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