Electromagnetism Archives - Time Travel, Quantum Entanglement and Quantum Computing https://stationarystates.com/category/electromagnetism/ Not only is the Universe stranger than we think, it is stranger than we can think...Hiesenberg Tue, 30 Sep 2025 19:35:07 +0000 en-US hourly 1 https://wordpress.org/?v=6.9.1 Fields of an accelerating charge https://stationarystates.com/electromagnetism/fields-of-an-accelerating-charge/?utm_source=rss&utm_medium=rss&utm_campaign=fields-of-an-accelerating-charge Tue, 30 Sep 2025 15:57:30 +0000 https://stationarystates.com/?p=990   Electric Field of an Accelerating Electron For a point charge (electron) in motion, the electric field at an observation event can be decomposed into two parts: a velocity (near/Coulomb) […]

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Electric Field of an Accelerating Electron

For a point charge (electron) in motion, the electric field at an observation event can be decomposed into two parts:

  • a velocity (near/Coulomb) field that scales like 1/R^2, and
  • an acceleration (radiation) field that scales like 1/R.

Below, we first present the low-velocity (nonrelativistic) result for intuition, and then the fully relativistic (Liénard–Wiechert) fields.

Geometry and Retarded Kinematics

Let the source (electron) be at retarded position \mathbf{r}_s(t_r) with velocity \mathbf{v}(t_r) and acceleration \mathbf{a}(t_r). Define:

    \[ \mathbf{R}=\mathbf{r}-\mathbf{r}_s(t_r),\qquad R=\|\mathbf{R}\|,\qquad \mathbf{n}=\frac{\mathbf{R}}{R},\qquad \boldsymbol{\beta}=\frac{\mathbf{v}}{c},\qquad \dot{\boldsymbol{\beta}}=\frac{\mathbf{a}}{c}. \]

All source quantities are to be evaluated at the retarded time t_r, defined implicitly by t_r = t - R(t_r)/c.

Nonrelativistic Limit (\|\boldsymbol{\beta}\|\ll 1)

Keeping terms to leading order in \beta, the electric field separates cleanly into a velocity piece \mathbf{E}_{\text{vel}}\propto 1/R^2 and an acceleration piece \mathbf{E}_{\text{acc}}\propto 1/R:

    \[ \boxed{\; \mathbf{E}(\mathbf{r},t) = \underbrace{\frac{q}{4\pi\varepsilon_0}\,\frac{\mathbf{n}}{R^{2}}}_{\displaystyle \mathbf{E}_{\text{vel}}} \;+\; \underbrace{\frac{q}{4\pi\varepsilon_0 c^{2}}\,\frac{\mathbf{n}\times\left(\mathbf{n}\times \mathbf{a}\right)}{R}}_{\displaystyle \mathbf{E}_{\text{acc}}} \Bigg|_{t_r}\;} \]

Key features:

  • \mathbf{E}_{\text{vel}} is essentially the instantaneous Coulomb field (evaluated at the retarded time).
  • \mathbf{E}_{\text{acc}} is transverse (\mathbf{n}\cdot \mathbf{E}_{\text{acc}}=0) and falls off as 1/R: it is the radiation field.

The magnetic field follows from \mathbf{B}=\mathbf{n}\times \mathbf{E}/c in this limit.

Fully Relativistic Fields (Liénard–Wiechert)

For arbitrary velocities and accelerations, the exact fields are the Liénard–Wiechert fields. They naturally split into a velocity part (near field) and an acceleration part (radiation field):

    \[ \boxed{\; \mathbf{E}(\mathbf{r},t) = \frac{q}{4\pi\varepsilon_0} \left[ \frac{\mathbf{n}-\boldsymbol{\beta}}{\gamma^{2}\left(1-\mathbf{n}\cdot\boldsymbol{\beta}\right)^{3}R^{2}} \;+\; \frac{\mathbf{n}\times\left[(\mathbf{n}-\boldsymbol{\beta})\times \dot{\boldsymbol{\beta}}\right]}{c\left(1-\mathbf{n}\cdot\boldsymbol{\beta}\right)^{3}R} \right]_{t_r} \;} \]

    \[ \boxed{\; \mathbf{B}(\mathbf{r},t)=\left[\mathbf{n}\times \mathbf{E}(\mathbf{r},t)\right]_{t_r} \;} \]

Here \gamma=(1-\beta^{2})^{-1/2}. The first bracketed term is the velocity field (\propto R^{-2}); the second is the acceleration (radiation) field (\propto R^{-1}). Both include the characteristic beaming and retardation factor \bigl(1-\mathbf{n}\cdot\boldsymbol{\beta}\bigr)^{-3}.

Checks and Limits

  • Low-velocity check: For \beta\to 0, \gamma\to 1, and keeping leading order in \beta, the expressions reduce to the nonrelativistic formulas above.
  • Transversality of radiation: The acceleration term is transverse to \mathbf{n}, as seen from the double cross product.
  • Beaming: At relativistic speeds, fields concentrate in a narrow cone around the instantaneous velocity direction due to the \bigl(1-\mathbf{n}\cdot\boldsymbol{\beta}\bigr)^{-3} factor.

