每日彙整: 2018-08-10

樹莓派、樹莓派之學習、樹莓派之教育

STEM 隨筆︰古典力學︰轉子【五】《電路學》 五【電感】 IV‧馬達‧二‧B

AC Motor
As in the DC motor case, a current is passed through the coil, generating a torque on the coil. Since the current is alternating, the motor will run smoothly only at the frequency of the sine wave. It is called a synchronous motor. More common is the induction motor, where electric current is induced in the rotating coils rather than supplied to them directly.

One of the drawbacks of this kind of AC motor is the high current which must flow through the rotating contacts. Sparking and heating at those contacts can waste energy and shorten the lifetime of the motor. In common AC motors the magnetic field is produced by an electromagnet powered by the same AC voltage as the motor coil. The coils which produce the magnetic field are sometimes referred to as the “stator”, while the coils and the solid core which rotates is called the “armature”. In an AC motor the magnetic field is sinusoidally varying, just as the current in the coil varies.

AC Generator

The turning of a coil in a magnetic field produces motional emfs in both sides of the coil which add. Since the component of the velocity perpendicular to the magnetic field changes sinusoidally with the rotation, the generated voltage is sinusoidal or AC. This process can be described in terms of Faraday’s law when you see that the rotation of the coil continually changes the magnetic flux through the coil and therefore generates a voltage.

Generator and Motor

A hand-cranked generator can be used to generate voltage to turn a motor. This is an example of energy conversion from mechanical to electrical energy and then back to mechanical energy.

As the motor is turning, it also acts as a generator and generates a “back emf“. By Lenz’s law, the emf generated by the motor coil will oppose the change that created it. If the motor is not driving a load, then the generated back emf will almost balance the input voltage and very little current will flow in the coil of the motor. But if the motor is driving a heavy load, the back emf will be less and more current will flow in the motor coil and that electric power being used is converted to the mechanical power to drive the load.

─── 摘自 HyperPhysics***** Electricity and Magnetism

 

深入一個簡單手搖『發電‧電動』機制，仔細思考其中關鍵概念︰

DC Motor Operation

This is an active graphic. Click on bold type for further illustration.
 

Faraday’s Law

Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be “induced” in the coil. No matter how the change is produced, the voltage will be generated. The change could be produced by changing the magnetic field strength, moving a magnet toward or away from the coil, moving the coil into or out of the magnetic field, rotating the coil relative to the magnet, etc.

Further comments on these examples

Faraday’s law is a fundamental relationship which comes from Maxwell’s equations. It serves as a succinct summary of the ways a voltage (or emf) may be generated by a changing magnetic environment. The induced emf in a coil is equal to the negative of the rate of change of magnetic flux times the number of turns in the coil. It involves the interaction of charge with magnetic field.

 

Lenz’s Law

When an emf is generated by a change in magnetic flux according to Faraday’s Law, the polarity of the induced emf is such that it produces a current whose magnetic field opposes the change which produces it. The induced magnetic field inside any loop of wire always acts to keep the magnetic flux in the loop constant. In the examples below, if the B field is increasing, the induced field acts in opposition to it. If it is decreasing, the induced field acts in the direction of the applied field to try to keep it constant.

 

的聯繫，將可一時瞥見『自然阻抗』之身影乎？

Lenz’s law

Lenz’s law (pronounced /ˈlɛnts/), named after the physicist Heinrich Friedrich Emil Lenz who formulated it in 1834,[1] states that the direction of the current induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes the initial changing magnetic field. Or as informally, yet concisely summarised by D.J. Griffiths:

Nature abhors a change in flux.[2]

Lenz’s law is shown by the negative sign in Faraday’s law of induction:

\displaystyle {\mathcal {E}}=-{\frac {\partial \Phi _{\mathbf {B} }}{\partial t}},

which indicates that the induced electromotive force \displaystyle {\mathcal {E}} and the rate of change in magnetic flux \displaystyle \Phi _{\mathbf {B} } have opposite signs.[3] It is a qualitative law that specifies the direction of induced current but says nothing about its magnitude. Lenz’s law explains the direction of many effects in electromagnetism, such as the direction of voltage induced in an inductor or wire loop by a changing current, or why eddy currents exert a drag force on moving objects in a magnetic field.

