Simple Nature - Light and Matter

of the inward field contributed by the outside sphere is{ 0, r < b|E − | =kq/r 2 , r > b.In the region outside the whole capacitor, the two fields are equalin magnitude, but opposite in direction, so they cancel. We thenhave for the total field⎧⎨ 0, r < a|E| = kq/r 2 , a < r < b ,⎩0, r > bso to calculate the energy, we only need to worry about the regiona < r < b. The energy density in this region isdU edv = 18πk E 2= kq28π r −4 .This expression only depends on r, so the energy density is constantacross any sphere of radius r. We can slice the regiona < r < b into concentric spherical layers, like an onion, andthe energy within one such layer, extending from r to r + dr isdU e = dU edv dv= dU e(area of shell)(thickness of shell)dv= ( kq28π r −4 )(4πr 2 )(dr)= kq22 r −2 dr .Integrating over all the layers to find the total energy, we have∫U e ==∫ badU ekq 22 r −2 dr= − kq22 r −1 ∣ ∣∣∣b= kq22a( 1a − 1 b)586 Chapter 10 Fields