Page:EB1911 - Volume 16.djvu/641

Let the indices p and n relate to the two principal cases in which the incident (and, consequently, the reflected) light is polarized in the plane of incidence, or normally to it, and let positive directions h and h′ be chosen for the disturbance (at the surface itself) in the incident and for that in the reflected beam, in such a manner that, by a common rotation, h and the incident ray prolonged may be made to coincide with h′ and the reflected ray. Then, if α₁ and α₂ are the angles of incidence and refraction, Fresnel shows that, in order to get the reflected disturbance, the incident one must be multiplied by

αp = −sin (α1 − α2) / sin (α1 + α2)

(9)

in the first, and by

αn = tan (α1 − α2) / tan (α1 + α2)

(10)

in the second principal case.

As to double refraction, Fresnel made it depend on the unequal elasticity of the aether in different directions. He came to the conclusion that, for a given direction of the waves, there are two possible directions of vibration (§ 6), lying in the wave-front, at right angles to each other, and he determined the form of the wave-surface, both in uniaxal and in biaxal crystals.

Though objections may be urged against the dynamic part of Fresnel’s theory, he admirably succeeded in adapting it to the facts.

16. Electromagnetic Theory.—We here leave the historical order and pass on to Maxwell’s theory of light.

James Clerk Maxwell, who had set himself the task of mathematically working out Michael Faraday’s views, and who, both by doing so and by introducing many new ideas of his own, became the founder of the modern science of electricity,^[1] recognized that, at every point of an electromagnetic field, the state of things can be defined by two vector quantities, the “electric force” E and the “magnetic force” H, the former of which is the force acting on unit of electricity and the latter that which acts on a magnetic pole of unit strength. In a non-conductor (dielectric) the force E produces a state that may be described as a displacement of electricity from its position of equilibrium. This state is represented by a vector D (“dielectric displacement”) whose magnitude is measured by the quantity of electricity reckoned per unit area which has traversed an element of surface perpendicular to D itself. Similarly, there is a vector quantity B (the “magnetic induction”) intimately connected with the magnetic force H. Changes of the dielectric displacement constitute an electric current measured by the rate of change of D, and represented in vector notation by

C = Ḋ

(11)

Periodic changes of D and B may be called “electric” and “magnetic vibrations.” Properly choosing the units, the axes of coordinates (in the first proposition also the positive direction of s and n), and denoting components of vectors by suitable indices, we can express in the following way the fundamental propositions of the theory.

(a) Let s be a closed line, σ a surface bounded by it, n the normal to σ. Then, for all bodies,

$\int$Hsds =	1	$\int$Cn dσ,   $\int$Es ds = −	1		d	$\int$Bn dσ,
	c		c		dt

where the constant c means the ratio between the electro-magnet and the electrostatic unit of electricity.

From these equations we can deduce:

(α) For the interior of a body, the equations

∂Hz

−

∂Hy

=

1

Cx,

∂Hx

−

∂Hz

=

1

Cy,

∂Hy

−

∂Hx

=

1

Cz

∂y

∂z

c

∂z

∂x

c

∂x

∂y

c

(12)

∂Ez

−

∂Ey

= −

1

∂Bz

,

∂Ex

−

∂Ez

= −

1

∂By

,

∂Ey

−

∂Ex

= −

1

∂Bz

;

∂y

∂z

c

∂t

∂z

∂x

c

∂t

∂x

∂y

c

∂t

(13)

(ß) For a surface of separation, the continuity of the tangential components of E and H;

(γ) The solenoidal distribution of C and B, and in a dielectric that of D. A solenoidal distribution of a vector is one corresponding to that of the velocity in an incompressible fluid. It involves the continuity, at a surface, of the normal component of the vector.

(b) The relation between the electric force and the dielectric displacement is expressed by

D_x = ε₁E_x,  D_y = ε₂E_y,  D_z = ε₃E_z,

(14)

the constants ε₁, ε₂, ε₃ (dielectric constants) depending on the properties of the body considered. In an isotropic medium they have a common value ε, which is equal to unity for the free aether, so that for this medium D = E.

(c) There is a relation similar to (14) between the magnetic force and the magnetic induction. For the aether, however, and for all ponderable bodies with which this article is concerned, we may write B = H. It follows from these principles that, in an isotropic dielectric, transverse electric vibrations can be propagated with a velocity

v = c / √ε.

