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EM & Vector calculus #3 Physical Systems, Tuesday 30 Jan 2007, EJZ. Vector Calculus 1.3: Integral Calculus Line, surface, volume integrals Fundamental theorems Integration by parts Ch.3a: Special Techniques (Electrostatics) Quick homework review
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EM & Vector calculus #3Physical Systems, Tuesday 30 Jan 2007, EJZ • Vector Calculus 1.3: Integral Calculus • Line, surface, volume integrals • Fundamental theorems • Integration by parts • Ch.3a: Special Techniques (Electrostatics) • Quick homework review • Poisson’s and Laplace’s equations (Prob. 3.3 p.116) • Uniqueness • Method of images (Prob. 3.9 p.126)
Consequences Gauss’s law and fundamental theorem for divergences: Ampere’s Law and fundamental theorem for curls:
E&M Ch.3: Techniques for finding V • Why? • Easy to find E from V • Scalar V superpose easily • How? • Poisson’s and Laplace’s equations (Prob. 3.3 p.116) • Guess if possible: unique solution for given BC • Method of images (Prob. 3.9 p.126) • Separation of variables (next week)
Poisson’s equation Gauss: Potential: combine to get Poisson’s eqn: Laplace equation holds in charge-free regions: Prob.3.3 (p.116): Find the general solution to Laplace’s eqn. In spherical coordinates, for the case where V depends only on r. Do the same for cylindrical coordinates, assuming V(s). (See Laplacian on p.42 and 44)
Method of images • A charge distribution r induces s on a nearby conductor. • The total field results from combination of r and s. • + - • Guess an image charge that is equivalent to s. • Satisfy Poisson and BC, and you have THE solution. • Prob.3.9 p.126 (cf 2.2 p.82)