Basiswissen

P = a²·q²/(6·π·ε₀·c³): verliert ein (nicht-relativistisch) beschleunigtes Teilchen durch die Wirkung eines elektromagnetischen Feldes Energie, muss diese in Form elektromagnetischer Strahlung emittiert werden: die Larmor-Formel ist eine Formel aus der klassischen Elektrodynamik (nicht relativistisch), aus der die abgestrahlte Leistung P eines mit a beschleunigten Teilchens mit der Ladung q berechnet werden kann. ε₀ ist die sogenannte elektrische Feldkonstante und c die Geschwindigkeit von Licht im Vakuum.

Bildbeschreibung und Urheberrecht — Die Larmor-Formel für den Energieverlust einer beschleunigtn Punktladung im Vakuum: P = Strahlungsleistung, q = Punktladung, π = Kreiszahl, c = Lichtgeschwindigkeit, ε₀ = elektrische Feldkonstante. Ob diese Formel für alle beschleunigten Ladungen generell gilt, ist umstritten. © Gunter Heim/ChatGPT ☛

Schreibweisen

P = a²·q²/(6·π·ε₀·c³)^[4]

P = µ₀·a²·q²/(6·π·c)^[4]

Legende

P = die abgestrahle Leistung [z. B. in Watt] ↗

a = die Beschleunigung [der Ladung q] ↗

q = die Ladungsmenge [die beschleunigt wird] ↗

π = die Kreiszahl [etwa 3,14] ↗

ε₀ = die elektrische Feldkonstante ↗

µ₀ = die magnetische Feldkonstante ↗

c = die Vakuumlichtgeschwindigkeit ↗

· = ein Malzeichen ↗

/ = ein Geteiltzeichen ↗

Fußnoten

[1] Joseph Larmor: Dynamical Theory of the Electric and Luminiferous. Medium. Part III. Relations with Material Media. 1897. Online: https://royalsocietypublishing.org/doi/epdf/10.1098/rspl.1897.0036

[2] Larmor formuliert als grundlegende Fragestellung die Art und Weise wie bewegte Ladungen und der hypothetische Äther aufeinander wirken: "A formula for the distribution of the electric energy throughout the aether was suggested by various considerations, and from it an attempt was made by the methods of general dynamics to establish the laws of the force exerted by the aether on the differentparts of conductors conveying currents : and it was natural that the same procedure should be extended to an attempt to place the fundamental formal equations of the aether itself on a dynamical basis. But what was lacking for the satisfactory accomplishment of this purpose was a definite and consistent idea of how the electric charges and currents in the matter established a hold on the aether." In: Joseph Larmor: Dynamical Theory of the Electric and Luminiferous. Medium. Part III. Relations with Material Media. 1897.

[3] Larmor fasst kurz Probleme mit der zeitgenössischen Vorstellung von Elektrizität zusammen. Die Lösung liegt - unter anderem - in der Vorstellung von Teilchen, den Elektronen, als Trägern elektrischer Ladung: "This [die erwähnten Probleme] led to the introduction of the mobile electron or atomic charge of electricity as the true physical element, and to a dynamical theory of molecular type which is held to be self-consistent and in full agreement with experimental knowledge, and which may be regarded as in a manner final development of the Weberian notion of moving electric particles." In: Joseph Larmor: Dynamical Theory of the Electric and Luminiferous. Medium. Part III. Relations with Material Media. 1897.

[4] Die Schreibweise stammt von Michael Fowler, ehemals Professor an der University of Virginia (USA). In seiner Vorlesung über Galilei und Einstein behandelt er die Larmor-Formel ausführlich im Zusammenhang mit dem elektrischen Feld einer beschleunigten Ladung. Abgerufen am 23. Juni 2023. Online: https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_66_Radiation_Accelerating_Charge.html

[5] Eingeschränkte Gültigkeit? "The theory of linear acceleration emission is developed for a large amplitude electrostatic wave in which all particles become highly relativistic in much less than a wave period. An Airy integral approximation is shown to apply near the phases where the electric field passes through zero and the Lorentz factors of all particles have their maxima. The emissivity is derived for an individual particle and is integrated over frequency and solid angle to find the power radiated per particle. The result is different from that implied by the generalized Larmor formula which, we argue, is not valid in this case. " In: . B. Melrose, Q. Luo: Linear acceleration emission: 2 Power spectrum. Astrophys.J.698:124-130,2009. arXiv:0903.2879. DOI: https://doi.org/10.1088/0004-637X/698/1/124

[6] Das Spektrum der ausgestrahlten Photonen wird in einer Veröffentlichung untersucht. In der Zusammenfassung (abstract) heißt es: "The nonrelativistic Larmor radiation formula, giving the power radiated by an accelerated charged point particle, is generalized for a spatially extended particle in the context of the classical charged harmonic oscillator. The particle is modeled as a spherically symmetric rigid charge distribution that possesses both translational and spinning degrees of freedom. The power spectrum obtained exhibits a structure that depends on the form factor of the particle, but reduces, in the limit of an infinitesimally small particle and for the charge distributions considered, to Larmor’s familiar result. It is found that for finite-duration small-enough accelerations as well as perpetual uniform accelerations the power spectrum of the spatially extended particle reduces to that of a point particle. It is also found that when the acceleration is violent or the size parameter of the particle is very large compared to the wavelength of the emitted radiation the power spectrum is highly suppressed. Possible applications are discussed." In: Edwin A. Marengo and Mohamed R. Khodja: Generalized power-spectrum Larmor formula for an extended charged particle embedded in a harmonic oscillator. Phys. Rev. E 74, 036611 – Published 15 September, 2006. DOI: https://doi.org/10.1103/PhysRevE.74.036611