一九四八年時，荷蘭物理學家『亨德里克‧卡西米爾』 Hendrik Casimir 提出了『真空不空』的『議論』。因為依據『量子場論』，『真空』也得有『最低能階』，因此『真空能量』不論因不因其『實虛』粒子之『生滅』，總得有一個『量子態』。由於已知『原子』與『分子』的『主要結合力』是『電磁力』，那麼該『如何』說『真空』之『量化』與『物質』的『實際』是怎麽來『配合』的呢？因此他『計算』了這個『可能效應』之『大小』，然而無論是哪種『震盪』所引起的，他總是得要面臨『無窮共振態』 $\langle E \rangle = \frac{1}{2} \sum \limits_{n}^{\infty} E_n$ 的『問題』，這也就是說『平均』有『多少』各種能量的『光子？』所參與 $h\nu + 2h\nu + 3h\nu + \cdots$ 的『問題』？據知『卡西米爾』用『歐拉』等之『可加法』，得到了 ${F_c \over A} = -\frac {\hbar c \pi^2} {240 a^4}$ 。

此處之『負』 $-$ 代表『吸引力』，而今早也已經『證實』的了，真不知『宇宙』是果真先就有『計畫』的嗎？還是說『人們』自己還在『幻想』的呢？？

─── 《【SONIC Π】電聲學之電路學《四》之《 V!》‧下》

也曾醉心於大自然之神奇！想當初聽聞『真空不空』之論時，仔細思考過借此能否邏輯歸結出『色不異空，空不異色』命題耶？終究因為『空』、『色』難定『確指』而作罷！！

反倒記起更早前好奇那『光』怎麼會是『電磁波』呢？它還可以借

偶極子

在電磁學裏，有兩種偶極子（dipole）：電偶極子是兩個分隔一段距離，電量相等，正負相反的電荷。磁偶極子是一圈封閉循環的電流，例如一個有常定電流運行的線圈，稱為載流迴路。偶極子的性質可以用它的偶極矩描述。

電偶極矩（ $\displaystyle \mathbf {p}$ ）由負電荷指向正電荷，大小等於正電荷量乘以正負電荷之間的距離。磁偶極矩（ $\displaystyle \mathbf {m}$ ）的方向，根據右手法則，是大拇指從載流迴路的平面指出的方向，而其它手指則指向電流運行方向，磁偶極矩的大小等於電流乘以線圈面積。

除了載流迴路以外，電子和許多基本粒子都擁有磁偶極矩。它們都會產生磁場，與一個非常小的載流迴路產生的磁場完全相同。但是，現時大多數的科學觀點認為這個磁偶極矩是電子的自然性質，而非由載流迴路生成。

永久磁鐵的磁偶極矩來自於電子內稟的磁偶極矩。長條形的永久磁鐵稱為條形磁鐵，其兩端稱為指北極和指南極，其磁偶極矩的方向是由指南極朝向指北極。這常規與地球的磁偶極矩恰巧相反：地球的磁偶極矩的方向是從地球的地磁北極指向地磁南極。地磁北極位於北極附近，實際上是指南極，會吸引磁鐵的指北極；而地磁南極位於南極附近，實際上是指北極，會吸引磁鐵的指南極。羅盤磁針的指北極會指向地磁北極；條形磁鐵可以當作羅盤使用，條形磁鐵的指北極會指向地磁北極。

根據當前的觀察結果，磁偶極子產生的機制只有兩種，載流迴路和量子力學自旋。科學家從未在實驗裏找到任何磁單極子存在的證據。

地球磁場可以近似為一個磁偶極子的磁場。但是，圖內的 N 和 S 符號分別標示地球的地理北極和地理南極。這標示法很容易引起困惑。實際而言，地球的磁偶極矩的方向，是從地球位於地理北極附近的地磁北極，指向位於地理南極附近的地磁南極；而磁偶極子的方向則是從指南極指向指北極。

電極偶子的等值線圖。等值曲面清楚地區分於圖內。

凌空『輻射』勒？？

Dipole radiation

In addition to dipoles in electrostatics, it is also common to consider an electric or magnetic dipole that is oscillating in time. It is an extension, or a more physical next-step, to spherical wave radiation.