Lenz’s law can be seen as analogous to Newton’s third law in classic mechanics.[4]

 

或許此刻引用『勞侖茲力』

在電動力學裏，若考慮一帶電粒子在電磁場中的受力，可以用以下的勞侖茲力定律表示：

\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} ) ；

其中，\displaystyle \mathbf {F} 是勞侖茲力，\displaystyle q 是帶電粒子的電荷量，\displaystyle \mathbf {E} 是電場，\displaystyle \mathbf {v} 是帶電粒子的速度，\displaystyle \mathbf {B} 是磁場。

勞侖茲力定律是一個基本公理，不是從別的理論推導出來的定律 。

這方程式右邊有兩項，第一項是電場力 \displaystyle \mathbf {F} _{E}=q\mathbf {E} ，第二項是磁場力 \displaystyle \mathbf {F} _{B}=q\mathbf {v} \times \mathbf {B} 。

當兩個帶電粒子都以相同速度 \displaystyle \mathbf {v} 移動時，帶正電粒子 \displaystyle +q 會感受到電場力\displaystyle \mathbf {F} _{E} 、磁場力 \displaystyle \mathbf {F} _{M} 與淨力\displaystyle \mathbf {F} _{T} ，帶負電粒子 \displaystyle -q 會感受到電場力 \displaystyle -\mathbf {F} _{E} 、磁場力 \displaystyle -\mathbf {F} _{M} 與淨力 \displaystyle -\mathbf {F} _{T} 。注意到作用力 \displaystyle \mathbf {F} _{T} 和反作用力 \displaystyle -\mathbf {F} _{T} 不同線。在本圖內，速度 \displaystyle \mathbf {v} 的大小不按比例繪製。

 

藉著『德汝德模型』來作計算，仍難一窺全貌吧！？

幸而有

能量守恆定律

能量守恆定律（英語：law of conservation of energy）闡明，孤立系統的總能量 \displaystyle E 保持不變。如果一個系統處於孤立環境，即不能有任何能量或質量從該系統輸入或輸出。能量不能無故生成，也不能無故摧毀，但它能夠改變形式，例如，在炸彈爆炸的過程中，化學能可以轉化為動能。

從能量守恆定律可以推導出第一類永動機永遠無法實現。沒有任何孤立系統能夠持續對外提供能量[1]。

測量熱功當量的焦耳裝置：緊繫於繩子向下移動的砝碼會使得沉浸於水裏的槳輪轉動。

歷史

早從約西元前五百年時，古希臘哲學家泰勒斯就認為在所有物質之中，有某種潛藏的物質會守恆不變化，不過當時泰勒斯當時認為守恆的 物質是水，而這和現在認知的質量或質能都沒有關係，恩培多克勒（490–430 BCE）認為在宇宙是由四元素（火、風、水、地）組成，「沒有一様會增加或是減少。」[2]，不過這些元素會不斷的重組。

1638年時伽利略發表了許多研究，包括著名的單擺的實驗，可以表示為位能和動能之間不停的轉換。

戈特弗里德·萊布尼茨在1676年至1689年間，首先試著將和運動有關的能量以數學公式表示，萊布尼茨發現在許多力學系統中（有多個質量mi，各自的速度為vi ），只要各質量之間沒有碰撞，以下物理量會守恆：

\displaystyle \sum _{i}m_{i}v_{i}^{2}

他將此物理量稱為系統的「活力」。此定律精確的描述了在沒有摩擦力時動能的守恆。當時許多物理學家發現動量守恆，也就是在一個沒有摩擦力的系統中，以下式表示的動量會守恆：

\displaystyle \,\!\sum _{i}m_{i}v_{i}

後來發現在適當條件（例如彈性碰撞）時，動能和動量都會守恆。

像約翰·斯米頓、彼得·尤爾特、 卡爾·霍爾茨曼、古斯塔夫-阿道夫·希恩及馬克·塞甘等工程師反對只使用動量守恆定律，他們使用萊布尼茨的公式。而約翰·普萊費爾就指出動能明顯的不平衡，在現在利用熱力學第二定律為基礎，可以得到上述的結果，但在18世紀及19世紀，還不曉得失去的能量去哪裡了。最後大家開始懷疑在有摩擦力時，產生的熱是一種活力的型式。1783年時安托萬-洛朗·德·拉瓦錫和皮耶爾-西蒙·拉普拉斯重新確認二種互相競爭的理論：熱質說及活力。[3]班傑明·湯普森，倫福德伯爵在1798年觀察到加農炮鋿孔時一直發熱，表示力學的運動可以轉換為熱能，而且（重要的）其轉換是可以量化的，可以預測其發熱量（因此有一個有關熱和能量的通用轉換係數。）「活力」開始稱為energy（能量），第一個提出的是1807年的托馬斯·楊。

