# 論物理力線

## 分子渦流理論

${\displaystyle \mathrm {d} F_{c}=\rho r^{2}\omega ^{2}\mathrm {d} r\mathrm {d} \theta \mathrm {d} z}$

${\displaystyle \mathrm {d} p_{c}={\frac {\mathrm {d} F_{c}}{r\mathrm {d} \theta \mathrm {d} z}}=\rho r\omega ^{2}\mathrm {d} r}$

${\displaystyle p_{c_{R}}=\int _{0}^{r}\rho r\omega ^{2}\mathrm {d} r=\rho R^{2}\omega ^{2}/2=\rho v^{2}/2}$

${\displaystyle p_{R}=p_{0}+\mu v^{2}/2\pi }$

${\displaystyle {\overline {p}}=p_{0}+\mu v^{2}/4\pi }$

## 比擬磁場現象

${\displaystyle p_{xx}={\frac {1}{4\pi }}\mu \alpha ^{2}-p_{R}}$ ${\displaystyle \qquad p_{yy}={\frac {1}{4\pi }}\mu {\beta }^{2}-p_{R}}$ ${\displaystyle \qquad p_{zz}={\frac {1}{4\pi }}\mu {\gamma }^{2}-p_{R}}$
${\displaystyle p_{xy}={\frac {1}{4\pi }}\mu \alpha {\beta }}$ ${\displaystyle \qquad p_{xz}={\frac {1}{4\pi }}\mu \alpha {\gamma }}$ ${\displaystyle \qquad p_{yz}={\frac {1}{4\pi }}\mu {\beta }{\gamma }}$

${\displaystyle X={\frac {\partial p_{xx}}{\partial x}}+{\frac {\partial p_{xy}}{\partial y}}+{\frac {\partial p_{xz}}{\partial z}}}$

${\displaystyle X={\frac {\alpha }{4\pi }}\left[{\frac {\partial \mu \alpha }{\partial x}}+{\frac {\partial \mu {\beta }}{\partial y}}+{\frac {\partial \mu {\gamma }}{\partial z}}\right]+{\frac {\mu }{8\pi }}\ {\frac {\partial (\alpha ^{2}+{\beta }^{2}+{\gamma }^{2})}{\partial x}}-{\frac {\mu {\beta }}{4\pi }}\left({\frac {\partial {\beta }}{\partial x}}-{\frac {\partial \alpha }{\partial y}}\right)+{\frac {\mu {\gamma }}{4\pi }}\left({\frac {\partial \alpha }{\partial z}}-{\frac {\partial {\gamma }}{\partial x}}\right)-{\frac {\partial p_{R}}{\partial x}}}$

${\displaystyle {\frac {\partial \mu \alpha }{\partial x}}+{\frac {\partial \mu {\beta }}{\partial y}}+{\frac {\partial \mu {\gamma }}{\partial z}}=q_{m}}$

${\displaystyle X}$  關係式右手邊的第一個項目是磁感應強度乘以磁荷，也就是磁荷感受到的磁場力。由於磁單極子並不存在，這項目等於零。

${\displaystyle X}$  關係式右手邊的第三個項目和第四個項目的括號內部的表達式，分別比擬為電流密度的z-分量 ${\displaystyle {\mathfrak {p}}_{z}}$  和y-分量 ${\displaystyle {\mathfrak {p}}_{y}}$

${\displaystyle {\mathfrak {p}}_{z}={\frac {1}{4\pi }}\left({\frac {\partial {\beta }}{\partial x}}-{\frac {\partial {\alpha }}{\partial y}}\right)}$
${\displaystyle {\mathfrak {p}}_{y}={\frac {1}{4\pi }}\left(-{\frac {\partial {\gamma }}{\partial x}}+{\frac {\partial {\alpha }}{\partial z}}\right)}$

