Simple Nature - Light and Matter

that are closest to the central axis.) There is no obvious, pedestrianway to determine the field or voltage in the central vacuum region,but there’s a trick that works: with a little mathematical insight,we see that the voltage V = V o b −2 xy is consistent with all thegiven information. (Mathematicians could prove that this solutionwas unique, but a physicist knows it on physical grounds: if therewere two different solutions, there would be no physical way forthe system to decide which one to do!) (a) Use the techniques ofsubsection 10.2.2 to find the field in the vacuum region, and (b)√sketch the field as a “sea of arrows.”21 (a) A certain region of three-dimensional space has a voltagethat varies as V = br 2 , where r is the distance from the origin. Use√the techniques of subsection 10.2.2 to find the field.(b) Write down another voltage that gives exactly the same field.22 (a) Example 10 on page 574 gives the field of a charged rod inits midplane. Starting from this result, take the limit as the lengthof the rod approaches infinity. Note that λ is not changing, so as Lgets bigger, the total charge Q increases. ⊲ Answer, p. 930(b) In the text, I have shown (by several different methods) that thefield of an infinite, uniformly charged plane is 2πkσ. Now you’regoing to rederive the same result by a different method. Supposethat it is the x − y plane that is charged, and we want to find thefield at the point (0, 0, z). (Since the plane is infinite, there is noloss of generality in assuming x = 0 and y = 0.) Imagine that weslice the plane into an infinite number of straight strips parallel tothe y axis. Each strip has infinitesimal width dx, and extends fromx to x + dx. The contribution any one of these strips to the fieldat our point has a magnitude which can be found from part a. Byvector addition, prove the desired result for the field of the plane ofcharge.23 Consider the electric field created by a uniformly chargedcylindrical surface that extends to infinity in one direction.(a) Show that the field at the center of the cylinder’s mouth is 2πkσ,which happens to be the same as the field of an infinite flat sheet ofcharge!(b) This expression is independent of the radius of the cylinder.Explain why this should be so. For example, what would happen ifyou doubled the cylinder’s radius?24 In an electrical storm, the cloud and the ground act like aparallel-plate capacitor, which typically charges up due to frictionalelectricity in collisions of ice particles in the cold upper atmosphere.Lightning occurs when the magnitude of the electric field builds upto a critical value, E c , at which air is ionized.(a) Treat the cloud as a flat square with sides of length L. If it is ata height h above the ground, find the amount of energy released inProblem 23.Problems 637