I first asked this question with much less detail in the recent thread about the Abraham-Lorentz formula but it was sort of passed over, and since it was only tangentially related to the central issue in that thread anyway, I am creating this new one about it specifically. Perhaps it won't need much attention.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose I have a particle with electric charge q that moves according to the equation

[tex]

x = \frac {1}{2}a t ^2

[/tex]

in which the trajectory doesn't begin at t=0 but at some time before that. I consider two points on the trajectory that are very close to each other (x_{1}, t_{1}) and (x_{2}, t_{2}) and I want to know what force is being applied to the particle in the interval between these two points to maintain the constant acceleration a. If we let

[tex]

k=\frac {q^2}{6 \pi \epsilon_0 c^3}

[/tex]

then according to the Larmor formula the power emitted by the particle is [tex] ka^2[/tex] and so the energy emitted in this interval is [tex] k a^2 (t_2 - t_1) [/tex]. If I assume that energy is conserved and therefore the energy invested into the particle is the same as the energy emitted from it, and that

[tex]

W = \int \bold {F} \cdot d \bold {s}

[/tex]

then the average force being applied to the particle in this interval is

[tex]

ka^2 \big {(} \frac {t_2 - t_1}{x_2 - x_1} \big {)} = ka^2 \big {(} \frac {t_2 - t_1}{(a/2) (t_2 ^2 - t_1 ^2)} \big {)} = ka^2 \big {(} \frac {t_2 - t_1}{(a/2) (t_2 - t_1)(t_2 + t_1)} \big {)}

[/tex]

which as [tex] t_1 \rightarrow t_2 [/tex]

[tex]

= \frac {ka^2}{(a/2)(2t)} = \frac {ka^2}{at}

[/tex] which in this case [tex]

= \frac {ka^2}{v}

[/tex].

This is just another way of saying that force = power / speed. Obviously, this cannot be correct for this trajectory when t=0. The thread about the Abraham-Lorentz formula pointed out limits to its applications, but I haven't seen any mention of limits to the application of the Larmor formula other than that it is non-relativistic, which doesn't explain why there should be a problem using it when v=0.

So what is the problem?

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# About the Larmor power formula

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