活力後來又定義為

\displaystyle {\frac {1}{2}}\sum _{i}m_{i}v_{i}^{2}

可以用來了解功和動能之間的轉換，這大部份是賈斯帕-古斯塔夫·科里奧利和讓-維克托·彭賽列在1819至1839年之間的貢獻，前者稱之為「quantité de travail」（功的量），後者稱之為「travail mécanique」（力學功）。

1837年時卡爾·弗里德里希·莫爾在歐洲物理期刊發表的《Über die Natur der Wärme》用以下的文字表示能量守恆，是最早期的敘述之一：「在54種已知的化學元素以外，在物理世界中還有一種量稱為Kraft（功或是能）。依照運動、化學親和力、凝聚、電力、光或是磁力的條件不同，這種量可能會出現，也可能會改變為其他形式。 」

機械能和熱的等效性

在能量守恆定律發展過程中，熱功當量的發現是其中重要的階段。熱質說認為熱不會增加也不會減少，而能量守恆定律認為熱和機械能是可以互相轉換的。

在18世紀中，俄國科學家米哈伊爾·瓦西里耶維奇·羅蒙諾索夫提出熱和動能的理論，反對熱質說的概念。在分析實驗的結果後，羅蒙諾索夫認為熱不是由熱質流體所傳播。

1798年班傑明·湯普森量測了加農炮鏜孔時因摩擦所產生的熱，提供了熱是一種動能形式的概念，其量測結果也反對熱質說，不過精確度不夠，因此造成當時的懷疑。

機械能和熱等價的概念最早是由德國外科醫師尤利烏斯·馮·邁爾在1842年提出[4]，這是他在去荷屬東印度航行途中發現的，他發現他的病人在天氣較熱時，其血液呈較深的紅色，因為消耗較少的氧氣（也就是較少的能量）來維持體溫。邁爾發現熱和機械能都是能量的形式，在物理知識進步後，他在1845年發表聲明，說明兩者之間的量化關係[5]。

同時，詹姆斯·普雷斯科特·焦耳在1843年藉由一連串的實驗，獨立的發現機械能和熱等價[6]。在著名的「焦耳設備」中，一個漸漸下降的重物連接一個繩子，繩子會使水中的槳旋轉，他證明重物下降減少的重力位能等於因槳在水中的摩擦力，帶來水內能的增加。

在1840至1843年之間，丹麥工程師路德維格·奧古斯特·柯丁也進行了類似的實驗[7]，但在丹麥以外的國家很少有人知道。

邁爾和焦耳的研究在當時都受到很大的阻力及忽視，不過最後焦耳還是得到較多的認可。

1844年時威廉·羅伯特·格羅夫提出有關機械能、熱能、光、電及磁的關係，處理方式是將它們全部視為單一種「力」（以現在的觀點來看，是能量）的表現，在1874年時格羅夫在《The Correlation of Physical Forces》中提及他的理論[8]。1847年赫爾曼·馮·亥姆霍茲藉著焦耳、尼古拉·卡諾及埃米爾·克拉佩龍的早期研究，得到了和格羅夫類似的結論，發表在《’Über die Erhaltung der Kraft》（保守力）一書中[9]。此次出版代表此定律已得到一般性的認可。

1850年時，威廉·約翰·麥誇恩·蘭金首次使用「熱力學第一定律」來描述此定律[10]。

1877年時，彼得·泰特在有創意的讀了《自然哲學的數學原理》中的命題40和41後，聲稱此定律起源自牛頓。後來這被視為是輝格史的一個例子[11]。

質能等價

主條目：質能等價

物質是由原子、電子、中子和質子等粒子所組成，有靜止質量。以19世紀的認知，這類的靜止質量是守恆的，但愛因斯坦在1905年的相對論認為上述的質量對應「靜止能量」，也就是說質量可以轉換為其他等效（非質量）的能量形式，例如動能、位能及電磁輻射能 。當發生上述情形時，靜止質量是不守恆的。只有考慮質量及能量的總能量才會守恆。

像電子和中子都有靜止質量，兩者碰撞後會湮滅，將其質量轉換為光子的電磁輻射能，沒有靜止質量。若這發生在一個封閉系統中，光子及能量都沒有釋放在外界的環境，其總能量或轉換為質量的總質量都不會變化。產生的電磁輻射能恰好和電子和中子的靜止質量相等。相對的，非物質的能量形式也可以產生有靜止質量的物質。

