Consider the potential to arise from the charges themselves. The factor 1/2 is necessary to avoid double counting. To appreciate this factor consider just two-point charges: We only need to count the energy due to one charge in the presence of the other’s potential to obtain the energy required to bring the charges together. A substitution of ρ (r) gives
We can consider the volume V (enclosed by S) to be as large as we like, say a sphere of radius R so that all charges are contained well within it. Then the surface integral over S vanishes as R → ∞ since φ ∼ 1/R and E ∼ 1/R 2, whereas d an ∼ R 2 . Then the total energy is expressed solely in terms of the electric field.
Energy Density of Magnetic Fields
n a derivation similar, we consider the energy associated with magnetic fields The magnetic vector potential A(r) (in units of energy per charge velocity) describes the potential energy that a charge moving with velocity v would experience if placed in the field. The magnetic field and the vector potential are connected through
From for a gas with negligible absorption (i.e. γ ∼= 0, valid far from resonance ω0), where λ0, vat corresponds to frequency ω0 and A is a constant. Many materials (e.g. glass, air) have strong resonances in the ultraviolet. In such materials, do you expect the index of refraction for the blue light to be greater than that for red light Make a sketch of n as a function of wavelength for visible light down to the ultraviolet (where λ0, vac is located).
In the Lorentz model, take N = 1028 m−3 for the density of bound electrons in an insulator, and a single transition at ω0 = 6×1015 rad/sec (in the UV), and damping γ = ω0/5 (quite broad). Assume E0 is 104 V/m. For three frequencies ω = ω0−2γ, ω = ω0, and ω = ω0+2γ find the magnitude and phase (relative to the phase of E0e i(k·r−ωt) ) of the following quantities. Give correct SI units with each quantity. You don’t need to worry about vector directions
For silver, the complex refractive index is characterized by n = 0.13 and κ = 4.0.24 Find the distance that light travels inside of silver before the field is reduced by a factor of 1/e. Assume a wavelength of van = 633 nm. What is the speed of the wave crests in the silver (written as a number times c)? Are you surprised? P2.6 Use and expressions that follow to calculate the index of silver at λ = 633nm. The density of free electrons in silver is N = 5.86×1028m−3 and the DC conductivity is σ = 6.62×107C 2 /(Jam· s). 25 Compare with the actual index given in P2.5.
Last word
In the case of a linearly-polarized plane wave, where the phase of each vector component of E0 is the same, re-derive directly from the real field. For simplicity, you may ignore absorption (i.e. κ ∼= 0). HINT: The time average of cos2 ¡ k· rit +φ ¢ is 1/2. P2.10 (a) Find the intensity (in W/cm2 ) produced by a short laser pulse with duration ∆t = 2.5×10−14 s and energy E = 100 MJ, focused in vacuum to a round spot with radius r = 5 µm.
