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CHAPTER 3 ELECTRIC FLUX DENSITY, GAUSS'S LAW, AND DIVERGENCE After drawing a few of the fields described in the previous chapter and becoming familiar with the concept of the streamlines which show the direction of the force on a test charge at every point, it is difficult to avoid giving these lines a physical significance and thinking of them as flux lines. No physical particle is projected radially outward from the point charge, and there are no steel tentacles reaching out to attract or repel an unwary test charge, but as soon as the streamlines are drawn on paper there seems to be a picture showing ``something'' is present. It is very helpful to invent an electric flux which streams away symmetri- cally from a point charge and is coincident with the streamlines and to visualize this flux wherever an electric field is present. This chapter introduces and uses the concept of electric flux and electric flux density to solve again several of the problems presented in the last chapter. The work here turns out to be much easier, and this is due to the extremely symmetrical problems which we are solving. 53
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3.1 ELECTRIC FLUX DENSITY About 1837 the Director of the Royal Society in London, Michael Faraday, became very interested in static electric fields and the effect of various insulating materials on these fields. This problem had been bothering him during the past ten years when he was experimenting in his now famous work on induced elec- tromotive force, which we shall discuss in Chap. 10. With that subject completed, he had a pair of concentric metallic spheres constructed, the outer one consisting of two hemispheres that could be firmly clamped together. He also prepared shells of insulating material (or dielectric material, or simply dielectric ) which would occupy the entire volume between the concentric spheres. We shall not make immediate use of his findings about dielectric materials, for we are restrict- ing our attention to fields in free space until Chap. 5. At that time we shall see that the materials he used will be classified as ideal dielectrics. His experiment, then, consisted essentially of the following steps: 1. With the equipment dismantled, the inner sphere was given a known positive charge. 2. The hemispheres were then clamped together around the charged sphere with about 2 cm of dielectric material between them. 3. The outer sphere was discharged by connecting it momentarily to ground. 4. The outer space was separated carefully, using tools made of insulating material in order not to disturb the induced charge on it, and the negative induced charge on each hemisphere was measured. Faraday found that the total charge on the outer sphere was equal in magnitude to the original charge placed on the inner sphere and that this was true regardless of the dielectric material separating the two spheres. He con- cluded that there was some sort of ``displacement'' from the inner sphere to the outer which was independent of the medium, and we now refer to this flux as displacement ,
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