THE ELECTROMAGNETIC EQUATIONS It was Maxwell who made the great discovery that the equations governing the behaviour of electric waves are equally applicable to light, and this provides a strict formulation for the whole the ory, including of course polarization. For the derivation of the theory of electric waves see the article ELECTRICITY ; we shall here take it as given and show how it applies for light.
Consider first the case of free space. At every point there may be an electric force E and a magnetic force H. Each has both direction and magnitude and they can be most conveniently de scribed by the components Ey, and Hy, Hz along the directions x, y, z. The vector notation is well adapted to expressing their relations. In this notation is written div E and called the divergence, while the three quan tities are the components of curl E. A vector equation involving curl is thus three equations, when written in terms of the components. The equations are In these equations c is originally a rather abstruse quantity, the ratio of the electromagnetic to the electrostatic units. It is a velocity, and one of the strongest evidences for the electromag netic theory of light is that, when purely electric methods are used to determine the ratio, it is found to be the same as the velocity of light. We shall give a few examples of solutions, but before doing so must complete the theory by giving the rule for intensity. In our previous account of intensity we left a factor of proportionality undetermined, and this was right because we had then no other physical phenomenon to link with light, so that there was no way of fixing the absolute values. Now, how ever, we have a much more precise formulation, because we can imagine that we might measure (very ideally of course) the elec tric force in the light by means of an electrometer, and we can therefore make our definition absolute. Electrical theory assigns a value to the flux of energy, i.e., to the rate at which energy is carried across unit area in unit time, and this is a suitable measure for intensity. It is called the "Poynting vector," after its discov erer, and is [E, H] where [E, is a vector product with 4ir component 11,—E, Hy, along x, etc.
We now consider some solutions of the electromagnetic equa tions. One such solution may be verified to be We see in the first place that we have a wave travelling along z with velocity c, and purely electrical experiments have shown c to be equal to the velocity of light. Secondly we see that it is a
transverse wave, but it is ambiguous whether the electric or mag netic force is the light-vector. A similar solution is and this evidently represents the other polarized component. A third solution can be formed by superposing these two, or by superposing them with a phase difference between E. and E.
The intensity in such a case would be (A' + .
8r We must now consider whether the electric or magnetic force is the light-vector. Since both always occur in the wave, a theory could be constructed in which either was so taken, and it is a mat ter of convenience which we choose. A number of phenomena show that the electric force is the more important, primarily be cause matter is constructed out of electrons and not out of mag netic particles. We may give as one example the case of standing waves. When a plane wave of monochromatic light falls perpen dicularly on a mirror and is reflected straight back, the incident and reflected waves interfere with one another and produce a sys tem of stationary oscillations. These may be described by how the light will pass from one medium to another. Electric theory again provides the answer; when light goes through a boundary between two media, at every point of the surface and at every instant of time, the tangential components of electric and magnetic force just on one side of the boundary must each be equal to the tangential components of the corresponding forces just on the other side. (See ELECTRICITY.) Refraction in Transparent Media.—In a non-conducting isotropic medium, the equations assume the form We see that at points where z is any multiple of half a wave-length vanishes all the time, whereas Hy vanishes all the time at points where z is an odd multiple of quarter of a wave-length. Suppose that our mirror is coated with a nearly transparent photo graphic film, of some depth, which is afterwards developed and examined in section. Then we shall find places where it is fogged by the action of the light-vector, and others where it is unaffected and the positions of these tell us that it is the electric force that is effective and not the magnetic. From this and similar cases we conclude that it is best to take the electric force as the light vector.