§2.4 Electric work and energy Christopher Crawford PHY 416 2014-10-13

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)

Flux × Flow: energy/power density

Electric flux and flow FLUX FLOW FLUX x FLOW = ENERGY 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 Energy density (boxes) counts energy in any volume D  E ~ charge x energy/charge FLUX x FLOW = ? 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

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