduction
electricity (also
called "true
electricity").
The
third and the fourth
equations
read
analogously,
in accordance with the

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Extracted Text (may have errors)

18
DOC.
1
MANUSCRIPT
ON
SPECIAL RELATIVITY
[p. 14]
normal
n,
then,
according
to
the definition of
a,
the
product ano
is
equal
to
the
quantity
of
electricity traversing
o
per
unit time. If
we
succeed in
finding
this for
an
arbitrarily
oriented surface
element,
then
we
also have
along
with it the
expression
for
a.
Now
we inquire
about
the
quantity
of
electricity
that, per
unit
time,
traverses
that
particular
surface element
moving
with
the
matter
that
coincides with
o
at
the
beginning
of the unit of
time under
consideration,
but takes
up
the
position
o'
at
the time
t+dt.
The total
amount
of conduction
electricity passing through
this
element
is
inodt.
But the total
amount
of
polarization electricity passing through
the
comoving
element
is
equal
to
the
change
that the
expression
pno
undergoes during
the unit time
considered,
and thus
equal
to
pn'o'-pno.
This
change according
to
the calculation
rule
derived in the
appendix
()
is
equal to[31]
(Jl
+
q
div
p
-
curl
[Qp])n
odt,
where the index
at the bracket
means
that
one
is
to
take the
component along
the
normal of
o
of
the
vector
given
in the
bracket.
The
sum
of these
two terms is equal
to the total electrical
current
flowing through
the
comoving
surface element. The total
current
through
the
element
at rest
o
exceeds the former
by
the total
quantity
of
electricity
present
in the
space nno
that is
swept by
the
moving
element
o
in unit
time. It is
given by
div
eo.
Thus,
the total
current
passing through
the element
at rest
o
is
equal
to
i
+
p +
t
div
-
curl
[q,$J]
+
cj
div
C)no
or,
since div e
+
div
p
=
div
b
=
p,
it
is
equal
to
(i +
p
-
curl
[np] + np)no.
Thus,
the
expression
within the bracket
is
equal
to
the
vector
of
the total
current
that
was previously designated,
for the sake of
brevity, by
a.
Thus in
our
case,
the first
of
equations
(I)
assumes
the form
curl
h
=
-
(i
+ p
c
curl
[i\,p]
+
i
+
jp).
For the second
of
equations
(I) one
obtains
again
div
e
=
-div
p
+ p,
but
here,
of
course,
p
does
not
denote the total
density
of
electricity as
in
(I)
but
only
the
density
of the conduction
electricity (also
called "true
electricity").
The
third and the fourth
equations
read
analogously,
in accordance with the

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DOC.
1
MANUSCRIPT ON SPECIAL RELATIVITY
19
principle
of
duality.
One
has
only
to
bear in mind that there is
no
magnetic analog
to
the electrical conduction
current.
Accordingly,
the field
equations
read
c
curl
1) -
C
=
(ji
-
curl
[q,p]
+ (i +
Hp)
div
e
=
-
div
p
+
p
c
curl
C
+
1)
=
-
(ttt
-
curl
[Hw])
div
f)
=
~
div
m
(Ib)
The form
of
these
equations
tempts one
to introduce
the
vectors h +
-
e
+
p,
c
1
[nm], h + m
as
field
vectors
and,
in this
way,
to
simplify
the
equations
considerably.
In this
way
one
obtains Minkowski's form of the field
equations,
which
agrees
with the form of the field
equations
for bodies
at
rest,
up
to
the
term
np.[32]
But
we
do
not
do
this,
for the
vectors
that
are
to
be introduced in this
way possess
no
simple physical
meaning.
What
one
gains
in the
simplicity
of the fundamental
equations through
their
introduction,
one
loses
again through
the fact that the
equations
that
give
the
connection between the
states
of the bodies and the field
vectors
become
more complicated. Taking
into
account
that the force
acting
on a
moving
unit of
electricity according
to
§2
is
equal
to e
+
1/c
[n,h] ("electromotive
force")
and that the force
acting on
the
moving magnetic charge
unit[33]
is
equal
to
h
-
1/c
[ne] ("magnetomotive
force"),[34] one
obtains, analogously
to
equations (8'),
p
=
(e
-
1)
(c
+
I
ft,!)])
c
m
=
(|i
-
1)
(1)
-
-
ft,C])
c
i
=
A(C
+
1
ft,ft)
c
(12)
Considered from
the
standpoint
of
the
relativity theory,
which will be
developed later,
[p.
15]
these
equations
can
hold
only
to
a
first
degree
of
approximation, i.e., only
if
one
neglects
those
terms
in the
equations
that
are multiplied by
n/c
to
the second
or
even
higher
power.[35]
Since such
terms
possess practical meaning only
in
a
very
few
cases,
because the velocities
occurring
in
practice
are
almost
always very
small in
comparison
with
c,
one
can
base one's
calculations
on
these Lorentz
equations
in
almost all of the
cases
that
correspond
to
observable
phenomena.
From them and
equations
(12),
for
example,
one can
easily provide
the
theory
for the
previously

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DOC.
1
MANUSCRIPT
ON SPECIAL RELATIVITY
17
force the
physically
intuitive
formula[29]
f
=
ep
+1
[i,
ffl
-
Igrad
(C
p)
+
(pv) c
+
I
[M]
c
2
c
-
i-grad
(J),
m)
+
(mVj
1)
-
-
[m,c]
2
c
(11a)
1
In this formula
only
the
terms
-1/2
grad (ep)
and
-grad(hm)
were
physically
opaque. They
owe
their
appearance
to
the
assumption,
which
we
introduced
above,
that
during
an
infinitesimally
small
displacement
of the
matter
the dielectric
constant
of
the material
particle always
remains
unchanged,
even
in the
case
where the
particle
changes
its volume in
the
course
of the
displacement.
Thus,
these
terms
correspond
[p.
13]
to
a
physically unjustified
assumption.
But
they are
also of subordinate interest
because
they
are
not
capable
of
producing
motion in
incompressible
substances,
but
only
changes
in
pressure.
§4.
Lorentz's
Equations
for
(Slowly)
Moving
Media
As in the
case
of bodies
at
rest,
we again
start out
from the
fundamental
equations
(I)
in
§1,
which, however,
for
reasons
analyzed
in
§2, we
must
modify by assuming
the
presence
of several
continua
of
electrical
density,
which
are
to
be viewed
partly
as
carriers of
polarization
and
partly
as
carriers of the electrical conduction
currents
and
corresponding charges.
In
addition,
the third and the fourth of
equations (I)
are
to
be
supplemented by adding
terms
corresponding
to
the
magnetic polarization
currents. But because of the
duality
of electric and
magnetic processes,
it suffices
to
set
up
the
first two
equations
for
our case.
They
read
curl
h=
=
1/
(C
+
£
(ijgpg
+
£
H/P/)
c
div
If,
for the sake of
brevity,
we
denote
the
sum
of
the
second and the third
term
within
the bracket
on
the left-hand side
by
a,
then
a is
the
vector
of the total
electrical
current
with
respect
to
an
observer
at rest.
Our task
is to
express
a
by
means
of the
vectors
n,
i
(conduction current
vector)
and
p
(polarization vector),
the last
two
of
which
can
be
sharply
defined
only
for bodies
at rest,
that
is,
in
our
case,
for
an
observer
moving
with the material element under consideration.
If
one
chooses
an
infinitesimally small, plane
surface element
o
at rest,
with
a

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