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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)
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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)