${\displaystyle -\mu \beta {\mathfrak {p}}_{z}+\mu \gamma {\mathfrak {p}}_{y}}$

${\displaystyle X=q_{m}\alpha +{\frac {\mu }{8\pi }}\ {\frac {\partial v^{2}}{\partial x}}-\mu {\beta }{\mathfrak {p}}_{z}+\mu {\gamma }{\mathfrak {p}}_{y}-{\frac {\partial p_{R}}{\partial x}}}$ (2)
${\displaystyle Y=q_{m}{\beta }+{\frac {\mu }{8\pi }}\ {\frac {\partial v^{2}}{\partial y}}-\mu {\gamma }{\mathfrak {p}}_{x}+\mu \alpha {\mathfrak {p}}_{z}-{\frac {\partial p_{R}}{\partial y}}}$ (3)
${\displaystyle Z=q_{m}{\gamma }+{\frac {\mu }{8\pi }}\ {\frac {\partial v^{2}}{\partial z}}-\mu \alpha {\mathfrak {p}}_{y}+\mu {\beta }{\mathfrak {p}}_{x}-{\frac {\partial p_{R}}{\partial z}}}$ (4)
• 第一個項目是處於磁場的磁荷感受到的磁場力
• 第二個項目是由於磁能量不均勻分佈，和電介質與物體之間不同的磁導率，共同耦合而產生的作用力。
• 第三個項目和第四個項目是處於磁場的載流導線所感受到的安培力
• 第五個項目是表示流體壓強不均勻分佈所產生的作用力。

## 比擬電流現象

${\displaystyle {\frac {1}{2}}(u_{x},u_{y},u_{z})={\frac {1}{2}}(n_{z}{\beta }-n_{y}{\gamma },n_{x}{\gamma }-n_{z}\alpha ,n_{y}\alpha -n_{x}{\beta })}$

{\displaystyle {\begin{aligned}{\mathfrak {P}}_{x}&=-{\frac {\rho _{e}}{2}}{\sum }_{\mathcal {S}}u_{x}\Delta s\\&=-{\frac {\rho _{e}}{2}}{\sum }_{\mathcal {S}}(n_{z}{\beta }-n_{y}{\gamma })\Delta s\\&=-{\frac {\rho _{e}}{2}}{\sum }_{\mathcal {S}}(\Delta s_{z}{\beta }-\Delta s_{y}{\gamma })\\\end{aligned}}}

{\displaystyle {\begin{aligned}{\mathfrak {P}}_{x}&=-{\frac {\rho _{e}}{2}}\left\{\left[-\left(\beta _{0}-{\frac {\partial {\beta }}{\partial z}}\Delta z\right)\Delta s_{z}+\left(\beta _{0}+{\frac {\partial {\beta }}{\partial z}}\Delta z\right)\Delta s_{z}\right]-\left[-\left(\gamma _{0}-{\frac {\partial {\gamma }}{\partial y}}\Delta y\right)\Delta s_{y}+\left(\gamma _{0}+{\frac {\partial {\gamma }}{\partial y}}\Delta y\right)\Delta s_{y}\right]\right\}\\&=-\rho _{e}\left({\frac {\partial {\beta }}{\partial z}}\Delta z\Delta s_{z}-{\frac {\partial {\gamma }}{\partial y}}\Delta y\Delta s_{y}\right)\\&={\frac {\rho _{e}}{2}}\left({\frac {\partial {\gamma }}{\partial y}}-{\frac {\partial {\beta }}{\partial z}}\right)\Delta V\\\end{aligned}}}

${\displaystyle {\mathfrak {p}}_{x}={\frac {\rho _{e}}{2}}\left({\frac {\partial {\gamma }}{\partial y}}-{\frac {\partial {\beta }}{\partial z}}\right)}$

${\displaystyle {\mathfrak {p}}_{x}={\frac {1}{4\pi }}\left({\frac {\partial {\gamma }}{\partial y}}-{\frac {\partial {\beta }}{\partial z}}\right)}$ (5)
${\displaystyle {\mathfrak {p}}_{y}={\frac {1}{4\pi }}\left(-{\frac {\partial {\gamma }}{\partial x}}+{\frac {\partial {\alpha }}{\partial z}}\right)}$ (6)
${\displaystyle {\mathfrak {p}}_{z}={\frac {1}{4\pi }}\left({\frac {\partial {\beta }}{\partial x}}-{\frac {\partial {\alpha }}{\partial y}}\right)}$ (7)

