110 likes | 126 Views
Explore the fundamentals of electric work and energy, including field lines, Poisson’s equation, energy density, and Green’s functions. Discover the philosophical questions behind energy storage in electric fields and superposition principles.
E N D
§2.4Electric work and energy Christopher Crawford PHY 416 2014-10-13
Outline • Electric work and energyEnergy of a charge distributionEnergy density in terms of E field • Field lines and equipotentialsDrawing field linesFlux x flow analogy • Poisson’s equationCurvature of functionGreen’s functionsHelmholtz theorem
Energy of a charge distribution • Reminder of meaning: potential x charge = potential energy • Integrating energy over a continuous distribution • Continuous version
Energy of the electric field • Integration by parts • Derivative chain • Philosophical questions: • is the energy stored in the field, or in the force between the charges? • is the electric field real, or just a calculational device? potential field? • if a tree falls in the forest ...
Superposition • Force, electric field, electric potential all superimpose • Energy is quadratic in fields, not linear • the cross term is the `interaction energy’ between two distributions • the work required to bring two systems of charge together • W1 and W2 are infinite for point charges – self-energy • E1E2 is negative for a dipole (+q, -q)
Electric flux and flow FLUX • Field lines (flux tubes)counts charges inside surfaceD = ε0E = flux density ~ charge FLOW • Equipotential (flow) surfacescounts potential diffs. ΔV from a to bE = flow density ~ energy/charge Closed surfaces because E is conservative FLUX x FLOW = ENERGY • Energy density (boxes)counts energy in any volumeD E ~ charge x energy/charge FLUX x FLOW = ? B.C.’s:Flux lines bounded by charge Flow sheets continuous (equipotentials)
Green’s function G(r,r’) • The potential of a point-charge • A simple solution to the Poisson’s equation • Zero curvature except infinite at one spot
Green’s functions as propagators • Action at a distance: G(r’,r) `carries’ potentialfrom source at r' to field point (force) at r • In quantum field theory, potential is quantizedG(r’,r) represents the photon (particle) that carries the force • How to measure `shape’ of the proton?
Putting it all together… • Solution of Maxwell’s equations