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Electromagnetism Ch.3 Methods of Math. Physics, Thus. 11 Feb. 2011, EJZ Special Techniques in Electrostatics • Review HW? • Poisson’s and Laplace’s equations (Prob. 3.3 p.116) • Uniqueness • Method of images (Prob. 3.9 p.126) • Separation of variables (Prob. 3.12, 3.23) Techniques for finding V (potential, voltage) 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 (Spherical harmonics, Fourier series…) Poisson’s equation Gauss: E 0 Potential: E V combine to get Poisson’s eqn: Laplace’s equation holds in charge-free regions: d 2V 0 in 1-D (x): Solve 2 dx Laplace’s equation V 0 2 → ∂V =____ → ∂V =____ Method of images A charge distribution induces s on a nearby conductor. The total field results from combination of 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) Method of Images: #3.9 p.126 Now put this line of charge above a conducting plane Guess that the induced charge on the plane has the same field as a line of (-) charge the same distance BELOW the sheet. By symmetry, V=0 at the conducting sheet. This satisfies BC. Simplify the ln terms, and write in terms of z, d, and y… Simplify, and write up your solution for homework. break time … Separation of Variables When to use separation of variables? • In charge-free regions • With well-specified boundary conditions (BC) • Without sufficient symmetry for Gauss’ law How to use separation of variables? • Guess form of solutions based on BC • Separate variables, insert guessed solutions with constants • Apply BC and solve for constants Review Poisson and Laplace equations Gauss: E 0 combine to get Poisson’s eqn: E V Potential: V 2 0 Laplace equation holds in charge-free regions: 2V 0 We have found the general solutions to Laplace’s eqn. in spherical and cylindrical coordinates when V=V(r) … Solving Laplace w/ Separation of Variables V 0 2 Worksheets for Problems 3.12 (136), 3.23 (145) Homework due next week: work through Ex.3.3, do 3.12 and 3.23. Extra credit: #13, 24. Other good problems: #16, 17, 18 (p.144)