169
J. H. POYNTING
THE MOMENTUM OF A BEAM OF LIGHT
The existence of Light Pressure, and therefore the existence of Momentum, in
a beam of light follows directly from NEWTON'S Corpuscular Theory; but that a pressure also exists when light is regarded as undulatory was first deduced by MAXWELL
from his Electromagnetic Theory of light.
It can be shown however that the pressure must follow as a consequence of
any wave motion, without making assumptions as to the nature of the type of wave
which constitutes light, provided only that the medium is such that a reflecting solid
can move through it without disturbing the medium except by reflecting the waves
which it meets. This important generalization was first indicated by B ARTO LI, and
afterwords more fully worked out by LARMOR. The proof given by the latter may
be put in a simple form as follows :
Consider a train of waves incident normally on a perfectly reflecting wall which
is moving towards the beam with velocity v, and let V be the wave velocity. Then
evidently a length V -j- v of the incident beam which is just beginning to meet the
reflector at any instant will after an interval of one second be transformed into a
length V — v of the reflected beam.
These two lengths must contain the same number of waves; hence if I is the
wave length of the incident beam, the reflected beam must have the shorter wave
V—v
length X = -==—.—. X which is in accordance with DOPPLER'S principle.
V -f- v
The perfect reflection requires that the resultant disturbance at the surface shall
always be zero, and the incident and reflected trains must therefore have equal amplitudes. We must now assume that for a given amplitude the average energy density
is inversely as the square of the wave length. This is evidently true for elastic waves,
and it holds also for Electromagnetic waves when the amplitude is that of the
Vector Potential.
The effect of the change in wave length will therefore be to increase the energy
density from E, the value in the incident beam, to E\ the value for the reflected
beam, where
*-m-
E' == ( T H - ^ I . E .
22
— 170 —
Regarding then a beam of unit cross section, the total energy before reflection
is (V + v) E, and after reflection it is (V — v) E f , so that in one second the energy
of wave motion in the medium is increased by an amount
(V — v)W — (V + y)E
_Y + v
Y—v
. 2y.E.
This gain in energy is to be accounted for by supposing that the reflecting surface
is pressed back by the waves, so that work has to be done in moving it forward. If
then p is the pressure on the reflector when moving with velocity v, the work done
per second on each square unit of surface will be pv, and we obtain at once :
Y —v
When the reflector is at rest v = 0, and hence
p0 = 2E
or, since the incident and reflected beams are now equal and therefore E' = E, we
have the simple result that the pressure is equal to the total energy density in
front of the reflector. This relation, it should be noticed, does not hold in the case
of motion, for the total energy density is then
E + E' = E + ( ^ ) \ E
Y2-\-v2
.2E
(V — vf
and hence
V2 — v2
i' = V 2 7F^- (E+E ' )
/
2v2\
= 11 — — J (E + E')
approximately.
This shows that the pressure is only equal to the energy density if the second power
v
of — can be neglected.
If the incident beam is not reflected but totally absorbed by a surface at rest
it produces half the pressure, and we still have the pressure equal to the energy
density.
The wave train can therefore be regarded as carrying with it a stream of
momentum, and since the absorption of this momentum gives rise to a pressure on
the absorbing surface equal in amount to the energy density E of the waves, we
— 171 —
E
may consider that the stream of momentum has a density — and is moving with
the velocity of light V.
The idea of momentum in the wave train enables us to see at once what is
the nature of the action of a beam of light on a surface at which it is reflected,
refracted, absorbed, or emitted, without making any further appeal to the theory
of the wave motion. Several important cases will now be considered from this point
of view.
If the surface upon which a beam falls is a perfect absorber, and is at rest,
the pressure pQ on it must equal the momentum received per second per unit area, or
* - ( f ) . V = E.
Now suppose that the absorber is moving towards the beam with velocity4 v, in
one second it will sweep up the momentum in a lenght V -f- v of the incident beam
and the pressure is therefore increased to
^ = (V + » ) . |
-('+*) E.
Consider now a surface at rest which is emitting a parallel beam of light normally ; there will be a back pressure on it which is again equal to the energy density of the beam. This follows from the fact that the momentum stream sent out
comes from the surface and therefore the latter must experience the corresponding
reaction. Or we may deduce the pressure from the case of perfect reflection already
discused by regarding the reflected light as emitted by the surface.
When the source is moving with velocity v in the direction in which it is radiating, let E' be the energy density of the beam emitted. Since the momentum
¥'
density is —, and a length V — v is emitted in each second, the pressure must be
given by
V— v ™
* = — - . E
We have as yet made no assumption as to the value of E'. Suppose that the source
when at rest emits waves having energy density E 0 , and assume that when in motion
the emitting surface always converts its internal energy into radient energy at the
same rate E0V as when at rest, but that the work pv done per second against the
pressure is also transformed into radiant energy. We have then
E 0 V + ^ = B'(V — v)
— 172 —
and hence from last equation
P = T=-rE«
or
V
The pressure is therefore increased by the forward motion.
