# 拉莫爾方程式

（重定向自拉莫尔方程

${\displaystyle P={\frac {e^{2}a^{2}}{6\pi \varepsilon _{0}c^{3}}}{\mbox{ }}}$（SI單位制）
${\displaystyle P={2 \over 3}{\frac {e^{2}a^{2}}{c^{3}}}{\mbox{ }}}$（CGS單位制）

## 推導

### 推導1：數學逼近

${\displaystyle {\vec {E}}({\vec {r}},t)=q\left({\frac {{\vec {n}}-{\vec {\beta }}}{\gamma ^{2}(1-{\vec {\beta }}\cdot {\vec {n}})^{3}R^{2}}}\right)_{\rm {ret}}+{\frac {q}{c}}\left({\frac {{\vec {n}}\times [({\vec {n}}-{\vec {\beta }})\times {\vec {\dot {\beta }}}]}{(1-{\vec {\beta }}\cdot {\vec {n}})^{3}R}}\right)_{\rm {ret}}}$

${\displaystyle {\vec {B}}={\vec {n}}\times {\vec {E}}}$

${\displaystyle \mathbf {\beta } }$ 是電荷速度和光速c的商
${\displaystyle \mathbf {\dot {\beta }} }$ 是電荷加速度和光速c的商
${\displaystyle \mathbf {n} }$ ${\displaystyle \mathbf {r} -\mathbf {r} _{0}}$ 方向上的單位向量
${\displaystyle R}$ ${\displaystyle \mathbf {r} -\mathbf {r} _{0}}$ 方向的純量

${\displaystyle t'=t-{R \over c}}$

${\displaystyle {\vec {S}}={\frac {c}{4\pi }}{\vec {E}}_{a}\times {\vec {B}}_{a}}$

${\displaystyle {\vec {S}}={\frac {q^{2}}{4\pi c}}\left|{\frac {{\vec {n}}\times ({\vec {n}}\times {\vec {\dot {\beta }}})}{R}}\right|^{2}.}$

${\displaystyle \beta \left(t_{r}\right)\neq 0}$ 時證明會較困難（參見Griffiths）

${\displaystyle {\vec {S}}={\frac {q^{2}}{4\pi c^{3}R^{2}}}\sin ^{2}{\theta }|{\vec {\dot {V}}}|^{2}{\hat {n}}.}$

${\displaystyle P={\frac {2}{3}}{\frac {q^{2}|{\vec {\dot {V}}}|^{2}}{c^{3}}}.}$

### 推導2：爱德华·珀塞尔逼近

${\displaystyle E_{t}={{ea\sin(\theta )} \over {4\pi \varepsilon _{0}c^{2}R}}.}$

${\displaystyle \mathbf {S} ={E_{t}^{2} \over \mu _{0}c}\mathbf {\hat {r}} ={{e^{2}a^{2}\sin ^{2}(\theta )} \over {16\pi ^{2}\varepsilon _{0}c^{3}R^{2}}}\mathbf {\hat {r}} }$

${\displaystyle P={{e^{2}a^{2}} \over {6\pi \varepsilon _{0}c^{3}}}}$

${\displaystyle P={{\mu _{0}e^{2}a^{2}} \over {6\pi c}}}$

## 相對論性狀況下

### 協變形式

${\displaystyle P={\frac {2}{3}}{\frac {q^{2}}{c^{3}m^{2}}}\left({\frac {d{\vec {p}}}{dt}}\cdot {\frac {d{\vec {p}}}{dt}}\right).}$

${\displaystyle P=-{\frac {2}{3}}{\frac {q^{2}}{m^{2}c^{3}}}{\frac {dP^{\mu }}{d\tau }}{\frac {dP_{\mu }}{d\tau }}.}$

${\displaystyle {\frac {dP^{\mu }}{d\tau }}{\frac {dP_{\mu }}{d\tau }}={\frac {v^{2}}{c^{2}}}\left({\frac {dP}{d\tau }}\right)^{2}-\left({\frac {d{\vec {p}}}{d\tau }}\right)^{2}}$

### 非協變形式

${\displaystyle {\frac {dp^{\mu }}{d\tau }}{\frac {dp_{\mu }}{d\tau }}=-\left({\frac {d{\vec {p}}}{d\tau }}\right)^{2}+{\frac {1}{c^{2}}}\left({\frac {dE}{d\tau }}\right)^{2}}$
${\displaystyle {\frac {dp^{\mu }}{d\tau }}{\frac {dp_{\mu }}{d\tau }}=-\gamma ^{2}\left({\frac {d\gamma m{\vec {v}}}{dt}}\right)^{2}+{\frac {\gamma ^{2}}{c^{2}}}\left({\frac {d\gamma mc^{2}}{dt}}\right)^{2}}$
${\displaystyle {\frac {dp^{\mu }}{d\tau }}{\frac {dp_{\mu }}{d\tau }}=-\gamma ^{2}[-(\gamma m{\vec {\dot {v}}}+\gamma ^{3}m{\vec {v}}({\vec {\beta }}\cdot {\vec {\dot {\beta }}}))^{2}+{\frac {1}{c^{2}}}(\gamma ^{3}{\vec {\beta }}\cdot {\vec {\dot {\beta }}}mc^{2})^{2}]}$
${\displaystyle {\frac {dp^{\mu }}{d\tau }}{\frac {dp_{\mu }}{d\tau }}=\gamma ^{8}m^{2}c^{2}[({\vec {\beta }}\cdot {\vec {\dot {\beta }}})^{2}-({\vec {\beta }}({\vec {\beta }}\cdot {\vec {\dot {\beta }}})+{\frac {\vec {\dot {\beta }}}{\gamma ^{2}}})^{2}]}$
${\displaystyle \Rightarrow {\frac {dp^{\mu }}{d\tau }}{\frac {dp_{\mu }}{d\tau }}=\gamma ^{8}m^{2}c^{2}\left(-{\frac {1}{\gamma ^{2}}}({\vec {\beta }}\cdot {\vec {\dot {\beta }}})^{2}-{\frac {{\vec {\dot {\beta }}}^{2}}{\gamma ^{4}}}\right).}$

${\displaystyle \gamma ^{6}m^{2}c^{2}[({\vec {\beta }}^{2}{\vec {\dot {\beta }}}^{2}-({\vec {\beta }}\cdot {\vec {\dot {\beta }}})^{2})-{\vec {\dot {\beta }}}^{2}].}$

${\displaystyle ({\vec {\beta }}\times {\vec {\dot {\beta }}})\cdot ({\vec {\beta }}\times {\vec {\dot {\beta }}})=({\vec {\beta }}^{2}{\vec {\dot {\beta }}}^{2}-({\vec {\beta }}\cdot {\vec {\dot {\beta }}})^{2}).}$

${\displaystyle P={\frac {2q^{2}\gamma ^{6}}{3c}}\left(({\vec {\dot {\beta }}})^{2}-({\vec {\beta }}\times {\vec {\dot {\beta }}})^{2}\right)}$

## 參考資料

### 書目

• J. Larmor, "On a dynamical theory of the electric and luminiferous medium", Philosophical Transactions of the Royal Society 190, (1897) pp. 205–300 (Third and last in a series of papers with the same name).
• Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald. Gravitation. San Francisco: W. H. Freeman. 1973. ISBN 0-7167-0344-0.
• R. P. Feynman, F. B. Moringo, and W. G. Wagner. Feynman Lectures on Gravitation. Addison-Wesley. 1995. ISBN 0-201-62734-5.