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8.022pset3 - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology Department of Physics 8.022 Spring 2004 Assignment 3: Electric potential Post date: Thursday, February 19th Due date: Thursday, February 26th Be sure to write your name and section number on your pset Please staple multiple pages together NOTE: Only 75 points are assigned on this pset. If you completed pset 0, 25 points will be added to your score. 1. Vector calculus identities (10 pts): (a) Curl of gradient is zero: show that ~ ∇ × ~ f · = 0 for any scalar function f . (This can be done very simply using Stokes’ theorem.) (b) Divergence of curl is zero: show that ~ ∇ · ~ ∇ × ~ F · = 0 for any vector function ~ F . (This can be done very simply using Gauss’s theorem — see Purcell problem 2.16.) (c) Associativity of divergence: show that ~ ∇ · f ~ F · = f ~ ∇ · ~ F + ~ F · ~ f . (d) Associativity of curl: show that ~ ∇ × f ~ F · = f ~ ∇ × ~ F + ~ f × ~ F . 2. Purcell 3.1: Charges in a spherical conductor (10 pts).
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