REFRACTION AND DOUBLE REFRACTION We have already made much use of the idea that the optical effect of a transparent medium can be represented by a refractive index. This is not of course an explanation of refraction ; for that we shall have to consider atomic processes, but without doing this we can discuss many of its features, and can describe the experiments which have been used in its investigation. We shall be content for the most part with the description of results; to work them out in detail would involve rather elaborate mathe matics.
The Electromagnetic Equations for Refracting Media.— The natural starting point for the discussion is the extension of the electromagnetic theory to cover electrical waves propagated through matter. When a static electric force acts on matter it may produce two effects If the matter is a conductor, a cur rent will flow according to Ohm's law; this is expressed by de fining a current density j (a vector with components jx, jy, jz) which is given by j= 0-E, where a is the specific conductivity of the medium. If the matter is an insulator, the electric force displaces the electricity in the atoms in a way that may be corn pared to the compression of a spring, and a new quantity has to be introduced to express this, which is called the dielectric dis placement. In isotropic media, such as water or glass, the dielec tric displacement bears a constant ratio to the electric force. We write D = €E, and call e the dielectric constant. General elec tromagnetic theory then shows that the equations for free space must be altered so as to accommodate either or both of these properties of matter. We now write Matter also has magnetic properties, and this suggests that the other equations should be changed as well. It is found how ever to be unnecessary, for the alternations of force in light are so rapid that the magnetic properties have no time to take effect. So we adhere to the equations These equations together determine the behaviour of light in most types of matter, but we must remember that we always have to make observations outside, and therefore require to know The electric force is transverse to the waves and the wave velocity is c/'/€, so that the medium has refractive index ye.
An apparent difficulty at once arises, for our result seems to imply that the refractive index should not depend on the colour of the light. This will be explained when we come to the atomic theory, where it will appear that the dielectric constant depends on the frequency of the inducing electric force.
We next consider the passage of light from one medium to an other. Suppose that they have refractive indices n and n', and first suppose n to be the lesser; in the case of free space it will be unity. To find the formulae for refraction and reflection we take a given incident wave, and assume that there are reflected and refracted waves, but without making any assumptions about their wave-lengths or directions. The boundary conditions then require that the sum of the tangential components of incident and reflected forces at the boundary in the first medium, are everywhere and at every time equal to the corresponding forces in the refracted wave on the other side of the boundary. These conditions lead in the first place to the law that the frequencies of all three waves must be equal, so that the wave-length of the refracted wave is Xn/n', and then to the law that the angle of reflection is equal to the angle of incidence 0, while the angle of refraction 0' is given by nsine---n'sin0'. These rules can be deduced merely from the fact that there are boundary conditions and would be the same whatever the form of those conditions. Next, with the special conditions of the electrical theory, we can find the amplitudes of the reflected and refracted waves. Taking the component in which the electric force is polarized perpen dicular to the plane of incidence we find as amplitude of the re flected wave sin(0-0')/sin(0-1- 0') , while the component polarized in the plane of incidence gives tan(0-0')/tan(0-1-0'). These are Fresnel's sine and tangent formulae; their squares give the in tensities of reflection. The intensities of the refracted wave may also be given, but are not so important.