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MEASUREMENT OF LIGHT PRESSURE ON A MIRROR PLACED IN A REFRACTING MEDIUM
The classical experiments of P. N. Lebedev, as well as later measurements of the pressure of light on a mirror, were carried out in vacuum. Meanwhile, according to theoretical predictions, the pressure of light should depend quite substantially on the refractive index of the medium in which the mirror is placed. The considerable difficulties involved in carrying out the corresponding experiment apparently account for the fact that, up to now (for half a century!), this theoretical prediction has not been confirmed by a direct experiment. True, by the experiments of Barlow[^1], who observed the appearance of a torque when a light beam was passed obliquely through a glass cube, the existence of light pressure at the boundary of two transparent dielectrics with different refractive indices was shown. However, these experiments were qualitative in character and, moreover, could serve only as indirect confirmation of the conclusion concerning the dependence of light pressure on a mirror on the refractive index of the medium.
Therefore, the author’s accomplishment, in the note under review[^2], of direct measurements of this dependence is of primary interest. It is, of course, impossible to agree with the author that they cannot serve as a trial stone for the choice between the already rejected corpuscular theories of light, the electromagnetic theory, and quantum-mechanical ideas. The question is not posed in that way. However, contrary to the author’s assertion, such experiments have not only illustrative but, in a certain respect, decisive significance. Within the framework of modern electrodynamics, until very recently there was no clarity in the choice of expres-
... for the momentum density of the electromagnetic field in a medium. For a long time, two expressions have been discussed—one proposed by Minkowski\(^3\):
\[ \mathbf{g}=\frac{1}{4\pi c}[\mathbf{D}\mathbf{B}], \tag{1} \]
and the other, proposed by Abraham:
\[ \mathbf{g}=\frac{1}{4\pi c}[\mathbf{E}\mathbf{H}]. \tag{2} \]
Only quite recently was I. E. Tamm\(^4\) able convincingly to show the untenability of Abraham’s argument and the validity of expression (1). However, to this day many authors, especially foreign ones, give preference to Abraham’s expression. Meanwhile, these expressions lead to a different dependence of the pressure of light on a mirror on the refractive index of the medium. Indeed, if the refractive indices of the mirror and of the medium are respectively \(n\) and \(n'\), then the pressure of light on the mirror is equal to
\[ f=g v(1+r), \tag{3} \]
where
\[ r=\left(\frac{n-n'}{n+n'}\right)^2 \]
is the reflection coefficient of the mirror and \(v\) is the velocity of light in the medium.
Taking into account that for a transparent isotropic medium
\[ \mathbf{D}=n^2\mathbf{E}\quad \text{and}\quad \mathbf{B}=\mathbf{H}, \]
we have: according to (1)
\[ P_M=\frac{1+r}{4\pi c}\, n S \tag{4} \]
and according to (2)
\[ P_A=\frac{1+r}{4\pi c n}\, S=\frac{1}{n^2}P_M, \tag{5} \]
where
\[ S=\frac{c}{4\pi}[\mathbf{E}\mathbf{H}] \]
is the Umov–Poynting vector of the light flux incident on the mirror in the medium.
Labels in the figure: upper suspension fastening; mirror serving to measure the angle of rotation of the suspension; liquid level; silver wings coated with rhodium; lower suspension fastening; \(5\ \mathrm{cm}\); A; B.
Thus, an experimental determination of the dependence of \(P\) on \(n\) provides a direct criterion for choosing between (1) and (2).
The measurements were carried out with the aid of the apparatus shown in the figure. A light rod carrying a pair of miniature wings \((5\times 2\times 0.1\ \mathrm{mm})\), made of silver and coated (in order to increase the reflection coefficient) with rhodium, was attached to a short-period (less than 1 sec) unifilar suspension. The threads supporting the rod were kept taut, and both at the bottom and at the top they were rigidly fixed, in order to avoid horizontal displacements of the rod under the action of the surface-tension forces of the liquid filling the vessel.
In all experiments on measuring light pressure, the principal difficulties are connected with the need to eliminate the influence of secondary effects—
effects, namely: the radiometric effect and convection currents. In the present case, reduction of the disturbances caused by convection currents was achieved by a possible reduction in the dimensions both of the wings themselves and of the vessel in which they were placed. On the other hand, the author relied on the investigations of Teer1, which showed that the radiometric effect for most gases at atmospheric pressure becomes negligibly small.
In the experiment the angle \(\theta\) of torsion of the suspension was measured directly under the action of light pressure of constant intensity, incident on the wings; here the part of the vessel in which the wings were located was alternately filled either with air or with a liquid of known refractive index. The angle of torsion, amounting to about \(10^{-5}\) radian, was measured photoelectrically from the deflection of a beam reflected from the upper mirror (see Fig.). To increase the measured effect, a pair of opposing beams was used, directed initially as shown in the figure by the arrows \(A'\), \(B\), and then reversed (switched to the position \(A, B'\)); the resulting change of the angle \(\theta\) was determined.