因此能量守恆（總能量，包括靜止能量）及質量守恆（總質量，不止是靜止質量）在相對論下仍然成立，而且是等效的定律，但以19世紀的觀點，這是兩個不同的定律。

β衰變下的能量守恆

1911年時發現β衰變發射的電子有連續光譜，而不是離散光譜，當時β衰變只是單純由核子中發射一個電子，上述的現象認為看似不符合能量守恆定律。此問題後來在1933年由恩里科·費米用費米交互作用描述β衰變，認為β衰變時除了發射電子，還發射帶有許多能量的反電子微中子，才解決上述的問題。

 

以及熱力學︰

First law of thermodynamics

Main article: First law of thermodynamics

 

For a closed thermodynamic system, the first law of thermodynamics may be stated as:

\displaystyle \delta Q=\mathrm {d} U+\delta W , or equivalently, \displaystyle \mathrm {d} U=\delta Q-\delta W,

where \displaystyle \delta Q is the quantity of energy added to the system by a heating process, \displaystyle \delta W is the quantity of energy lost by the system due to work done by the system on its surroundings and \displaystyle \mathrm {d} U is the change in the internal energy of the system.

The δ’s before the heat and work terms are used to indicate that they describe an increment of energy which is to be interpreted somewhat differently than the \displaystyle \mathrm {d} U increment of internal energy (see Inexact differential). Work and heat refer to kinds of process which add or subtract energy to or from a system, while the internal energy \displaystyle U is a property of a particular state of the system when it is in unchanging thermodynamic equilibrium. Thus the term “heat energy” for \displaystyle \delta Q means “that amount of energy added as the result of heating” rather than referring to a particular form of energy. Likewise, the term “work energy” for \displaystyle \delta W means “that amount of energy lost as the result of work”. Thus one can state the amount of internal energy possessed by a thermodynamic system that one knows is presently in a given state, but one cannot tell, just from knowledge of the given present state, how much energy has in the past flowed into or out of the system as a result of its being heated or cooled, nor as the result of work being performed on or by the system.

Entropy is a function of the state of a system which tells of limitations of the possibility of conversion of heat into work.

For a simple compressible system, the work performed by the system may be written:

\displaystyle \delta W=P\,\mathrm {d} V,

where \displaystyle P is the pressure and \displaystyle dV is a small change in the volume of the system, each of which are system variables. In the fictive case in which the process is idealized and infinitely slow, so as to be called quasi-static, and regarded as reversible, the heat being transferred from a source with temperature infinitesimally above the system temperature, then the heat energy may be written

\displaystyle \delta Q=T\,\mathrm {d} S,

where \displaystyle T is the temperature and \displaystyle \mathrm {d} S is a small change in the entropy of the system. Temperature and entropy are variables of state of a system.

If an open system (in which mass may be exchanged with the environment) has several walls such that the mass transfer is through rigid walls separate from the heat and work transfers, then the first law may be written:[22]

\displaystyle \mathrm {d} U=\delta Q-\delta W+u'\,dM,

where \displaystyle dM is the added mass and \displaystyle u' is the internal energy per unit mass of the added mass, measured in the surroundings before the process.

 

得以借『巨觀』描述耶？！

更好的是，那個『能量守恆定律』還有

Noether’s theorem

Main article: Noether’s theorem

Emmy Noether (1882-1935) was an influential mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics.

The conservation of energy is a common feature in many physical theories. From a mathematical point of view it is understood as a consequence of Noether’s theorem, developed by Emmy Noether in 1915 and first published in 1918. The theorem states every continuous symmetry of a physical theory has an associated conserved quantity; if the theory’s symmetry is time invariance then the conserved quantity is called “energy“. The energy conservation law is a consequence of the shift symmetry of time; energy conservation is implied by the empirical fact that the laws of physics do not change with time itself. Philosophically this can be stated as “nothing depends on time per se”. In other words, if the physical system is invariant under the continuous symmetry of time translation then its energy (which is canonical conjugate quantity to time) is conserved. Conversely, systems which are not invariant under shifts in time (an example, systems with time dependent potential energy) do not exhibit conservation of energy – unless we consider them to exchange energy with another, external system so that the theory of the enlarged system becomes time invariant again. Conservation of energy for finite systems is valid in such physical theories as special relativity and quantum theory (including QED) in the flat space-time.

First page of Emmy Noether‘s article “Invariante Variationsprobleme” (1918), where she proved her theorem.

 

數理證明呦☆☆