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EM & Vector calculus #3 Physical Systems, Tuesday 30 Jan 2007, EJZ

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 #3 Physical Systems, Tuesday 30 Jan 2007, EJZ

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  1. 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)

  2. 1.2.6 Product rules

  3. 1.2.7 Second derivatives

  4. 1.3.1 Line, Surface, Volume integrals

  5. 1.3.2 Fundamental theorem of calculus

  6. 1.3.3 Fundamental theorem for Gradients

  7. 1.3.4 Fundamental theorem for Divergences

  8. 1.3.5 Fundamental theorem for Curls

  9. Consequences Gauss’s law and fundamental theorem for divergences: Ampere’s Law and fundamental theorem for curls:

  10. 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)

  11. 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)

  12. 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)

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