1. §2.4 Electric work and energy
Christopher Crawford
PHY 416

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. Flux × Flow: energy/power density

7. 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)

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. 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?

11. Putting it all together…
Solution of Maxwell’s equations