340

PONDEROMOTIVE FORCES

§1.

Forces that

do

not

depend

on

the

velocities

of elementary

particles

In

this derivation

we

will

consistently

base ourselves

on

the

standpoint

[4]

of

the electron

theory1;

hence

we

put

(2)

®

=

5

+

p,

(3)

= S)

+

0,

where

?ß

denotes

the electric

and

0 the

magnetic

polarization

vector.

We

think

of

electric

and

magnetic

polarizations,

respectively,

as

consisting of

spatial

displacements

of

electric

and

magnetic mass

particles

of dipoles that

are

bound to equilibrium

positions.

In addition,

we

also

assume

the

presence

of mobile

electric

particles

not

bound

to dipoles (conduction

electrons).

Let

Maxwell's

equations

for

empty

space be

valid

in

the

space

between

the

above

particles,

and let,

as

in Lorentz,

the

interactions between matter and

elec-

tromagnetic

field

be

exclusively

brought

about

by

these particles.

Accord-

ingly,

we assume

that the forces exerted

by

the

electromagnetic

field

on

the

volume

element of the

matter equal

the resultant

of

the

ponderomotive

forces

exerted

by

this field

on

all

elementary

electric

and

magnetic

particles

in the

volume

element

considered.

By

a

volume

element of the

matter

we

always

understand

a

space

so

large

that it contains

a

very

large

number

of electric

and

magnetic

particles.

The

boundaries

of

a

volume

element

must

always

be

imagined

as

drawn

such that

the

boundary

surface

does

not cut through

any

electric

or

magnetic

dipoles.

[6]

First

we

calculate

that

force

acting

on a

dipole which is due

to

the

field

strength

£

not being exactly

the

same

at

the

locations

of

the

elemen-

tary

masses

of the

dipole. If

p

denotes the

vector

of

the

dipole

moment,

one

obtains the

following expression

for the

x-component

of the force

sought:

?(£

9£

M

c

_

y.

X

_

X_

_

X

~

^x

dx

^y

dy

^z

dz

[5] 1However,

we

stick

to

the dual

treatment

of electric

and

magnetic phenomena

for the

sake of

a

simpler

presentation.