The existence of the pressure when a beam of light falls normally upon a surface has been fully proved by the experiments of LEBEDEW and of NICHOLS and
HULL.
In the experiments which we shall now describe, illustrating the momentum in
a beam of light, we shall for convenience consider the energy Ylx per unit length
of the beam, and the total pressure force, so avoiding any necessity for taking into
account the cross section of the beam. The beam of light then carries a quantity of
E
momentum -~r in unit length, and exerts a force equal to E t on any totally absorbing surface at rest, and this force is always in the direction of the beam.
Thus for oblique incidence on an absorbing surface at an angle 0 with the
normal, the force on the surface has components
Ei cos 0 normally
E! sin 0 tangentially.
In the more general case when the surface has reflection coefficient r, it is easily
seen that the components are (1-f-r) Ei coso and (1 — r) Ej sin 0 respectively
\
FIG. 1. -
Plan.
The tangential stress is of especial interest as it may be detected more easily
than the normal effect. For generally the disturbing action of the surrounding gas
gives a resultant force along the normal to the surface, and it is always difficult
to separate this from the normal light pressure. On the other hand it is merely
necessary to arrange that the absorbing surface is free to move only in its own plane
in order to eliminate the gas action and exhibit the tangential component of the
light pressure.
In the first experiment to show this effect, two circular glass disks, 2 cm. diameter, were fixed to the ends of a horizontal glass rod, length 5.3 cm., the disks
being perpendicular to the rod (Fig. 1. plan). One of the disks was coated with
— 173 —
camphor black, and the other brightly silvered. The whole system was then suspended
by a fine quartz fibre within an exhausted chamber. A parallel beam of light from
a Nernst Lamp was directed horizontally on either disk at an angle of 45° as shown
by the arrow in Fig. 1. When the beam fell on the black disk the system was
pushed round by the tangential force, but when it fell on the silvered disk the
push round was very small. The deflections of the system were observed by the usual
telescope and mirror method. The energy of the beam was determined, after the
manner of NICHOLS and HULL, by allowing it to be absorbed by a blackened
silver disk of known thermal capacity and measuring the rise in temperature by
means of a thermoelectric junction. Some irregular disturbances due to gas action
were never entirely eliminated, but these were found to be at a minimum, as in the
experiments of NICHOLS and HULL, for gas pressures near 1 cm. of mercury. The
calculated and observed deflections were in good agreement.
Another, and much more successful, form of the experiment has been recently
carried out. A single circular disk of camphor blacked mica, 3.5 cm. diameter, was
suspended with its plane horizontal by a quartz fibre from its centre (Fig. 2). Two
equal beams of light, A and B, were thrown down at an angle of 45° on exacthy
the same part of the disk: these beams were in the same vertical plane, and therefore mutually at right angles, and hence the two tangential stresses on the disk
were equal and tended to rotate it in opposite directions. The amount of rotation
was determined (a) for A alone (b) for B alone and (c) for A and B together. The
algebraical difference between the deflections (a) and (b) gives twice the tangential
effect, for the heating of the mica disc, and therefore any uneliminated gas action,
may be assumed to be the same in both cases. On the other hand the deflection (c)
will give merely twice the gas action. Observations made with widely varying gas
pressures in both air and hydrogen (in which gas the disturbing action is much less
than in air) demonstrated beyond all doubt the existence of a tangential stress of
the magnitude, always within 15 per cent, required by theory.
If by means of any system of reflecting or refracting surfaces a beam of light
be displaced parallel to its original direction, we see at once from the momentum
— 174 —
idea that the system must be acted upon by a couple. Two such cases have been
examined experimentally.
A rectangular block of glass (Fig. 3. Plan) 3 cm. X 1 cm. X 1 cm., was suspended
by a quartz fibre so that the long axis of the block was horizontal. A parallel beam
FIG. 3. — Plan.
of light was directed horizontally on to one end of the block and at an angle of
incidence of 55°. After two internal total reflections it emerged from the other end
in a direction parallel to the incident beam, as indicated by the arrows in Fig. 3.
The couple on the block produced by this lateral shift of the beam of light
was found to be in close agreement with the value calculated from measurements
of the energy of the beam.
In the other experiment two glass prisms (Fig. 4. Plan) each having a refracting
angle of 34° and with refracting edge 1.6 cm. long, were fixed at the ends of a
-*->-
FIG. 4. — Plan.
thin brass torsion arm suspended at its middle point from a quartz fibre. The two
inner faces of the prisms were perpendicular to the torsion arm and 3 cm. apart.
When a parallel beam of light was sent symmetrically throngh the system, as shown
by the arrows in the diagram, the lateral diplacement of the beam, 1.64 cms. in this
experiment, produced an easily measurable couple in the direction and within a few
per cent of the magnitude indicated by theory. This experiment was also successfully
repeated with another and smaller pair of prisms.
These experiments were performed with reduced air pressure, but the absence of
serious gas action was probably due to the fact that in such a system of reflections
and refractions at glass surfaces only a very small fraction of the energy of the light
is absorbed by the system.
The experiments were made in collaboration with my colleague Dr. GUY BARLOW.