The author indicates that in air, after the beams had been moved from the position \(A'\), \(B\) to the position \(AB'\), the new equilibrium position of the suspension was established as rapidly as its mechanical parameters allowed. When, however, the wings were immersed in a liquid, the process of establishing the equilibrium position after switching the beam proceeded in three phases: 1) during approximately 1 sec. the angle \(\theta\) reached a certain value characteristic of the given conditions, then 2) over several seconds the magnitude of the angle decreased to a value lying in various cases between 20 and 30% of the initial value, and finally, 3) the angle again increased, reaching values exceeding the initial one by 1–5 times, depending on the conditions. The author believes that the second and third phases were caused, respectively, by the appearance of a temperature difference between the front and rear sides of the wings and by convection effects. Since the effect of light pressure is instantaneous, whereas thermal effects require a certain time for their development, the initial equilibrium value of the angle \(\theta\) had to be chosen as the measure of the light pressure in the liquid.
As a light source there was used a tungsten incandescent lamp; moreover, to ensure the same spectral composition (the same value of the Poynting vector) in measurements in air and in the liquid, a vessel with the same liquid was placed in the path of the beams, serving as a light filter.
The author indicates that the accuracy of the measurements was not high and regards his measurements as preliminary. In addition to the smallness of the measured angle \(\theta\), one of the sources of error was Brownian motion, which produced errors of the order of 10%. A factor that substantially simplified the measurements and increased their accuracy was that the author confined himself to relative measurements and determined not the magnitude of the light pressure in the liquid itself, but its ratio to the pressure in air, i.e. the ratio
\[ \frac{\theta\ \text{(in liquid)}}{\theta\ \text{(in air)}} . \]
The results of the measurements are given in the table (on p. 466).
It is not difficult to see that under the conditions of this experiment, according to (4),
\[ \frac{P_{\text{liquid}}}{P_{\text{air}}} = n^{2} \left( \frac{1+n_{0}}{n+n_{0}} \right)^{2} \frac{ 1+\left(\dfrac{n-n'}{n+n'}\right)^{3} }{ 1+\left(\dfrac{1-n'}{1+n'}\right)^{2} }, \tag{6} \]
| Medium | Refractive index | $\dfrac{\theta\text{ (in liquid)}}{\theta\text{ (in air)}}$ measured | $\dfrac{\theta\text{ (in liquid)}}{\theta\text{ (in air)}}$ for identical $S$ in the medium |
|---|---|---|---|
| Water | 1.33 | 1.36 | 1.31 |
| Ethyl ether | 1.35 | 1.44 | 1.39 |
| Ethyl alcohol | 1.36 | 1.42 | 1.37 |
| Xylene | 1.50 | 1.50 | 1.44 |
| Benzene | 1.50 | 1.53 | 1.47 |
| Nitrobenzene | 1.55 | 1.50 | 1.44 |
| Carbon disulfide | 1.63 | 1.65 | 1.59 |
where $n_0$ is the refractive index of the material from which the window is made that admits the light rays into the vessel.
Putting
\[ |n'| \gg n,\quad |n-n_0| \ll 1 \quad \text{and} \quad |1-n_0| \ll 1, \]
we have:
\[ \frac{\theta_{\text{liq}}}{\theta_{\text{air}}} = \frac{P_{\text{liq}}}{P_{\text{air}}} \cong n. \tag{7} \]
If $\dfrac{\theta_{\text{liq}}}{\theta_{\text{air}}}$ is referred to equal values of the Poynting vector $S$ in the medium (these quantities are given in the last column of the table), then according to (4) (to within the difference in the reflection coefficient) equality (7) is satisfied exactly:
\[ \left(\frac{\theta_{\text{liq}}}{\theta_{\text{air}}}\right)_S = n. \tag{8} \]
At the same time, according to (5) this quantity should be equal to $\dfrac{1}{n}$.
Comparison with the observational results unambiguously testifies in favor of expression (4): the light pressure on the mirror proves to be directly proportional to the refractive index of the medium into which the mirror is immersed. Thus the question of the correct expression for the momentum density of the electromagnetic field is resolved unambiguously, in accordance with the considerations expressed by I. E. Tamm. It is curious that the author himself did not pay attention to this aspect of the question.
G. Rozenberg
References Cited
- G. Barlow, Proc. Roy. Soc. A 87, 1 (1912).
- R. V. Jones, Nature 167, 439 (1951).
- See, for example, W. Pauli, Theory of Relativity. Gostekhizdat, 1947.
- I. E. Tamm, Foundations of the Theory of Electricity. Gostekhizdat, 1949.
- J. D. Teag, J. Opt. Soc. Am. 11, 135 (1925).
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Teer. ↩