direction 0,.-the direction it used to be-the so-called retarded direction-and at the retarded distance r. That would be easy enough to understand, too, but it is also wrong. The whole thing is much more complicated There are several more terms. The next term is as though nature were trying allow for the fact that the effect is retarded, if we might put it very crudely. It suggests that we should calculate the delayed coulomb field and add a correction to it, which is its rate of change times the time delay that we use. Nature seems to be attempting to guess what the field at the present time is going to be, by taking the rate of change and multiplying by the time that is delayed. But we are not yet through. There is a third term-the second derivative, with respect to t, of the unit vector in the direction of the charge. Now the formula is finished, and that is all there is to the electric field from an arbitrarily moving charge The magnetic field is given by X E/c We have written these down only for the purpose of showing the beauty of nature or, in a way, the power of mathematics. We do not pretend to understand why it is possible to write so much in such a small space, but(28.3)and (28.4)contain the machinery by which electric generators work, how light operates, all the phe omena of electricity and magnetism. Of course, to complete the story we also need to know something about the behavior of the materials involved-the prop erties of matter-which are not described properly by (28.3) To finish with our description of the world of the 19th century we must mention one other great synthesis which occurred in that century, one with which Maxwell had a great deal to do also, and that was the synthesis of the phenomena of heat and mechanics. We shall study that subject soon What had to be added in the 20th century was that the dynamical laws of Newton were found to be all wrong, and quantum mechanics had to be introduced to correct them. Newton's laws are approximately valid when the scale of things is sufficiently large. These quantum-mechanical laws, combined with the laws of electricity, have only recently been combined to form a set of laws called quantum electrodynamics. In addition, there were discovered a number of new phenomena, of which the first was radioactivity, discovered by Becquerel in 1898-he just sneaked it in under the 19th century. This phenomenon of radioactivity was followed up to produce our knowledge of nuclei and new kinds of forces that are not gravitational and not electrical, but new particles with different interactions a subject which has still not been unravelled For those purists who know more(the professors who happen to be reading this), we should add that when we say that(28.3 )is a complete expression of the knowledge of electrodynamics, we are not being entirely accurate. There was a problem that was not quite solved at the end of the 19th century. When we try to calculate the field from all the charges including the charge itself that we want the field to act on, we get into trouble trying to find the distance, for example, of a charge from itself, and dividing something by that distance, which is zero. The problem of how to handle the part of this field which is generated by the very charge on which we want the field to act is not yet solved today. So we leave it here;we do not have a complete solution to that puzzle yet, and so we shall avoid the puzzle for as long as we can 28-2 Radiation That, then, is a summary of the world picture it to dis mena called radiation to dis st select from Eq.( 28.3)only that piece which varies inversely as the distance and not as the square of the distance. It turns out that when we finally do find that piece, it is so simple in its form that it is legitimate to study optics and electrodynamics in an lementary way by taking it as"the law of the electric field produced by a moving charge far away. We shall take it temporarily as a given law which we will learn about in detail next yes