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PHYS195 NotesChapter4 - Chapter 4 19th Century Physics 4.1...

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Chapter 4 19th Century Physics 4.1 Action at a Distance and Field Dynamics The previous construction of Fresnel/Young/Huygens tell us how to con- struct an amplitude for light at any point in space given the amplitude at some other point in space. This is the first part of the construction of a field. A field is something, generally a measured quantity, that is defined at every point in space. At each point in space you can measure the entity. In addition, as you move from one point to a nearby point the value of the something changes smoothly; it varies as you change places. There will even be a rule on how the change as you move from point to point is manifest. To appreciate these rather abstract comments let’s look at several examples. There are numerous examples of fields. The temperature in a room is a field. Temperature is measured for instance by a mercury bulb thermometer. As you move the thermometer from point to point, you will get different values for the temperature. If the room is not too drafty, the temperature at nearby points will be similar; the temperature varies smoothly as you move to nearby points. You can even intuit certain rules for how the temperature changes as you move from point to point. For instance, you can guess that a point at the center of a surrounding group of points, the temperature will be the average of the temperatures of the surrounding points. It is because of rules like this that you expect that the temperature varies smoothly as you go among nearby points. Other obvious examples of fields are air pressure in a room, height above or below the normal height of water in a pool, or the transverse displacement of a stretched string. With some amount of smoothing you can make a field from such things as population density on the earth. Any system that is defined over a continuous manifold is a field. 119
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120 CHAPTER 4. 19TH CENTURY PHYSICS The discussion of the previous examples generally did not deal with the time variation. It is not until we endow something with a time dependence that the something becomes interesting. In fact, as we will see, Section 5.4.4, we cannot really talk about energy until we have temporal evolution. In the Fresnel/Young/Huygens construction of the amplitude for light, we elimi- nated the effect of the time variation by “seeing” only the brightness, the amplitude squared, and averaging for long times so that the short time oscillations of the phasers cancelled out, Section 3.4.5. Thus although the brightness as a field can be interpreted as slowly varying there is an intrinsic time variation that makes light especially interesting. In other words, a field is something that is defined over some manifold, usually space, that has a temporal evolution. The rules for the behavior of the field are usually local in the sense that its variation in space and time is determined by what is going on at those points of space at those times.
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