We know from our study of vectors that 2 variant under the transformation, since k, is a four-vector. But this quantity is precisely whata inside the cosine for a plane wave, and it ought to be variant under a transformation. We cannot have a formula with something that changes inside the cosine, since we know that the phase of the wave cannot change when we change the coordinate system. In deriving Eqs. (34. 17)and (34. 18), we have taken a simple example where k happened to be in a direction of motion, but of course we can generalize it to othe cases also. For example, suppose there is a source sending out light in a certain direction from the point of view of a man at rest, but we are moving along on the earth, say(Fig. 34-12). From which direction does the light appear to come? To find out, we will have to write down the four of ku and apply Lorentz transformation. The answer, however, can be found by the followir argument: we have to point our telescope at an angle to see the light. Why? Because light is coming down at the speed c, and we are moving sidewise at the speed v, so the telescope has to be tilted forward so that as the light comes down it goes"straight "down the tube. It is very easy to see that the horizontal distance is vt when the vertical distance is ct, and therefore, if e is the angle of tilt, tan gr v/c. How nice! How nice, indeed-except for one little thing: 0 is not the angle it which we would have to set the telescope relative to the earth, because we made a) our analysis from the point of view of a"fixed"observer. When we said the hori- zontal distance is vf, the man on the earth would have found a different distance Fig 34-12. a distant source S is since he measured with a"squashed"ruler. It turns out that, because of that con viewed by (a)a stationary telescope, and traction effect (b) a laterally moving telescope tan 8= 1-n2/e2 (3422) which is equivalent to (3423) It will be instructive for the student to derive this result, using the Lorentz trans formation This effect, that a telescope has to be tilted is called aberration, and it has been observed. How can we observe it? Who can say where a given star should be? Suppose we do have to look in the wrong direction to see a star: how do we know it is the wrong direction? Because the earth goes around the sun. today we have to point the telescope one way; six months later we have to tilt the telescope the ther way. That is how we can tell that there is such an effect 34-9 The momentum of light Now we turn to a different topic. We have never, in all our discussion of the past few chapters, said anything about the effects of the magnetic field that is associated with light. Ordinarily, the effects of the magnetic field are very small but there is one interesting and important effect which is a consequence of the agnetic field. Suppose that light is coming from a source and is acting on a Fig34-13.The magnetic force on a charge and driving that charge up and down. We will suppose that the electric ch is driven by the elect field is in the direction of the light beam. has a position x and a velocity v, as shown in Fig. 34-13. The magnetic field is at right angles to the electric field. Now as the electric field acts on the charge and moves it up and down, what does the magnetic field do? The magnetic field acts on the charge(say an electron)only when it is moving but the electron is moving it is driven by the electric field, so the two of them work together: While the thing out in which direction is this force? It is in the direction oftt, B times v times q going up and down it has a velocity a d there is gation of light Therefore, when light is shining on a charge and it is oscillating in response to that 34-10