§2.4 Electric work and energy

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1 §2.4 Electric work and energy
Christopher Crawford PHY 311

2 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

3 Energy of a charge distribution
Reminder of meaning: potential x charge = potential energy Integrating energy over a continuous distribution Continuous version

4 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 ...

5 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)

6 Velocity field: flux, flow, [and fish]

7 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)

8 Plotting field lines and equipotentials

9 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

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

11 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?

12 Putting it all together…
Solution of Maxwell’s equations