## 比擬電場現象

${\displaystyle Q_{x}=-4\pi E^{2}h_{x}}$

${\displaystyle {\mathfrak {p}}_{x}'={\frac {\partial h_{x}}{\partial t}}}$

${\displaystyle {\mathfrak {p}}_{x}={\frac {1}{4\pi }}\left({\frac {\partial {\gamma }}{\partial y}}-{\frac {\partial {\beta }}{\partial z}}-{\frac {1}{E^{2}}}\ {\frac {\partial Q_{x}}{\partial t}}\right)}$ (8)
${\displaystyle {\mathfrak {p}}_{y}={\frac {1}{4\pi }}\left({\frac {\partial \alpha }{\partial z}}-{\frac {\partial {\gamma }}{\partial x}}-{\frac {1}{E^{2}}}\ {\frac {\partial Q_{y}}{\partial t}}\right)}$ (9)
${\displaystyle {\mathfrak {p}}_{z}={\frac {1}{4\pi }}\left({\frac {\partial {\beta }}{\partial x}}-{\frac {\partial \alpha }{\partial y}}-{\frac {1}{E^{2}}}\ {\frac {\partial Q_{z}}{\partial t}}\right)}$ (10)

${\displaystyle {\frac {\partial {\mathfrak {p}}_{x}}{\partial x}}+{\frac {\partial {\mathfrak {p}}_{y}}{\partial y}}+{\frac {\partial {\mathfrak {p}}_{z}}{\partial z}}+{\frac {\partial e}{\partial t}}=0}$ (11)

${\displaystyle e={\frac {1}{4\pi E^{2}}}\left({\frac {\partial Q_{x}}{\partial x}}+{\frac {\partial Q_{y}}{\partial y}}+{\frac {\partial Q_{z}}{\partial z}}\right)}$

## 比擬電磁能量現象

${\displaystyle {\mathfrak {E}}_{m}=k_{m}({\alpha }^{2}+{\beta }^{2}+{\gamma }^{2})}$

${\displaystyle {\mathfrak {E}}_{e}=k_{e}({h_{x}}^{2}+{h_{y}}^{2}+{h_{z}}^{2})=k_{Q}({Q_{x}}^{2}+{Q_{y}}^{2}+{Q_{z}}^{2})}$

${\displaystyle {\frac {\partial {\mathfrak {E}}}{\partial t}}=2k_{m}\left({\alpha }{\frac {\partial \alpha }{\partial t}}+{\beta }{\frac {\partial {\beta }}{\partial t}}+{\gamma }{\frac {\partial {\gamma }}{\partial t}}\right)+2k_{Q}\left({Q_{x}}{\frac {\partial Q_{x}}{\partial t}}+{Q_{y}}{\frac {\partial Q_{y}}{\partial t}}+{Q_{z}}{\frac {\partial Q_{z}}{\partial t}}\right)}$ (12)

${\displaystyle {\frac {\partial W}{\partial t}}=-{\frac {\rho _{e}}{2\Delta V}}{\sum }_{\mathcal {S}}(Q_{x}u_{x}+Q_{y}u_{y}+Q_{z}u_{z})\Delta s}$

${\displaystyle {\frac {\partial W}{\partial t}}\Delta V=-{\frac {1}{4\pi }}{\sum }_{\mathcal {S}}(Q_{x}(n_{z}{\beta }-n_{y}{\gamma })+Q_{y}(n_{x}{\gamma }-n_{z}\alpha )+Q_{z}(n_{y}\alpha -n_{x}{\beta }))\Delta s}$

${\displaystyle {\frac {\partial W_{x}}{\partial t}}\Delta V=-{\frac {1}{4\pi }}{\sum }_{\mathcal {S}}Q_{x}(n_{z}{\beta }-n_{y}{\gamma })\Delta s=-{\frac {1}{4\pi }}{\sum }_{\mathcal {S}}(Q_{x}{\beta }\Delta s_{z}-Q_{x}{\gamma }\Delta s_{y})}$

