In electromagnetic theory, the force on a given volume element within a dielectric is expressible in terms of the electric and magnetic fields at the surface of the volume element and the associated concept of surface stress tensor is well understood [1]. The calculation of the radiation pressure exerted on a moving interface between two media forms an interesting aspect of study in relativistic electrodynamics. Motivation for such a study arises in the context of explaining the excess of radiated power from a medium moving normal to the interface. Daly and Gruenberg [2] have considered this problem while studying the energy balance for perpendicularly polarized plane waves reflected from moving media. Later, Yeh [3] also touched upon this aspect in his investigation of the Brewster angle for a moving dielectric medium.

A detailed consideration of the radiation pressure due to a parallel or perpendicularly polarized plane electromagnetic wave obliquely incident (xz-plane of incidence) on a moving conducting half-space and the mechanical power (P_m) supplied to keep it uniformly moving is presented for the entire velocity range. The analogous study for a dielectric medium entends Yeh’s work [3] by including the velocity range beyond the critical medium velocity [4] β = C (C = cos θ, θ being the angle of incidence). The symbols used have their usual meaning [4].

Using the concept of the Maxwell stress tensor [5] and following the procedure outlined by Daly and Gruenberg [2], one may obtain the force per unit area on the interface:

and

The subscripts ǁ and ⊥ indicate the type of wave polarization involved.

The mechanical power supplied to the conductor to overcome this radiation pressure is then given by

where $S_{\text{z}}^I$ is the normal component of the Poynting vector associated with the incident wave given by

and

It is clear from Equation 14B.1 that the field force increases with an increase in C, being maximum at normal incidence. Further, the normalized mechanical power, $P_{\text{m}}/S_{\text{z}}^I$, is seen to be the same for both types of wave polarization. Figure 14B.1 shows the variation of this normalized mechanical power supplied with medium velocity. It is seen that the mechanical power supplied is positive in the range −1 < β < 0. The conductor is moving toward the wave, and, the field force on it being one of compression, mechanical power has to be supplied to the conductor to keep it uniformly moving. This power supplied will appear in the form of radiation, that is, the reflected power is more than the incident power. For positive β, that is, conductor moving away from the incident wave, there are three ranges to be considered. In the first range given by 0 < β < (1 – S)/C, the field does work on the conductor and so P_m is negative. The magnitude of the normalized rate of change of mechanical power per unit area increases and reaches a maximum. Due to the work done by the fields and the increase in the stored energy in the system, the reflected power is less than the incident power, that is, the power reflection coefficient is less than unity. The reflection coefficient eventually reduces to zero at β = (1 − S)/C. In the second range (1 − S)/C < β < C, the reflected wave propagates toward [4] the conductor and the reflection coefficient continues to be less than unity. At β = C, the conductor has no interaction with the wave and so P_m is zero. The last range, C < β < 1, corresponds to the problem of a conductor impinging on an existing plane wave above and the variation of P_m with β is similar to that in the range −1 < β < 0 except for the change in sign. The reason for the negative sign becomes clear from the orientation of the free-space waves as depicted in Figure 5 of Reference 4.

Assume that the fields in the dielectric medium (Σ′) have an exponential variation of the form

The characteristic root q′ may be determined in the usual manner by solving Maxwell’s equations in Σ′:

One may calculate the radiation pressure exerted on the moving dielectric surface following Daly and Gruenberg [2]. Alternatively, the radiation pressure may be calculated in the rest frame of the moving medium (Σ′) and then transformed to the laboratory system [3] (Σ) according to the transformation

Thus,

Finally,

For a perpendicularly polarized wave, one obtains

Our result (Equation 14B.8) for perpendicular polarization agrees with that of Daly and Gruenberg derived by another method. This may be verified by setting Q of their expression equal to (1 − βC)q′. Figure 14B.2 shows the variation of the normalized P_m with β for e′ = 2 for both wave polarizations. The general features of these curves are same as that for the relativistic conductor except for the change in sign. This is because the force on a conductor and that on a dielectric are in opposite directions [2]. This is depicted in Figure 14B.3 where $cF/S_{\text{z}}^I$ is plotted against β. The radiation pressure pushes on a conducting surface, whereas it has a sucking effect on a dielectric. Another conclusion drawn from Figure 14B.3 is that the radiation pressure on the dielectric is less for a perpendicularly polarized wave than for the parallel polarization.