the field inside the glass appears to be moving at a different speed That is the idea which we would like to work out quantitatively Now this is, in the exact case, pretty complicated, because although we have id that all the other moving charges are driven by the source field, that is not quite true. If we think of a particular charge, it feels not only the source, but like anything else in the world, it feels all of the charges that are moving. It feels, in particular, the charges that are moving somewhere else in the glass. So the total field which is acting on a particular charge is a combination of the fields from th can see that it would take a complicated set of equations to get l es doing!You complete and exact formula. It is so complicated that we postpone this problem until next year. Instead we shall work out a very simple case in order to understand all the physical principles very clearly. We take a circumstance in which the effects from the other atoms is very small relative to the effects from the source. In other words e take a material in which the total field is not modified very much by the motion of the other charges. That corresponds to a material in which the index of refraction is very close to 1, which will happen, for example, if the density of the atoms is very low. Our calculation will be valid for any case in which the index is for any reason very close to 1. In this way we shall avoid the complications of the most general, complete solution Incidentally, you should notice that there is another effect caused by the motion of the charges in the plate. These charges will also radiate waves back toward the source S. This backward-going field is the light we see reflected from the surfaces transparent materials. It does ne t the surface The backw radiation comes from everywhere in the interior, but it turns out that the total effect is equivalent to a reflection from the surfaces. These reflection effects are beyond our approximation at the moment because we shall be limited to a calculation for a material with an index so close to l that very little light is reflected Before we proceed with our study of how the index of refraction comes about we should understand that all that is required to understand refraction is to under stand why the apparent wave velocity is different in different materials. The bending of light rays comes about just because the effective speed of the waves is different in the materials. To remind you how that comes about we have drawn in Fig. 31-2 several successive crests of an electric wave which arrives fre vacuum onto the surface of a block of glass. The arrow perpendicular to the wave crests indicates the direction of travel of the wave. Now all oscillations in the wave must have the same frequency. (We have seen that driven oscillations have the same frequency as the driving source. ) This means, also, that the wave crests for the waves on both sides of the surface must have the same spacing along the surface because they must travel together, so that a charge sitting at the boundary will Fig. 31-2. Relation between refrac- feel only one frequency. The shortest distance between crests of the wave, however, on and velocity change is the wavelength which is the velocity divided by the frequency. On the vacuum side it is o=2Tc/w, and on the other side it is x= 2y/w or 2c/ is the velocity of the wave. From the figure we can see that the only way for the vaves to"fit"properly at the boundary is for the waves in the material to be travelling at a different angle with respect to the surface. From the geometry of the figure you can see that for a"fit"we must have Ao/sin 00=A/sin B, or sin 0o/sin 0= n, which is Snell's law. We shall, for the rest of our discussion consider only why light has an effective speed of c/n in material of index n, and the bending of the We go back now to the situation shown in Fig. 31-1. We see that what we have to do is to calculate the field produced at P by all the oscillating charges in the glass plate. We shall call this part of the field Ea, and it is just the sum written as the second term in Eq (31.2). When we add it to the term Es, due to the source, we will have the total field at P