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Review of electrostatics, Gauss’ Law, conservative fields, electrostatic energy, capacitors, Poisson’s and Laplace’s equations, method of images, vector analysis, and techniques for finding potentials. Homework and quiz focusing on boundary conditions, potential, and unique solutions. Discusses scalar potential differences, electric fields, forces, energy, and central forces. Topics include parallel plate configurations, electrostatic boundary conditions, capacitance, potential differences, and separation of variables.
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Electrostatic potential and energyfall 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
Therefore depends only on endpoints. Conservative fields admit potentials • 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
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)
