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Finding electrostatic potential Griffiths Ch.3: Special Techniques week 3 fall EM lecture, 14.Oct.2002, Zita, TESC • Review electrostatics: E, V, boundary conditions, energy • Homework and quiz • Ch.3: Techniques for finding potentials: Why and how? • 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 Cartesian, 3.23 Cylindrical) • Your minilectures on vector analysis (choose one prob. each) Review of electrostatics Electrostatic BC and energy Boundary conditions across a surface charge: potential and E|| are continuous; discontinuity in E equal to 0 Electrostatic energy: W 1 2 d Homework and quiz 0 2 2 E d Ch.3: Techniques for finding electrostatic potential V Why? • Easy to find E from V • Scalar V superpose easily How? • Poisson’s and Laplace’s equations (Prob. 3.3 p.116 last week) • Guess if possible: unique solution for given BC • Method of images (Prob. 3.9 p.126) • Separation of variables Poisson’s equation Gauss: E 0 Potential: combine to get Poisson’s eqn: Laplace equation holds in charge-free regions: • V(r) = average of V of neighboring points • no local max or min in V(r) NB: proof of shell theorem in Section 3.1.4, p.114 E V Uniqueness theorems: (1) The solution to Laplace’s eqn. in some volume is uniquely determined if V is specified on the boundary surface. (cf Fig.3.5 p.117) (2) In a volume surrounded by conductors and containing a speciried charge density, the electric field is uniquely determined if the total charge on each conductor is given. (cf Fig.3.6 p.119) Elegant proof in Prob.3.5 p.121. (cf Z.34) Solution V depends on boundary conditions: V V 0 2 x 2 2 has solutions V(x) = mx+b specify two points Dirichlet and von Neumann BC or point + slope Method of images A charge distribution in space induces on a nearby conductor. The total field results from combination of and . + - • Guess an image charge that is equivalent to . • Satisfy Poisson and BC, and you have THE solution. Prob.3.9 p.126 (cf 2.2 p.82) Separation of variables Guess that solution to Laplace equation is a product of functions in each variable. If that works, the diffeq is separable, and boundary conditions will determine the unknown constants. V=0 y=a V=? y=0 V=0 Cartesian coordinates: Prob.3.12 (worksheet) Cylindrical coordinates: Prob.3.23 (worksheet) x 1.1.3 Triple Products, by Andy Syltebo 1.2.1+1.2.2: Ordinary derivatives + Gradient, by Don Verbeke 1.2.3: Del operator, by Andrew White