In particular, consider a harmonically oscillating electric dipole, with angular frequency ω and a dipole moment p₀ along the ẑ direction of the form

\displaystyle \mathbf {p} (\mathbf {r} ,t)=\mathbf {p} (\mathbf {r} )e^{-i\omega t}=p_{0}{\hat {\mathbf {z} }}e^{-i\omega t}.

In vacuum, the exact field produced by this oscillating dipole can be derived using the retarded potential formulation as:

\displaystyle {\begin{aligned}\mathbf {E} &={\frac {1}{4\pi \varepsilon _{0}}}\left\{{\frac {\omega ^{2}}{c^{2}r}}\left({\hat {\mathbf {r} }}\times \mathbf {p} \right)\times {\hat {\mathbf {r} }}+\left({\frac {1}{r^{3}}}-{\frac {i\omega }{cr^{2}}}\right)\left(3{\hat {\mathbf {r} }}\left[{\hat {\mathbf {r} }}\cdot \mathbf {p} \right]-\mathbf {p} \right)\right\}e^{\frac {i\omega r}{c}}e^{-i\omega t}\\\mathbf {B} &={\frac {\omega ^{2}}{4\pi \varepsilon _{0}c^{3}}}({\hat {\mathbf {r} }}\times \mathbf {p} )\left(1-{\frac {c}{i\omega r}}\right){\frac {e^{i\omega r/c}}{r}}e^{-i\omega t}.\end{aligned}}

For rω/c ≫ 1, the far-field takes the simpler form of a radiating “spherical” wave, but with angular dependence embedded in the cross-product:^[9]

\displaystyle {\begin{aligned}\mathbf {B} &={\frac {\omega ^{2}}{4\pi \varepsilon _{0}c^{3}}}({\hat {\mathbf {r} }}\times \mathbf {p} ){\frac {e^{i\omega (r/c-t)}}{r}}={\frac {\omega ^{2}\mu _{0}p_{0}}{4\pi c}}({\hat {\mathbf {r} }}\times {\hat {\mathbf {z} }}){\frac {e^{i\omega (r/c-t)}}{r}}=-{\frac {\omega ^{2}\mu _{0}p_{0}}{4\pi c}}\sin(\theta ){\frac {e^{i\omega (r/c-t)}}{r}}\mathbf {\hat {\phi }} \\\mathbf {E} &=c\mathbf {B} \times {\hat {\mathbf {r} }}=-{\frac {\omega ^{2}\mu _{0}p_{0}}{4\pi }}\sin(\theta )\left({\hat {\phi }}\times \mathbf {\hat {r}} \right){\frac {e^{i\omega (r/c-t)}}{r}}=-{\frac {\omega ^{2}\mu _{0}p_{0}}{4\pi }}\sin(\theta ){\frac {e^{i\omega (r/c-t)}}{r}}{\hat {\theta }}.\end{aligned}}

The time-averaged Poynting vector

\displaystyle \langle \mathbf {S} \rangle =\left({\frac {\mu _{0}p_{0}^{2}\omega ^{4}}{32\pi ^{2}c}}\right){\frac {\sin ^{2}(\theta )}{r^{2}}}\mathbf {\hat {r}}

is not distributed isotropically, but concentrated around the directions lying perpendicular to the dipole moment, as a result of the non-spherical electric and magnetic waves. In fact, the spherical harmonic function (sin θ) responsible for such toroidal angular distribution is precisely the l = 1 “p” wave.

The total time-average power radiated by the field can then be derived from the Poynting vector as

\displaystyle P={\frac {\mu _{0}\omega ^{4}p_{0}^{2}}{12\pi c}}.

Notice that the dependence of the power on the fourth power of the frequency of the radiation is in accordance with the Rayleigh scattering, and the underlying effects why the sky consists of mainly blue colour.

A circular polarized dipole is described as a superposition of two linear dipoles.

請問誰能先知道『真空竟然也有阻抗』哩☻？

Impedance of free space

The impedance of free space, $Z 0$ , is a physical constant relating the magnitudes of the electric and magnetic fields of electromagnetic radiation travelling through free space. That is, $Z 0 = | E | / | H |$ , where $| E |$ is the electric field strength and $| H |$ is the magnetic field strength. It has an exactly defined value

\displaystyle Z_{0}=(119.916~983~2)\pi ~\Omega \approx 376.730~313~461~77\ldots ~\Omega .