${\displaystyle {\frac {\partial W_{x}}{\partial t}}\Delta V=-{\frac {1}{4\pi }}\left\{\left[-\left(Q_{x0}-{\frac {\partial Q_{x}}{\partial z}}\Delta z\right)\left(\beta _{0}-{\frac {\partial {\beta }}{\partial z}}\Delta z\right)+\left(Q_{x0}+{\frac {\partial Q_{x}}{\partial z}}\Delta z\right)\left(\beta _{0}+{\frac {\partial {\beta }}{\partial z}}\Delta z\right)\right]\Delta s_{z}\right.}$
${\displaystyle \left.-\left[-\left(Q_{x0}-{\frac {\partial Q_{x}}{\partial y}}\Delta y\right)\left(\gamma _{0}-{\frac {\partial {\gamma }}{\partial y}}\Delta y\right)+\left(Q_{x0}+{\frac {\partial Q_{x}}{\partial y}}\Delta y\right)\left(\gamma _{0}+{\frac {\partial {\gamma }}{\partial y}}\Delta y\right)\right]\Delta s_{y}\right\}}$

${\displaystyle {\frac {\partial W_{x}}{\partial t}}=-{\frac {1}{4\pi }}\left[\left(\beta _{0}{\frac {\partial Q_{x}}{\partial z}}-\gamma _{0}{\frac {\partial Q_{x}}{\partial y}}\right)-Q_{x0}\left({\frac {\partial {\gamma }}{\partial y}}-{\frac {\partial {\beta }}{\partial z}}\right)\right]}$

${\displaystyle {\frac {\partial W_{y}}{\partial t}}=-{\frac {1}{4\pi }}\left[\left(-\alpha _{0}{\frac {\partial Q_{y}}{\partial z}}+\gamma _{0}{\frac {\partial Q_{y}}{\partial x}}\right)-Q_{y0}\left(-{\frac {\partial {\gamma }}{\partial x}}+{\frac {\partial \alpha }{\partial z}}\right)\right]}$
${\displaystyle {\frac {\partial W_{z}}{\partial t}}=-{\frac {1}{4\pi }}\left[\left(\alpha _{0}{\frac {\partial Q_{z}}{\partial y}}-\beta _{0}{\frac {\partial Q_{z}}{\partial x}}\right)-Q_{z0}\left({\frac {\partial {\beta }}{\partial x}}-{\frac {\partial \alpha }{\partial y}}\right)\right]}$

${\displaystyle {\frac {\partial W}{\partial t}}=-{\frac {1}{4\pi }}\left\{\left[\alpha _{0}\left({\frac {\partial Q_{z}}{\partial y}}-{\frac {\partial Q_{y}}{\partial z}}\right)+\beta _{0}\left(-{\frac {\partial Q_{z}}{\partial x}}+{\frac {\partial Q_{x}}{\partial z}}\right)+\gamma _{0}\left({\frac {\partial Q_{y}}{\partial x}}-{\frac {\partial Q_{x}}{\partial y}}\right)\right]\right.}$
${\displaystyle \left.-\left[Q_{x0}\left({\frac {\partial {\gamma }}{\partial y}}-{\frac {\partial {\beta }}{\partial z}}\right)+Q_{y0}\left(-{\frac {\partial {\gamma }}{\partial x}}+{\frac {\partial \alpha }{\partial z}}\right)+Q_{z0}\left({\frac {\partial {\beta }}{\partial x}}-{\frac {\partial \alpha }{\partial y}}\right)\right]\right\}}$

${\displaystyle {\frac {\partial {\mathfrak {E}}}{\partial t}}=2k_{m}\left({\alpha _{0}}{\frac {\partial \alpha }{\partial t}}+{\beta _{0}}{\frac {\partial {\beta }}{\partial t}}+{\gamma _{0}}{\frac {\partial {\gamma }}{\partial t}}\right)+2k_{Q}\left({Q_{x0}}{\frac {\partial Q_{x}}{\partial t}}+{Q_{y0}}{\frac {\partial Q_{y0}}{\partial t}}+{Q_{z0}}{\frac {\partial Q_{z}}{\partial t}}\right)}$

