# 位能的迷思

## 無法解釋的困境與解決之道

$$\Delta E=W+Q$$

$$dE=W+Q\tag{1}$$

$$\because Q=0\tag{2}$$

$$\because W=\left(-F_{\text{Ag}}\right)dr_A+\left(+F_{\text{Bg}}\right)dr_B\tag{3}$$

$$\because F_{\text{Ag}}=F_{\text{Bg}}=\frac{Gm_\text{A}m_\text{B}}{r_\text{AB}^2}\tag{4}$$

\begin{aligned}\therefore W&=\frac{Gm_\text{A}m_\text{B}}{r_\text{AB}^2}\left(dr_B-dr_A\right)\\[2ex]&=\frac{Gm_\text{A}m_\text{B}}{r_\text{BA}^2}\end{aligned}\tag{5}

\begin{aligned}\therefore dE&=\frac{Gm_\text{A}m_\text{B}}{r_\text{BA}^2}\left(dr_\text{BA}\right)\\[2ex]&=d\left(-\frac{Gm_\text{A}m_\text{B}}{r_\text{BA}}\right)\end{aligned}\tag{6}

$$\therefore E\equiv-\frac{Gm_\text{A}m_\text{B}}{r_\text{BA}}+C\tag{7}$$

## 場源為物體與地球的重力場能量形式

$$\therefore E\equiv\frac{1}{2}\sum_{i=1}^N\sum_{j=1,j\neq i}^N\left(-\frac{Gm_i m_j}{r_{ij}}\right)+C\tag{8}$$

$$E\approx\sum_{i=1}^N\left(-\frac{GMm_i}{r_{i,M}}\right)+C\text{ ; }r_{\text{i,M}}=R_e+h_i\tag{9}$$

$$0\approx\sum_{i=1}^N\left(-\frac{GMm_i}{R_e}\right)+C\tag{10}$$

$$C\approx\sum_{i=1}^N\frac{GMm_i}{R_e}\tag{11}$$

$$E\approx-\sum_{i=1}^N\left(\frac{GMm_i}{R_e+h_i}-\frac{GMm_i}{R_e}\right)\tag{12}$$

$$\therefore E\approx\sum_{i=1}^N\frac{GMm_i}{R_e^2}h_i\tag{13}$$

$$\therefore E\approx\sum_{i=1}^N m_i gh_i\text{ ; }g\equiv\frac{GM}{R_e^2}\tag{14}$$

[1] 只有在慣性座標系才能使用牛頓定律推得單質點系統之能量形式為動能，關於這點，之後我會於另文論述。簡單來說，非慣性座標系是無法討論任何基於「牛頓力學」的物理概念的。
[2] Marc Lange (2002). An introduction to the philosophy of physics : locality, fields, energy and mass. Wiley-Blackwell; 1 edition.
[3] 這種解釋與電路學中的電磁能傳遞方式是相同的，之後我會在發表一篇電路傳遞能量的迷思。簡言之，電磁能並非透過電流傳遞，電流反而是消耗電磁能的因素，而是透過佔據空間的電磁場。另外，引述 Robert P. Baumann 於〈Physics that textbook writers usually get wrong: I. Work〉的論述：

If we attribute potential energy to the ball at the top, we cannot properly calculate the work done on it, and hence energy transferred to, the ball in its descent. … It is, of course, convenient to think of objects in gravitational (or electric) fields as having potential energy. We can do that by redefining the system as ball plus field. Then the field can do no work on our system, as the ball goes up or down.

[4] Robert P. Bauman (1992). Physics that textbook writers usually get wrong: II. Heat and energy. THE PHYSICS TEACHER, 30, 353–356.
[5] 注意，兩團黑霧並非指兩個重力場，由A、B場源形成的重力場只有一個。此外，黑點密度表示重力場強度，而非重力場位置。

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