§2.4 Electric work and energy Christopher Crawford PHY 311 2014-02-12

Outline Electric work and energy Energy of a charge distribution Energy density in terms of E field Field lines and equipotentials Drawing field lines Flux x flow analogy Poisson’s equation Curvature of function Green’s functions Helmholtz 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 E1E2 is negative for a dipole (+q, -q)

Velocity field: flux, flow, [and fish]

Electric flux and flow FLUX FLOW FLUX x FLOW Field lines (flux tubes) counts charges inside surface D = ε0E = flux density ~ charge FLOW Equipotential (flow) surfaces counts potential diffs. ΔV from a to b E = flow density ~ energy/charge Closed surfaces because E is conservative FLUX x FLOW Energy density (boxes) counts energy in any volume D  E ~ charge x energy/charge B.C.’s: Flux lines bounded by charge Flow sheets continuous (equipotentials)

Plotting field lines and 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

General solution to Poisson’s equation Expand f(x) as linear combination of delta functions Invert linear Lapacian on each delta function individually

Green’s functions as propagators Action at a distance: G(r’,r) `carries’ potential from source at r' to field point (force) at r In quantum field theory, potential is quantized G(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