中央研究院物理研究所高中物理科學培育計畫

電磁學主要觀念的演進

Evolution of Concepts

任慶運東吳大學物理學系

電磁理論所需的高等數學

• 多變數向量函數• 多重積分• 偏微分方程式

兩種表述方式：

概念陳述數學表式

第一分概念陳述

Track One

Conceptual

Development

• 從「瞬時隔空作用」到「場」 法拉第 M. Faraday

from “action-at-a-distance” to “field”

• 從「電」「磁」到「電磁」 厄斯特 H. C. Örsted from “electricity and magnetism’’ to “electromagnetism”

• 從「光」到「電磁波」 馬克斯威 J. C. Maxwell

from “light” to “electromagnetic wave”

• 從「以太說」到「相對論」 愛因斯坦 A. Eistein

from “ether theory” to “relativity”

從「瞬時隔空作用」到「場」

從牛頓到法拉第之前

• 牛頓（ Newton ）：萬有重力（引力）

重力：

• 庫倫（ Coulomb ）：電力、磁力

庫倫力：

平方反比律 inverse-square law

gravitaty 應譯為「重力」而非「引力」

互相吸引是重力的屬性

庫倫力則兼有吸引與排斥兩種屬性

若以屬性命名，庫倫力豈不成為「引斥力」

關於譯名

「平方反比律」是怎麼知道的？

科學方法 演繹法（外籀術） deduction

歸納法（內籀術） induction

觀測（ observation ） 天文

實驗（ experiment ） 物理

考察（ investigation ） 生物

• Tycho Brahe 第谷 觀測• Johannes Kepler 刻卜勒 行星運動三定律• Issac Newton 牛頓 萬有重力：平方反比定律

重力的平方反比律

The electric force is about a billion-billion-billion-billion times stronger than gravitation.

109 × 109 × 109 × 109 = 1036

庫倫電力約為萬有重力的一兆兆兆倍

（萬萬曰億，萬億約兆） 104 108 1012

「黃帝為法，數有十等。」「十等者，億、兆、京、垓、秭、壤、溝、澗、正、載。」

大小怎麼比？

估計數量級 estimating order of magnitude

比較兩個粒子的重力與庫倫力：

重力：

庫倫力：

Fe / Fg = k q2/Gm2

以氦原子核（即 粒子）為例

m = 6.6 × 10-27 kg q = 3.2 × 10-19 C k = 9.0 × 109 C m2 N/C2

G = 6.7 × 10-11 m2 N/kg2

Fe / Fg = 9.0 × 109 ×(3.2 × 10-19 ) 2 / 6.7 × 10-11 ×(6.6 × 10-27 ) 2

只數次方 ( 數量級 )

= 10 1 ＋ 9 ＋ 1 － 38 － 1 ＋ 11 － 2 ＋ 54

= 10 35

• 重力極其微弱， 故用巨大天體。

• 庫倫電力雖甚大於萬有重力， 仍須使用精密儀器仔細測量。

扭秤（ torsion balance ）• J. Michell 發明 測量小質點間重力• H. Cavendish 利用 測量小質點間重力• C. A. Coulomb 發明 測量電力與磁力

Maxwell, Treatise, Art. 38 and 215

electric forcebetween two charges Q1 and Q2

separated by a distance r

action-at-a-distance瞬時隔空作用

名詞翻譯action-at-a-distance

• 超距作用（物理學名詞第二次增訂本）• 瞬間作用（劉源俊）• 隔空抓藥（新聞詞彙）• 隔空力• 隔空瞬時力• 瞬時隔空力

瞬時隔空作用

Although Newton had expressed doubt

about the idea of action at a distance,

most subsequent physics,

as well as Newton’s,

was based on that idea.

Emilio Segrè

對瞬時隔空作用的檢討

Before Faraday, electric and magnetic,

as well as gravitational, forces were usually

considered as acting across space,

separating the interacting objects.

To his simple mind, however,

such an “action in distance”

did not seem to make physical sense,

and seeing a load being moved from place to place

he wanted also to see a rope which drags it

or a stick with which it is being pushed.

George Gamow

對瞬時隔空作用的檢討

關於法拉第

數學知識不足 lack of formal mathematical knowledge

十九世紀最了不起的實驗物理學家 the greatest experimental physicist of the

nineteenth century

It was perhaps for the advantage of science that Faraday, though thoroughly conscious of the fundamental forms of space, time, and force, was not a professed mathematician. He was not tempted to enter into the many interesting researches in pure mathematics which his discoveries would have suggested if they had been exhibited in a mathematical form, and he did not feel called upon either to force his results into a shape acceptable to the mathematical taste of the time, or to express them in a form which mathematicians might attack. He was thus left at leisure to do his proper work, to coordinate his ideas with his facts, and to express them in natural untechnical language.

