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Rethinking of Poynting’s Theorem
DOI: 10.12677/MP.2018.82006, PDF, HTML, XML, 下载: 2,315  浏览: 6,118

Abstract: In this paper, we study the energy transmission process of direct current and low frequency alternating current. We point out that in addition to the electromagnetic energy flow defined by Poynting vector, there is a “classical” energy transfer process: free electrons driven by the electromotive force of the power supply. The electromotive force of the power supply can be converted into kinetic energy. Outside wires, no current is formed because there are no free electrons. In the case of DC circuits and low-frequency AC circuits, the transfer of electric energy in wires is not related to the displacement current, so it is also independent of the Poynting theorem. The model of lumped element circuit does not calculate the length of the wire. Indeed, it implies an assumption: the speed of low-frequency alternating signals in the wire is infinite. Our experimental results show that under certain conditions, the speed of alternating electrical signals in the wire can be much greater than the speed of light, which is an improvement on the model of lumped parameter circuit.

1. 引言

Figure 1. Schematic diagram of electromagnetic energy flow in a DC circuit

d 位英国科学家(J.P. Poynting)，他从Maxwell方程组推导出了坡印亭定理 [2] [3] 。这个定理类似于经典力学中的能量守恒定理，在数学上类似于连续性方程。

2. 坡印亭定理的推导

$\nabla ×H={j}_{c}+\frac{\partial D}{\partial t}$(1)

$E\cdot \nabla ×H=E\cdot {j}_{c}+E\cdot \frac{\partial D}{\partial t}$ (2)

$\nabla \cdot \left(E×H\right)=-E\cdot \nabla ×H+H\cdot \nabla ×E$ (3)

$H\cdot \nabla ×E-\nabla \cdot \left(E×H\right)=E\cdot {j}_{c}+E\cdot \frac{\partial D}{\partial t}$ (4)

$-\nabla \cdot \left(E×H\right)=E\cdot {j}_{c}+E\cdot \frac{\partial D}{\partial t}+H\cdot \frac{\partial B}{\partial t}$ (5)

$E\cdot \frac{\partial D}{\partial t}=\frac{\partial }{\partial t}\left(\frac{\epsilon {E}^{2}}{2}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}H\cdot \frac{\partial H}{\partial t}=\frac{\partial }{\partial t}\left(\frac{\mu {H}^{2}}{2}\right)$ (6)

$-\nabla \cdot \left(E×H\right)=E\cdot {j}_{c}+\frac{\partial }{\partial t}\left(\frac{\epsilon {E}^{2}}{2}+\frac{\mu {H}^{2}}{2}\right)$ (7)

3. 对坡印廷定理的再思考

$\nabla ×H={j}_{c}$ (8)

$E\cdot \nabla ×H=E\cdot {j}_{c}$ (9)

$-\nabla \cdot \left(E×H\right)=\frac{\partial }{\partial t}\left(\frac{\epsilon {E}^{2}}{2}+\frac{\mu {H}^{2}}{2}\right)$ (10)

1) 如果它们是由闭合电路中电源的电动势产生的，那么这个物理过程不涉及到交变磁场产生交变电场，也就不涉及坡印亭定理。

2) 如果是电磁波传输到接收天线，在接收天线的导线内产生了电场强度E和电流密度j，那么这个物理过程需要坡印亭定理。所以，坡印亭定理(7)是有一定适用范围的。

4. RL电路中低频交流电的速度

$U\left(t\right)=I\left(t\right)R+L\text{d}I\left(t\right)/\text{d}t$ (11)

Figure 2. A simple RL circuit

$L=2l\left[\mathrm{ln}\left(2l/r\right)-0.75\right]×{10}^{-7}$ (12)

$I\left(t\right)={I}_{o}\mathrm{exp}\left(j\omega t\right)$ (13)

$U\left(t\right)=I\left(t\right)R+j\omega LI\left(t\right)$ (14)

$\omega L\ll R$ 的条件下，式(14)可以改写成为

$U\left(t\right)=I\left(t\right)R\mathrm{exp}\left(j\omega t+j\Phi \right)$ (15)

$\Phi =\omega L/R=\omega \Delta t$ (16)

$\Delta t=L/R$ (17)

${V}_{AC}=l/\Delta t=lR/L$ (18)

5. 结论

1) 如果它们是由闭合电路中电源的电动势产生的，那么这个物理过程不涉及到交变磁场产生交变电场，也就不涉及坡印亭定理。

2) 如果是电磁波传输到接收天线，在接收天线的导线内产生了电场强度E和电流密度j，那么这个物理过程需要坡印亭定理。

NOTES

*通讯作者。

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