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§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
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Energy of a charge distribution
• Reminder of meaning: potential x charge = potential energy
• Integrating energy over a continuous distribution
• Continuous version
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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 ...
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Superposition
• Force, electric field, electric potential all superimpose
• Energy is quadratic in fields, not linear
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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)
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Velocity field: flux, flow, [and fish]
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Electric flux and flow
FLUX
FLOW
• Field lines (flux tubes)
• Equipotential (flow) surfaces
counts potential diffs. ΔV from a to b
E = flow density ~ energy/charge
counts charges inside surface
D = ε0E = flux density ~ 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)
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Plotting field lines and equipotentials
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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
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General solution to Poisson’s equation
• Expand f(x) as linear combination of delta functions
• Invert linear Lapacian on each delta function individually
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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?
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Putting it all together…
• Solution of Maxwell’s equations
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