- #1
snoopies622
- 846
- 28
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.
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 (x1, t1) and (x2, t2) 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?
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 (x1, t1) and (x2, t2) 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?