The impedance of free space (more correctly, the wave impedance of a plane wave in free space) equals the product of the vacuum permeability $μ 0$ and the speed of light in vacuum $c 0$ . Since the values of these constants are exact (they are given in the definitions of the ampere and the metre respectively), the value of the impedance of free space is likewise exact.

Terminology

The analogous quantity for a plane wave travelling through a dielectric medium is called the intrinsic impedance of the medium, and designated $η$ (eta). Hence $Z 0$ is sometimes referred to as the intrinsic impedance of free space,^[1] and given the symbol $η 0$ .^[2] It has numerous other synonyms, including:

wave impedance of free space,^[3]
the vacuum impedance,^[4]
intrinsic impedance of vacuum,^[5]
characteristic impedance of vacuum,^[6]
wave resistance of free space.^[7]

Relation to other constants

From the above definition, and the plane wave solution to Maxwell’s equations,

\displaystyle Z_{0}={\frac {E}{H}}=\mu _{0}c_{0}={\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}={\frac {1}{\varepsilon _{0}c_{0}}},

where

μ 0

is the magnetic constant,

ε 0

is the electric constant,

c 0

is the speed of light in free space.^[8]^[9]

The reciprocal of $Z 0$ is sometimes referred to as the admittance of free space and represented by the symbol $Y 0$ .

Exact value

Since 1948, the definition of the SI unit ampere has relied upon choosing the numerical value of $μ 0$ to be exactly 4 $π$ × 10⁻⁷ H/m. Similarly, since 1983 the SI metre has been defined relative to the second by choosing the value of $c 0$ to be 299792458 m/s. Consequently,

\displaystyle Z_{0}=\mu _{0}c_{0}=119.916\,9832\,\pi ~\Omega

exactly,

\displaystyle Z_{0}\approx 376.730\,313\,461\,77\ldots ~\Omega .

This chain of dependencies will change if the ampere is redefined in 2018. See New SI definitions.

然後明白立地於『事實』之『實驗』，才是真『科學精神』呦☺！

Wheatstone bridge

Wheatstone bridge circuit diagram. The unknown resistance R_x is to be measured; resistances R₁, R₂and R₃ are known and R₂ is adjustable. If the measured voltage V_G is 0, then R₂/R₁ = R_x/R₃.

A Wheatstone bridge is an electrical circuit used to measure an unknown electrical resistance by balancing two legs of a bridge circuit, one leg of which includes the unknown component. The primary benefit of the circuit is its ability to provide extremely accurate measurements (in contrast with something like a simple voltage divider).^[1] Its operation is similar to the original potentiometer.

The Wheatstone bridge was invented by Samuel Hunter Christie in 1833 and improved and popularized by Sir Charles Wheatstone in 1843. One of the Wheatstone bridge’s initial uses was for the purpose of soils analysis and comparison.^[2]

Operation

In the figure, $\displaystyle \scriptstyle R_{x}$ is the unknown resistance to be measured; $\displaystyle \scriptstyle R_{1},$ $\displaystyle \scriptstyle R_{2},$ and $\displaystyle \scriptstyle R_{3}$ are resistors of known resistance and the resistance of $\displaystyle \scriptstyle R_{2}$ is adjustable. The resistance $\displaystyle \scriptstyle R_{2}$ is adjusted until the bridge is “balanced” and no current flows through the galvanometer $\displaystyle \scriptstyle V_{g}$ . At this point, the voltage between the two midpoints (B and D) will be zero. Therefore the ratio of the two resistances in the known leg $\displaystyle \scriptstyle (R_{2}/R_{1})$ is equal to the ratio of the two in the unknown leg $\displaystyle \scriptstyle (R_{x}/R_{3})$ . If the bridge is unbalanced, the direction of the current indicates whether $\displaystyle \scriptstyle R_{2}$ is too high or too low.

