Field energy in a dispersive medium

Section 80

Review energy in non-dispersive dielectrics

U = internal energy difference for body with and without fields, holding entropy and density constant.

Dispersive media dissipate energy

• Mean evolved heat density per unit time Q = <-divS>t • Electromagnetic energy U is not constant.• Net inflow of energy is needed to sustain it.

Assume monochromatic fieldsE = E0e-iwt

dU = (E.dD +H.dB)/4p

dU/dt =

Need to use real expressions in non-linear functions

Result is a constant dU/dt = -divS= <-divS>t =Q

Dissipation of field energy per unit time is given by e” and m”

e” and m” are positive

• Second law of thermodynamicsdQ = TdS > 0

Which is true?

1. Real and imaginary parts of permittivity are always positive.

2. Real part of permittivity can be negative, but the imaginary part is always positive.

3. Both parts of the permittivity can be positive or negative.

Non-monochromatic fields

• Monochromatic fields are a fiction because their durations are finite.

• Instead of dissipation per unit time, consider time-integrated net dissipation.

• Amplitude of nearly monochromatic (e.g. laser) varies slowly.

Any time dependent field can be written as a sum of monochromatic fields

Electric part of net dissipation is a triple integral

Perform integral over w’. The delta function makes w’ -> -w

Transparency ranges

• e” and m” are never zero except at w = 0. • However, they may be very small e”<<|e’|• Then, neglect absorption, reintroduce internal

energy concept.

Purely monochromatic fields do not accumulate U

Amplitude varies slowly