Sketch of Derivation (from Retarded Potentials)

  1. Start with the retarded Liénard–Wiechert potentials

        \[ \phi(\mathbf{r},t)=\frac{q}{4\pi\varepsilon_0}\,\frac{1}{\bigl(1-\mathbf{n}\cdot\boldsymbol{\beta}\bigr)R}\Big|_{t_r},\qquad \mathbf{A}(\mathbf{r},t)=\frac{\boldsymbol{\beta}}{c}\,\phi(\mathbf{r},t). \]

  2. Compute fields via

        \[ \mathbf{E}=-\nabla \phi-\frac{\partial \mathbf{A}}{\partial t},\qquad \mathbf{B}=\nabla\times \mathbf{A}, \]

    carefully accounting for the implicit dependence of t_r in \mathbf{n},R,\boldsymbol{\beta}.

  3. Group terms by their distance scaling: R^{-2} (velocity field) and R^{-1} (acceleration field). The algebra yields the boxed expressions above.

Power and Radiation (for context)

The far (acceleration) field governs radiation. The instantaneous radiated power is given by the relativistic Liénard formula,
\[
P=\frac

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]]> Kramers Kronig Violation https://stationarystates.com/electromagnetism/kramers-kronig-violation/?utm_source=rss&utm_medium=rss&utm_campaign=kramers-kronig-violation Mon, 22 Sep 2025 15:02:24 +0000 https://stationarystates.com/?p=979 Kramers–Kronig Relations: Can Materials Violate Them? Do Any Materials Violate the Kramers–Kronig (KK) Relations? Short answer: No passive, linear, time-invariant material violates the KK relations. They follow from causality (response […]

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Kramers–Kronig Relations: Can Materials Violate Them?




Do Any Materials Violate the Kramers–Kronig (KK) Relations?

Short answer: No passive, linear, time-invariant material violates the KK relations.
They follow from causality (response cannot precede the stimulus), together with linearity and time invariance.
If any of those assumptions is broken, the textbook KK formulas need modification and may not apply in their simple form.

1) Why KK holds

  • The KK relations link the real and imaginary parts of a linear response function (e.g., refractive index \(n(\omega)\),
    dielectric function \(\epsilon(\omega)\), magnetic permeability \(\mu(\omega)\)).
  • They are derived from the Cauchy integral formula applied to a complex susceptibility \(\chi(\omega)\) that is
    (i) analytic in the upper half of the complex \(\omega\)-plane and
    (ii) vanishes sufficiently fast as \(\lvert \omega \rvert \to \infty\).
  • Those properties follow from causality + stability for any linear, time-invariant, passive medium.

2) Situations where the standard form may appear to fail

Scenario What happens Does KK really fail?
Active / gain media Laser amplifiers or negative-resistance systems can have poles in the upper half-plane; dispersion–absorption pairing needs extra pole (residue) terms. No fundamental violation; use generalized KK including pole contributions.
Nonlinear response KK assumes linear response to the probe. Strong-field or intensity-dependent effects do not obey the linear KK pair. KK applies to the linear susceptibility (small-signal limit) only.
Time-variant / modulated systems Material parameters evolve in time; the system is not time-invariant, so stationary KK relations are inapplicable directly. Not a violation—assumptions changed.
Spatial dispersion Response depends on both frequency and wavevector, e.g., \(\epsilon(\omega,k)\); the simple 1D KK in \(\omega\) needs generalization. Generalized relations exist; the textbook form is insufficient.
Experimental “violations” Limited-band / noisy data can appear inconsistent because KK integrals require information over all frequencies. Artefacts, not real violations.

3) Reported “KK-violating” claims

Occasional claims involve, for example, negative-index metamaterials near resonances, “superluminal” pulse propagation,
or gain-assisted transparency windows. Closer analysis typically shows that the medium has gain (pumped/active),
is nonlinear, or the dataset is incomplete. When these are accounted for, the appropriate generalized KK relations hold.

4) Bottom line

As long as a system is linear, causal, and stable, its response functions obey Kramers–Kronig.

  • Apparent violations usually indicate gain/active elements, nonlinearity, time variance, spatial dispersion, or incomplete data.
  • No reproducible, peer-reviewed experiment has shown a passive, linear medium whose susceptibility truly violates KK.

References (classics & reviews)

  • L. D. Landau & E. M. Lifshitz, Electrodynamics of Continuous Media, §85.
  • R. W. Boyd, Nonlinear Optics, ch. 2 (KK in the context of nonlinear susceptibilities).
  • N. Nussenzveig, Causality and Dispersion Relations.
  • V. Lucarini, J. J. Saarinen, K.-E. Peiponen, E. Vartiainen, Kramers–Kronig Relations in Optical Materials Research (Springer, 2005).


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