At the point of balance,

\displaystyle {\begin{aligned}{\frac {R_{2}}{R_{1}}}&={\frac {R_{x}}{R_{3}}}\\[4pt]\Rightarrow R_{x}&={\frac {R_{2}}{R_{1}}}\cdot R_{3}\end{aligned}}

Detecting zero current with a galvanometer can be done to extremely high precision. Therefore, if $\displaystyle \scriptstyle R_{1},$ $\displaystyle \scriptstyle R_{2},$ and $\displaystyle \scriptstyle R_{3}$ are known to high precision, then $\displaystyle \scriptstyle R_{x}$ can be measured to high precision. Very small changes in $\displaystyle \scriptstyle R_{x}$ disrupt the balance and are readily detected.
Alternatively, if $\displaystyle \scriptstyle R_{1},$ $\displaystyle \scriptstyle R_{2},$ and $\displaystyle \scriptstyle R_{3}$ are known, but $\displaystyle \scriptstyle R_{2}$ is not adjustable, the voltage difference across or current flow through the meter can be used to calculate the value of $\displaystyle \scriptstyle R_{x},$ using Kirchhoff’s circuit laws. This setup is frequently used in strain gauge and resistance thermometer measurements, as it is usually faster to read a voltage level off a meter than to adjust a resistance to zero the voltage.

Derivation

First, Kirchhoff’s first law is used to find the currents in junctions B and D:

\displaystyle {\begin{aligned}I_{3}-I_{x}+I_{G}&=0\\I_{1}-I_{2}-I_{G}&=0\end{aligned}}

Then, Kirchhoff’s second law is used for finding the voltage in the loops ABD and BCD:

\displaystyle {\begin{aligned}(I_{3}\cdot R_{3})-(I_{G}\cdot R_{G})-(I_{1}\cdot R_{1})&=0\\(I_{x}\cdot R_{x})-(I_{2}\cdot R_{2})+(I_{G}\cdot R_{G})&=0\end{aligned}}

When the bridge is balanced, then $I G = 0$ , so the second set of equations can be rewritten as:

\displaystyle {\begin{aligned}I_{3}\cdot R_{3}&=I_{1}\cdot R_{1}-(1)\\I_{x}\cdot R_{x}&=I_{2}\cdot R_{2}-(2)\end{aligned}}

Then, the equations (1)&(2) are divided(=equation(1)/equation(2)) and rearranged, giving:

\displaystyle R_{x}={{R_{2}\cdot I_{2}\cdot I_{3}\cdot R_{3}} \over {R_{1}\cdot I_{1}\cdot I_{x}}}

From the first law, $I 3 = I x$ and $I 1 = I 2$ . The desired value of $R x$ is now known to be given as:

\displaystyle R_{x}={{R_{3}\cdot R_{2}} \over {R_{1}}}

If all four resistor values and the supply voltage ( $V S$ ) are known, and the resistance of the galvanometer is high enough that $I G$ is negligible, the voltage across the bridge ( $V G$ ) can be found by working out the voltage from each potential divider and subtracting one from the other. The equation for this is:

\displaystyle V_{G}=\left({R_{2} \over {R_{1}+R_{2}}}-{R_{x} \over {R_{x}+R_{3}}}\right)V_{s}

where $V G$ is the voltage of node D relative to node B.

Significance

The Wheatstone bridge illustrates the concept of a difference measurement, which can be extremely accurate. Variations on the Wheatstone bridge can be used to measure capacitance, inductance, impedance and other quantities, such as the amount of combustible gases in a sample, with an explosimeter. The Kelvin bridge was specially adapted from the Wheatstone bridge for measuring very low resistances. In many cases, the significance of measuring the unknown resistance is related to measuring the impact of somephysical phenomenon (such as force, temperature, pressure, etc.) which thereby allows the use of Wheatstone bridge in measuring those elements indirectly.

The concept was extended to alternating current measurements by James Clerk Maxwell in 1865 and further improved by Alan Blumlein around 1926.

Modifications of the fundamental bridge

Kelvin bridge

The Wheatstone bridge is the fundamental bridge, but there are other modifications that can be made to measure various kinds of resistances when the fundamental Wheatstone bridge is not suitable. Some of the modifications are:

Carey Foster bridge, for measuring small resistances
Kelvin bridge, for measuring small four-terminal resistances
Maxwell bridge, and Wien bridge for measuring reactive components.

祇是那時又沒有『工具』︰

計算甚感無趣，今日且『補過』吧☆

FreeSandal

每日彙整: 2018-08-04

STEM 隨筆︰古典力學︰轉子【五】《電路學》五【電感】 III‧阻抗‧A‧下

偶極子