# 阿布拉罕-勞侖茲力

## 定義與描述

• ${\displaystyle \mathbf {F} _{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}} ={\frac {q^{2}}{6\pi \epsilon _{0}c^{3}}}\mathbf {\dot {a}} \,}$SI單位制
• ${\displaystyle \mathbf {F} _{\mathrm {rad} }={2 \over 3}{\frac {q^{2}}{c^{3}}}\mathbf {\dot {a}} \,}$cgs單位制

F是力，
${\displaystyle \mathbf {\dot {a}} }$加加速度加速度的時間导数）。
μ0ε0真空磁导率真空电容率
c是真空中的光速
q电荷量

## 背景

1. 問題中，產生場的電荷與電流源已指定，要計算出場；
2. 問題中，場已指定，要計算出電荷的運動。

1. 忽略「自身場（self-fields）」通常仍可得到足夠精確的答案，足以用在許多應用上；
2. 包含自身場會導致物理學中目前未解決的問題，關係到物質能量的本質。

## 推導

${\displaystyle P={\frac {\mu _{0}q^{2}a^{2}}{6\pi c}}}$.

${\displaystyle \int _{\tau _{1}}^{\tau _{2}}\mathbf {F} _{\mathrm {rad} }\cdot \mathbf {v} dt=\int _{\tau _{1}}^{\tau _{2}}-Pdt=-\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}a^{2}}{6\pi c}}dt=-\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}}{6\pi c}}{\frac {d\mathbf {v} }{dt}}\cdot {\frac {d\mathbf {v} }{dt}}dt}$.

${\displaystyle \int _{\tau _{1}}^{\tau _{2}}\mathbf {F} _{\mathrm {rad} }\cdot \mathbf {v} dt=-{\frac {\mu _{0}q^{2}}{6\pi c}}{\frac {d\mathbf {v} }{dt}}\cdot \mathbf {v} {\bigg |}_{\tau _{1}}^{\tau _{2}}+\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}}{6\pi c}}{\frac {d^{2}\mathbf {v} }{dt^{2}}}\cdot \mathbf {v} dt=-0+\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}} \cdot \mathbf {v} dt}$

${\displaystyle \mathbf {F} _{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}} }$

## 来自未来的讯号

${\displaystyle m{\dot {\mathbf {v} }}=\mathbf {F} _{\mathrm {rad} }+\mathbf {F} _{\mathrm {ext} }=mt_{0}{\ddot {\mathbf {v} }}+\mathbf {F} _{\mathrm {ext} }}$

${\displaystyle t_{0}={\frac {\mu _{0}q^{2}}{6\pi mc}}}$

${\displaystyle m{\dot {\mathbf {v} }}={1 \over t_{0}}\int _{t}^{\infty }\exp \left(-{t'-t \over t_{0}}\right)\,\mathbf {F} _{\mathrm {ext} }(t')\,dt'}$

${\displaystyle \exp \left(-{t'-t \over t_{0}}\right)}$

## 參考文獻

1. ^ F. Rohrlich. The dynamics of a charged sphere and the electron (PDF). Am J Phys. 1997, 65 (11): 1051.。其中文字："The dynamics of point charges is an excellent example of the importance of obeying the validity limits of a physical theory. When these limits are exceeded the predictions of the theory may be incorrect or even patently absurd. In the present case, the classical equations of motion have their validity limits where quantum mechanics becomes important: they can no longer be trusted at distances of the order of (or below) the Compton wavelength… Only when all distances involved are in the classical domain is classical dynamics acceptable for electrons."

### 書目

• Griffiths, David J. Introduction to Electrodynamics 3rd ed. Prentice Hall. 1998. ISBN 978-0-13-805326-0.
• Jackson, John D. Classical Electrodynamics (3rd ed.). Wiley. 1998. ISBN 978-0-471-30932-1.
• F. Rohrlich, Am. J. Phys. 65, 1051 (1997).