60.

In view of the extraordinarily simple and close relation between the radiation pressure and the energy of radiation, the question might be raised whether this relation is really a special consequence of the electromagnetic theory, or whether it might not, perhaps, be founded on more general energetic or thermodynamic considerations. To decide this question we shall calculate the radiation pressure that would follow by Newtonian mechanics from Newton's (emission) theory of light, a theory which, in itself, is quite consistent with the energy principle. According to it the energy radiated onto a surface by a light ray passing through a vacuum is equal to the kinetic energy of the light particles striking the surface, all moving with the constant velocity $c$. The decrease in intensity of the energy radiation with the distance is then explained simply by the decrease of the volume density of the light particles.

Let us denote by $n$ the number of the light particles contained in a unit volume and by $m$ the mass of a particle. Then for a beam of parallel light the number of particles impinging in unit time on the element $d\sigma$ of a reflecting surface at the angle of incidence $\theta$ is $nc\cos \theta d\sigma .\mathrm{\backslash Label}[eqn](67)\text{upshape (67)}$

Their kinetic energy is given according to Newtonian mechanics by $I=nc\cos \theta d\sigma \frac{m{c}^2}{2}=nm\cos \theta \frac{c^3}{2}d\sigma .\mathrm{\backslash Label}[eqn](68)\text{upshape (68)}$ Now, in order to determine the normal pressure of these particles on the surface, we may note that the normal component of the velocity $c\cos \theta$ of every particle is changed on reflection into a component of opposite direction. Hence the normal component of the momentum of every particle (impulse-coordinate) is changed through reflection by $-2mc\cos \theta$. Then the change in momentum for all particles considered will be, according to [eqn:(67)] (67), $-2nm{\cos }^2\theta c^{2}d\sigma .\mathrm{\backslash Label}[eqn](69)\text{upshape (69)}$

Should the reflecting body be free to move in the direction of the normal of the reflecting surface and should there be no force acting on it except the impact of the light particles, it would be set into motion by the impacts. According to the law of action and reaction the ensuing motion would be such that the momentum acquired in a certain interval of time would be equal and opposite to the change in momentum of all the light particles reflected from it in the same time interval. But if we allow a separate constant force to act from outside on the reflector, there is to be added to the change in momenta of the light particles the impulse of the external force, i.e., the product of the force and the time interval in question.

Therefore the reflector will remain continuously at rest, whenever the constant external force exerted on it is so chosen that its impulse for any time is just equal to the change in momentum of all the particles reflected from the reflector in the same time. Thus it follows that the force $𝖥$ itself which the particles exert by their impact on the surface element $d\sigma$ is equal and opposite to the change of their momentum in unit time as expressed in [eqn:(69)] (69) $𝖥=2nm{\cos }^2\theta c^{2}d\sigma$ and by making use of [eqn:(68)] (68), $𝖥=\frac{4\cos \theta }{c}I.$

On comparing this relation with equation [eqn:(64)] (64) in which all symbols have the same physical significance, it is seen that

Newton's radiation pressure is twice as large as Maxwell's for the same energy radiation. A necessary consequence of this is that the magnitude of Maxwell's radiation pressure cannot be deduced from general