This remarkable phenomenon is what we shall discuss in the present chapter At the beginning of this course in physics we outlined a broad picture of the world but we are now better prepared to understand some aspects of it and so we shall now go over some parts of it again in greater detail. We begin by describing the position of physics at the end of the 19th century. All that was then known about the fundamental laws can be summarized as follows First, there were laws of forces: one force was the law of gravitation, which we have written down several times; the force on an object of mass m, due to another of mass M, is given by F= GmMe/ where e, is a unit vector directed from m to M, and r is the distance between them Next, the laws of electricity and magnetism, as known at the end of the 19th century, are these: the electrical forces acting on a charge q can be described by two fields, called E and B, and the velocity v of the charge g, by the equation F=qE+×B) To complete this law, we have to say what the formulas for E and B are in a given circumstance: if a number of charges are present, E and the B are each the sum of contributions, one from each individual charge. So if we can find the e and B produced by a single charge, we need only to add all the effects from all the charges in the universe to get the total E and B! This is the principle of superposition What is the formula for the electric and magnetic field produced by one in dividual charge? It turns out that this is very complicated, and it takes a great deal of study and sophistication to appreciate it. But that is not the point. We write down the law now only to impress the reader with the beauty of nature, so to speak .e, that it is possible to summarize all the fundamental knowledge on one page with notations that he is now familiar with. This law for the fields of an individual charge is complete and accurate, so far as we know(except for quantum mechani but it looks rather complicated. We shall not study all the pieces now; we only write it down to give an impression to show that it can be written, and so that we can see ahead of time roughly what it looks like. As a matter of fact, the most useful way to write the correct laws of electricity and magnetism is not the way we shall now write them, but involves what are called field equations, which we shall learn about next year. But the mathematical notations for these are different and new,and so we write the law in an inconvenient form for calculation but in nota tions that we now know The electric field, E, is given by (283) What do the various terms tell us? Take the first term, e=-ge, 4TEo'.That of course, is Coulomb's law, which we already know: g is the charge that is pro- ducing the field; e, is the unit vector in the direction from the point P where e is measured, r is the distance from P to q. But, Coulomb's law is wrong The dis- coveries of the 19th century showed that influences cannot travel faster than a certain fundamental speed c, which we now call the speed of light. It is not correct at the first term is Coulombs law, not only because it is not possible to know where the charge is now and at what distance it is now, but also because the only thing that can affect the field at a given place and time is the behavior of the charges in the past. How far in the past? The time delay or retarded time, so-called, is the time it takes, at speed c, to get from the charge to the field point P. the delay So to allow for this time delay, we put a little prime on r, meaning how far away it was when the information now arriving at P left g. Just for a moment suppose that the charge carried a light, and that the light could only come to P at the speed c. Then when we look at q, we would not see where it is now, of course but where it was at some earlier time. What appears in our formula is the appar 28-2