Electrostatic potential and energy fall EM lecture, week 2, 7. Oct

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Electrostatic potential and energy fall EM lecture, week 2, 7. Oct Electrostatic potential and energy fall EM lecture, week 2, 7.Oct.2002, Zita, TESC Homework and quiz Review electrostatics, Gauss’ Law: charges  E field Conservative fields and path independence  potential V Boundary conditions (Ex. 2.5 p.74, Prob. 2.30 p.90) Electrostatic energy (Prob. 2.40 p.106), capacitors (Ex. 2.10 p.104) Start Ch.3: Techniques for finding potentials: V  E Poisson’s and Laplace’s equations (Prob. 3.3 p.116), uniqueness Method of images (Prob. 3.9 p.126) Your minilectures on vector analysis (choose one prob. each)

Ch.2: Electrostatics (d/dt=0): charges  fields  forces, energy Charges make E fields and forces charges make scalar potential differences dV E can be found from V Electric forces move charges Electric fields store energy (capacitance) F = q E = m a W = qV, C = q/V

Conservative fields admit potentials  depends only on endpoints. Therefore Easy to find E from V is independent of choice of reference point V=0 V is uniquely determined by boundary conditions Every central force (curl F = 0) is conservative (prob 2.25) Ex.2.5 p.74: parallel plates

Parallel plates

Electrostatic boundary conditions: E is discontinuous across a charge layer: DE = s/e0 E|| and V are continuous Prob 2.30 (a) p.90: check BC for parallel plates

Electrostatic potential: units, energy Prob. 2.40 p.106: Energy between parallel plates Ex. 2.10 p.104: Find the capacitance between two metal plates of surface area A held a distance d apart.

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) 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)

1.1.1 Vector Operations - by

1.1.2 Vector Algebra - by

1.1.3 Triple Products - by