Consider the electric field E(r) defined by equation (10). Note that the integrals in

Chapter 7, Problem 20

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Consider the electric field E(r) defined by equation (10). Note that the integrals in equation (10) are improper in the sense that they become infinite at points r D, where (r) is nonzero. In this exercise, you will show that, nonetheless, the integrals in equation (10) converge when D is a bounded region in R3 and is a continuous charge density function on D. (a) Write E(r) in terms of triple integrals for the individual components. Let r = (r1,r2,r3) and x = (x, y,z). (b) Show that if each component of E is written in the form D f (x) dV, then | f (x)| K/ r x 2, where K is a positive constant. (c) It follows from part (b) that if D K r x 2 dV converges, so must D f (x) dV. Show that D K r x 2 dV converges by considering an iterated integral in spherical coordinates with origin at r. (Hint: Look carefully at the integrand in spherical coordinates.)