${\displaystyle {\frac {\partial Q_{z}}{\partial y}}-{\frac {\partial Q_{y}}{\partial z}}=-\mu {\frac {\partial \alpha }{\partial t}}}$
${\displaystyle -{\frac {\partial Q_{z}}{\partial x}}+{\frac {\partial Q_{x}}{\partial z}}=-\mu {\frac {\partial {\beta }}{\partial t}}}$
${\displaystyle {\frac {\partial Q_{y}}{\partial x}}-{\frac {\partial Q_{x}}{\partial y}}=-\mu {\frac {\partial {\gamma }}{\partial t}}}$

${\displaystyle {\frac {\partial {\gamma }}{\partial y}}-{\frac {\partial {\beta }}{\partial z}}={\frac {1}{E^{2}}}{\frac {\partial Q_{x}}{\partial t}}}$
${\displaystyle -{\frac {\partial {\gamma }}{\partial x}}+{\frac {\partial \alpha }{\partial z}}={\frac {1}{E^{2}}}{\frac {\partial Q_{y}}{\partial t}}}$
${\displaystyle {\frac {\partial {\beta }}{\partial x}}-{\frac {\partial \alpha }{\partial y}}={\frac {1}{E^{2}}}{\frac {\partial Q_{z}}{\partial t}}}$

${\displaystyle {\mathfrak {E}}_{m}={\frac {\mu }{8\pi }}({\alpha }^{2}+{\beta }^{2}+{\gamma }^{2})}$

${\displaystyle {\mathfrak {E}}_{e}={\frac {1}{8\pi E^{2}}}({Q_{x}}^{2}+{Q_{y}}^{2}+{Q_{z}}^{2})}$

## 光波就是電磁波

${\displaystyle V={\sqrt {m/\rho }}}$

${\displaystyle m=k_{1}E^{2}}$  ;

${\displaystyle \rho }$  是介質密度，與渦胞物質密度 ${\displaystyle \mu }$  有關：

${\displaystyle \rho =k_{2}\mu }$  ;

${\displaystyle k_{1}}$ ${\displaystyle k_{2}}$  都是比例常數。

${\displaystyle V=k_{3}E/{\sqrt {\mu }}}$

${\displaystyle V=E/{\sqrt {\mu }}}$

## 注釋

1. ^ 假設粒子A和粒子B處於空間的某兩不同位置，則根據牛頓萬有引力定律，兩粒子互相直接施加於對方的引力，其大小 ${\displaystyle F}$  必定與距離 ${\displaystyle r}$  的平方成反比：
${\displaystyle F=G{\frac {m_{A}m_{B}}{r^{2}}}}$
其中，${\displaystyle G}$ 萬有引力常數${\displaystyle m_{A}}$ ${\displaystyle m_{B}}$  分別是粒子A和粒子B的質量。 從這方程式，可以觀察出萬有引力是一種超距作用，牛頓萬有引力定律只提到兩粒子互相直接作用於對方的引力，並沒有解釋傳播過程，而且這定律與時間無關，意味著瞬時直接地超距作用。

## 參考文獻

1. Jackson, John David. Classical Electrodynamic 3rd. USA: John Wiley & Sons, Inc. 1999. ISBN 978-0-471-30932-1.
2. ^ Simpson 1997，第143-144頁
3. ^ 馬克士威 1861，第161-162頁
4. Baigrie, Brian, Electricity and magnetism:a historical perspective illustrated, annotated, Greenwood Publishing Group: pp.97–98, 2007, ISBN 9780313333583
5. ^ Simpson 1997，第206-207, 231頁
6. ^ 湯姆森, 威廉, mechanical representation of electric, magnetic, and galvanic forces, The Cambridge and Dublin mathematical journal: 61–64, [1847]
7. ^ Simpson 1997，第147-149頁

## 進階閱讀

• Simpson, Thomas K., Maxwell on the electromagnetic field: a guided study, USA: Rutgers University Press, 1997, ISBN 9780813523637
• Crease, Robert, The Great Equations: Breakthroughs in Science from Pythagoras to Heisenberg, illustrated, W. W. Norton & Company: pp. 132ff, 2008, ISBN 9780393062045
• Siegel, Daniel M., Innovation in Maxwell's Electromagnetic Theory: Molecular Vortices, Displacement Current, and Light, Cambridge University Press: 240, [2003], ISBN 9780521533294