One resonance, and the two halves of a permittivity

At its defaults it draws one resonance, and the two halves of a permittivity. The real and imaginary parts of the permittivity of a single Lorentz oscillator, against frequency in units of its own resonance. Away from the resonance ε′ rises slowly with frequency and ε″ is negligible: that is ordinary dispersion. Between 0.938 and 1.058 of the resonant frequency, ε′ *falls* — the anomalous band, shaded — and it falls precisely where the absorption is large, which is not a coincidence but the content of the dispersion relation. ε″ peaks at 7.500 at the resonance itself. The two curves are not two properties of the material. They are one analytic function evaluated on the real axis, and either determines the other everywhere.

dielectric-response is one function in lib/figures/fields.js — charge, current, flux and the lines drawn between them. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

The real and imaginary parts of the permittivity of a single Lorentz oscillator, against frequency in units of its own resonance. Away from the resonance ε′ rises slowly with frequency and ε″ is negligible: that is ordinary dispersion. Between 0.938 and 1.058 of the resonant frequency, ε′ *falls* — the anomalous band, shaded — and it falls precisely where the absorption is large, which is not a coincidence but the content of the dispersion relation. ε″ peaks at 7.500 at the resonance itself. The two curves are not two properties of the material. They are one analytic function evaluated on the real axis, and either determines the other everywhere.

Field lines crossing the boundary between two dielectrics whose permittivities differ by a factor of 4, at incidences of 12°, 26°, 40°, 54°, 68°. Nothing is oscillating and nothing is travelling: these are static fields, and the only inputs are that the component of E along the surface is the same on both sides and that the component of D across it is. Dividing one condition by the other gives the ratio of the tangents of the two angles, and it comes out equal to the ratio of the permittivities — measured off the constructed directions as 4.000, 4.000, 4.000, 4.000, 4.000, against 4. The lines bend towards the surface on the side with the larger permittivity, which is the opposite sense from a light ray entering glass, and the chart on the right shows the whole relation: it is a version of Snell's law with the sines replaced by tangents, and there is no speed anywhere in it. Two things follow that are worth carrying. A field line meeting a surface at grazing incidence stays nearly parallel to it whatever the materials, so the ratio matters least where the field is largest along the surface. And in the limit of a very large ratio every line comes out very nearly perpendicular on the low side, which is the electrostatic ancestor of a conductor's boundary condition — a conductor is the ratio taken to infinity.

Field lines crossing the boundary between two dielectrics whose permittivities differ by a factor of 0.25, at incidences of 12°, 26°, 40°, 54°, 68°. Nothing is oscillating and nothing is travelling: these are static fields, and the only inputs are that the component of E along the surface is the same on both sides and that the component of D across it is. Dividing one condition by the other gives the ratio of the tangents of the two angles, and it comes out equal to the ratio of the permittivities — measured off the constructed directions as 0.250, 0.250, 0.250, 0.250, 0.250, against 0.25. The lines bend towards the surface on the side with the larger permittivity, which is the opposite sense from a light ray entering glass, and the chart on the right shows the whole relation: it is a version of Snell's law with the sines replaced by tangents, and there is no speed anywhere in it. Two things follow that are worth carrying. A field line meeting a surface at grazing incidence stays nearly parallel to it whatever the materials, so the ratio matters least where the field is largest along the surface. And in the limit of a very large ratio every line comes out very nearly perpendicular on the low side, which is the electrostatic ancestor of a conductor's boundary condition — a conductor is the ratio taken to infinity.

Field lines crossing the boundary between two dielectrics whose permittivities differ by a factor of 80, at incidences of 10°, 25°, 45°, 65°, 80°. Nothing is oscillating and nothing is travelling: these are static fields, and the only inputs are that the component of E along the surface is the same on both sides and that the component of D across it is. Dividing one condition by the other gives the ratio of the tangents of the two angles, and it comes out equal to the ratio of the permittivities — measured off the constructed directions as 80.000, 80.000, 80.000, 80.000, 80.000, against 80. The lines bend towards the surface on the side with the larger permittivity, which is the opposite sense from a light ray entering glass, and the chart on the right shows the whole relation: it is a version of Snell's law with the sines replaced by tangents, and there is no speed anywhere in it. Two things follow that are worth carrying. A field line meeting a surface at grazing incidence stays nearly parallel to it whatever the materials, so the ratio matters least where the field is largest along the surface. And in the limit of a very large ratio every line comes out very nearly perpendicular on the low side, which is the electrostatic ancestor of a conductor's boundary condition — a conductor is the ratio taken to infinity.

The refractive index n and the extinction coefficient κ of the same single-resonance material, obtained as the square root of the complex permittivity. Below the resonance n rises with frequency, which is why a prism spreads blue further than red. Above 1.058 of the resonant frequency n is *less than one*, so the phase velocity exceeds c — a fact about the speed of a crest of an infinite wave and not about the speed of anything that carries a signal. κ is large only in the band where n is behaving backwards, and a material is transparent exactly where the two curves are far apart.

Field lines crossing the boundary between two dielectrics whose permittivities differ by a factor of 12, at incidences of 10°, 25°, 45°, 65°, 80°. Nothing is oscillating and nothing is travelling: these are static fields, and the only inputs are that the component of E along the surface is the same on both sides and that the component of D across it is. Dividing one condition by the other gives the ratio of the tangents of the two angles, and it comes out equal to the ratio of the permittivities — measured off the constructed directions as 12.000, 12.000, 12.000, 12.000, 12.000, against 12. The lines bend towards the surface on the side with the larger permittivity, which is the opposite sense from a light ray entering glass, and the chart on the right shows the whole relation: it is a version of Snell's law with the sines replaced by tangents, and there is no speed anywhere in it. Two things follow that are worth carrying. A field line meeting a surface at grazing incidence stays nearly parallel to it whatever the materials, so the ratio matters least where the field is largest along the surface. And in the limit of a very large ratio every line comes out very nearly perpendicular on the low side, which is the electrostatic ancestor of a conductor's boundary condition — a conductor is the ratio taken to infinity.

physicscheck asserts something about dielectric-response that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.