Treatise, Art. 528.

關於法拉第 馬克斯威的說法

關於法拉第

• It was perhaps for the advantage of science that Faraday, though thoroughly conscious of the fundamental forms of space, time, and force, was not a professed mathematician.

馬克斯威的說法 Treatise, Art. 528.

• He was not tempted to enter into the many interesting researches in pure mathematics which his discoveries would have suggested if they had been exhibited in a mathematical form,

關於法拉第 馬克斯威的說法 Treatise, Art. 528.

（馬克斯威屬於這一類）

• and he did not feel called upon either to force his results into a shape acceptable to the mathematical taste of the time,

關於法拉第 馬克斯威的說法 Treatise, Art. 528.

• or to express them in a form which mathematicians might attack.

關於法拉第 馬克斯威的說法 Treatise, Art. 528.

• He was thus left at leisure to do his proper work, to coordinate his ideas with his facts, and to express them in natural untechnical language.

關於法拉第 馬克斯威的說法 Treatise, Art. 528.

leisure!

觀念的演進evolution of conceptsforce as action-at-a-distance

瞬時隔空作用

lines of force場線

field of force力場

Potential勢

法拉第的手稿

場線

F = q0 E

從「電」「磁」到「電磁」

電 磁

安培

法拉第

電流

磁場變化

「電」「磁」

• Electric charges did not influence the magnets in any way;

• Neither did the magnets influence the electric charges.

The honor of discovering the bridge between electricity and magnetism fell to a Danish physicist, Hans Christian Örsted.

Hans Christian Örsted (1777-1851)

Thus electromagnetism, as Örsted called it, became a reality!

July 21, 1820

安培的想法及發現 不僅電流對磁針有作用， 電流對電流也會有作用。 Not only does an electric current act

on magnetic needle but two electric currents act on one another.

電流之間的磁力

安培發現電流間之磁力的四個實驗

馬克斯威對安培的推崇：“電學上的牛頓”

馬克斯威對安培的推崇“電學上的牛頓”

Treatise, Art. 528

法拉第定律

Gamow p.150, Fig. V-15

法拉第的發現

• The magnetic action was at right angles to the direction of the current producing it.

• The central point of electromagnetic induction was the cutting of the lines of force by the conductor.

• Unity of Forces.

• 電流及其所感應產生之磁力互相垂直。

• 電磁感應的關鍵在於導線切割磁力線。

• 諸力一統。

from Segrè

馬克斯威：集大成者• It is mainly with

the hope of making these ideas the basis of a mathematical method that I have undertaken this treatise.

Treatise Art. 528

• 我之所以撰寫本書，主要就是希望以法拉第的這些觀念為基礎，（建立）一套數理方法 。

（電磁學的理論）

Segrè, p. 159

James Clerk Maxwell (1831-1879)

• The greatest theoretician of the 19th century （詳第二分）

• The founder of modern electrical theory （詳第二分）

• One of the founders of thermodynamics and statistical mechanics

從「光」到「電磁波」

空間中的電磁振盪

Gamow p. 153, Fig. V-17

(a)電能量貯存於空間中之電場

(b) 電能量轉為磁場中之磁能量

(c) 磁能量貯存於空間中之磁場

(d)磁能量轉為電場中之電能量

(e) 電能量貯存於空間中之電場

“Hence the agreement or disagreement of the values V and of v furnishes a test of the electromagnetic theory of light.”

Maxwell, Treatise, Art. 786.

（註：馬克斯威以 v 表示光速 ，以 V 表示電磁波傳播速度）

電磁振盪以波的形式在空間中傳播

Heinrich Hertz

電磁波1888

Segrè, p. 177

從「以太說」到「相對論」

• 光的傳播速度

• 光的傳播媒介

光速的測量

• Galileo Galilei (1638)

• O. Roemer (1675)

• A. H. L. Fizeau (1851)

• A. A. Michelson (1887)

Whittaker

•伽利略實驗 (1638)

兩個燈籠相距甚遠，閃光為訊號，測量時差•結果： 時差之小，難以測量

光速的測量

• Roemer 觀測 木星第一衛星 Io 之（掩）食• 平均繞行週期： 42 時 28 分 16 秒

• 半年後掩食時間比預期晚約二十一分鐘

光速的測量

光速的測量Roemer 觀測結果

延遲二十二分鐘的原因：木星至地球距離增加

地球公轉直徑 ＝ 2.83×1013 cm

光速的測量

光速的測量

Fizeau 實驗：運動介質中的光速光速的測量

邁克森 -毛立實驗 Michelson and Morley

光速的測量

邁克森 -毛立實驗 Michelson and Morley

光速的測量

邁克森 -毛立實驗 Michelson and Morley

光速的測量

邁克森 -毛立實驗 Michelson and Morley

光速的測量

以太說光介質

以太是甚麼？ether 的字源

to kindle, burn, shine• In ancient cosmological speculation conceiv

ed as an element filling all space beyond the sphere of the moon, and as the constituent substance of the stars and planets and of their spheres

• The clear sky; the upper regions of space beyond the clouds; the medium filling the upper regions of space, as the air fills the lower regions.