(15)

Indeed, all conditions are satisfied if we put

Dx = 0, Dy = a cos n (t − xv−1 + l),	Dz = 0,	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$
Hx = 0, Hy = 0,	Hz = avc−1 cos n (t − xv−1 + l)

(16)

For the free aether the velocity has the value c. Now it had been found that the ratio c between the two units of electricity agrees within the limits of experimental errors with the numerical value of the velocity of light in aether. (The mean result of the most exact determinations^[2] of c is 3,001·10¹⁰ cm./sec., the largest deviations being about 0,008·10¹⁰; and Cornu^[3] gives 3,001·10¹⁰ ± 0,003·10¹⁰ as the most probable value of the velocity of light.) By this Maxwell was led to suppose that light consists of transverse electromagnetic disturbances. On this assumption, the equations (16) represent a beam of plane polarized light. They show that, in such a beam, there are at the same time electric and magnetic vibrations, both transverse, and at right angles to each other.

It must be added that the electromagnetic field is the seat of two kinds of energy distinguished by the names of electric and magnetic energy, and that, according to a beautiful theorem due to J. H. Poynting,^[4] the energy may be conceived to flow in a direction perpendicular both to the electric and to the magnetic force. The amounts per unit of volume of the electric and the magnetic energy are given by the expressions

1/2 (E_xD_x + E_yD_y + E_zD_z),

(17)

and

1/2 (H_xB_x + H_yB_y + H_zB_z) = 1/2 H²,

(18)

whose mean values for a full period are equal in every beam of light.

The formula (15) shows that the index of refraction of a body is given by √ε, a result that has been verified by Ludwig Boltzmann’s measurements^[5] of the dielectric constants of gases. Thus Maxwell’s theory can assign the true cause of the different optical properties of various transparent bodies. It also leads to the reflection formulae (9) and (10), provided the electric vibrations of polarized light be supposed to be perpendicular to the plane of polarization, which implies that the magnetic vibrations are parallel to that plane.

Following the same assumption Maxwell deduced the laws of double refraction, which he ascribes to the unequality of ε₁, ε₂, ε₃. His results agree with those of Fresnel and the theory has been confirmed by Boltzmann,^[6] who measured the three coefficients in the case of crystallized sulphur, and compared them with the principal indices of refraction. Subsequently the problem of crystalline reflection has been completely solved and it has been shown that, in a crystal, Poynting’s flow of energy has the direction of the rays as determined by Huygens’s construction.

Two further verifications must here be mentioned. In the first place, though we shall speak almost exclusively of the propagation of light in transparent dielectrics, a few words may be said about the optical properties of conductors. The simplest assumption concerning the electric current C in a metallic body is expressed by the equation C = σE, where σ is the coefficient of conductivity. Combining this with his other formulae (we may say with (12) and (13)), Maxwell found that there must be an absorption of light, a result that can be readily understood since the motion of electricity in a conductor gives rise to a development of heat. But, though Maxwell accounted in this way for the fundamental fact that metals are opaque bodies, there remained a wide divergence between the values of the coefficient of absorption as directly measured and as calculated from the electrical conductivity; but in 1903 it was shown by E. Hagen and H. Rubens^[7] that the agreement is very satisfactory in the case of the extreme infra-red rays.

In the second place, the electromagnetic theory requires that a surface struck by a beam of light shall experience a certain pressure. If the beam falls normally on a plane disk, the pressure is normal too; its total amount is given by c⁻¹(i₁ + i₂ − i₃), if i₁, i₂ and i₃ are the quantities of energy that are carried forward per unit of time by the incident, the reflected, and the transmitted light. This result has been quantitatively verified by E. F. Nicholls and G. F. Hull.^[8]

Maxwell’s predictions have been splendidly confirmed by the experiments of Heinrich Hertz^[9] and others on electromagnetic waves; by diminishing the length of these to the utmost, some physicists have been able to reproduce with them all phenomena of reflection, refraction (single and double), interference, and polarization.^[10] A table of the wave-lengths observed in the aether now has

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1. Clerk Maxwell, A Treatise on Electricity and Magnetism (Oxford, 1st ed., 1873).
2. H. Abraham, Rapports présentés au congrès de physique de 1900 (Paris), 2, p. 247.
3. Ibid., p. 225.
4. Phil. Trans., 175 (1884), p. 343.
5. Ann. d. Phys. u. Chem. 155 (1875), p. 403.
6. Ibid. 153 (1874), p. 525.
7. Ann. d. Phys. 11 (1903), p. 873.
8. Phys. Review, 13 (1901), p. 293.
9. Hertz, Untersuchungen über die Ausbreitung der elektrischen Kraft (Leipzig, 1892).
10. A. Righi, L’Ottica delle oscillazioni elettriche (Bologna, 1897); P. Lebedew, Ann. d. Phys. u. Chem., 56 (1895), p. 1.