Finding electrostatic potential Griffiths Ch.3: Special Techniques week 3 fall EM lecture, 14.Oct.2002, Zita, TESC Review electrostatics: E, V, boundary.

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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 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 Electrostatic energy: Homework and quiz

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: 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

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: has solutions V(x) = mx+b specify two points or point + slope Dirichlet and von Neumann BC

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. Cartesian coordinates: Prob.3.12 (worksheet) Cylindrical coordinates: Prob.3.23 (worksheet)

1.1.3 Triple Products, by Andy Syltebo : Ordinary derivatives + Gradient, by Don Verbeke 1.2.3: Del operator, by Andrew White