光介質

from Oxford English Dictionary

• in modern physics: A substance of great elasticity and subtilty, believed to permeate the whole of planetary spaces, not only filling the interplanetary spaces, but also the interstices between the particles of air and other mattter on the earth; the medium through which the waves of light are propagated. Sometimes called the luminiferous ether.

光介質

from Oxford English Dictionary

from Ｍ ichelson and Morley experiment and Einstein’s theory of relativity :

The ether concept is unnecessary.

光介質

Clouds over Classical Physics

• The beauty and clearness of the dynamical theory, which asserts light and heat to be modes of motion, is at present obscured by two clouds.

Lord Kelvin (1900)

1. How the earth can move through an elastic solid such as essentially is the luminiferous ether in which electromagnetic waves were assumed to propagate.

Lord Kelvin (1900)

Clouds over Classical Physics

2. failure of the Mawell-Bolzmann doctrine regarding the equipartition of energy.

Lord Kelvin (1900)

Clouds over Classical Physics

第二分數學表式

Track Two

Mathematical

Formulation

目標馬克斯威方程式之數學表式

步驟所需數學

（回想馬克斯威對法拉第的看法）

馬克斯威方程式之數學表式所需之數學

時變向量場 time-varying vector field

內積 inner product or scalar product

外積 outer product or vector product

通量與散度 flux and divergence

環量與旋度 circulation and curl

數理新知 五門開解

一曰幾何直觀 geometrical intuition

二曰術語名相 nomenclatural terminology

三曰數理記號 mathematical notation

四曰定義界說 analytical definition

五曰演算公式 calculational formula

(non-verbatim correspondence)

（以此比較馬克斯威與法拉第）

• 描述電磁現象的物理量隨空間位置變化• 場：空間位置的函數• 有方向性• 向量：電場、磁場• 隨時間變化

時變向量場以空間時間為變數的向量值函數

內積• 幾何直觀：• 術語名相：• 數理記號：• 定義界說：

• 演算公式：

投影內積

外積• 幾何直觀：• 術語名相：• 數理記號：• 定義界說：

• 演算公式：

面積外積

通量 flux ：幾何直觀

通量 flux ：幾何直觀

（）（）

通量 flux

• 幾何直觀：• 術語名相：• 數理記號：• 定義界說：

• 演算公式：

通量 E

流體流量

逸出通量 outward flux ：幾何直觀

散度 divergence

• 幾何直觀：• 術語名相：• 數理記號：• 定義界說：

• 演算公式：

每單位體積逸出通量散度

線積分 line integral

線積分 line integral

線積分 line integral

線積分 line integral

• 幾何直觀：• 術語名相：• 數理記號：

• 定義界說：

• 演算公式：

線積分 line integral

沿線投影總和線積分

環量 circulation

• 幾何直觀：• 術語名相：• 數理記號：• 定義界說：

• 演算公式：

環量沿迴線投影總和

旋度 curl

• 幾何直觀：• 術語名相：• 數理記號：• 定義界說：

• 演算公式：

每單位面積沿邊環量旋度

高斯定理 Gauss’ Theorem

司托克士定理 Stokes’ Theorem

高斯定律 Gauss’ Law

位移電流 displacement current• 馬克斯威方程式中馬克斯威的貢獻

安培定律 Ampère’s Law

法拉第定律 Faraday’s Law

馬克斯威方程式 Maxwell’s Equations

波動方程式 wave equation

光速

Infallible Maxwell

• 馬克斯威電磁理論與相對論• 馬克斯威電磁理論與量子論

Segrè, p. 155

牛頓力學不符合勞倫茲 Lorentz 變換

馬克斯威電磁理論符合勞倫茲 Lorentz 變換

Infallible Maxwell

（馬比牛好？）

• Emilio Segrè ： From Falling Bodies to Radio Waves

• George Gamow： The Great Physicists from Galileo to Einstein

中央研究院物理研究所高中物理科學培育計畫

電磁學統一場論的範型

The Prototype of the Unified Field Theory

任慶運東吳大學物理學系