Electricity & Magnetism 電與磁
Chapter 20 Electrostatics
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Charge 電荷
- type:+and-
- unit: C
- neutral: no charge
Conservation of charge 電荷守恆定律
- charge can neither be created nor destroyed
電荷既不能憑空產生，也不能消滅 - in an isolated system, net charge = constant
在孤立系統內，淨電荷 = 定值 - charging: separating +q and -q
起電：將正、負電荷分開的過程
Conductor and insulator 導體與絕緣體
- conductor = material that has movable charge carriers
導體的載荷子可自由移動 - metal: free electrons (−)
金屬：自由電子（ − ） - salted water: ions (+ and −)
鹽水：離子（ + 或 − ） - insulator = material that has no movable charge carriers
絕緣體沒有可自由移動的載荷子
Transfer of electrons 轉移電子
- excess of electrons =⇒ −Q
擁有 額外電子 =⇒ −Q - shortage of electrons =⇒ +Q
缺乏電子 =⇒ +Q
Interaction between charges 電荷間的相互作用
direction: like charges repel; unlike charges attract
傾向：同性電荷互相排斥；異性電荷互相吸引
magnitude: Coulomb’s law
量值：庫倫定律
$$F=k\frac{Qq}{r^2}$$
$$k=\frac{1}{4\pi\varepsilon_{0}}\approx9\times10^{9}\mathrm{N}\mathrm{m}^{2}\mathrm{C}^{-2}$$
Charge redistribution 電荷再分佈
Earthing 接地
sharing charge with the Earth (huge body)
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Charging methods 起電方法
by friction 摩擦生電
by sharing 授受起電
by induction 感應起電
Attraction between charged and neutral objects
帶電與中性物體間的吸引力
neutral particle 中性粒子
charged object does not attract neutral particles
帶電物體不會吸引中性粒子
neutral object 中性物體
conductor 導電體：
charge redistribution =⇒ induced charges =⇒ attraction
電荷再分佈 =⇒ 感生電荷 =⇒ 產生吸引力
insulator 絕緣體：
polarization =⇒ induced charges =⇒ attraction
電極化 =⇒ 感生電荷 =⇒ 產生吸引力
Hazards and troubles due to electrostatic charge
靜電的不便及危害
- attracting dust
吸引塵埃 - electric shock
觸電 - sparks =⇒ fire or explosion
產生火花 =⇒ 引致火警，甚至爆炸
Application of electrostatics 靜電的應用
- precipitators
靜電除塵器 - spraying
靜電噴塗法 - photocopies and laser printers
影印機及激光打印機
Electric field 電場
region where charges experience an electric force
電荷受到電力作用的區域
Electric field strength 電場強度 E
E = electric force per unit +ve test charge at the position
E = 在該點上的 正檢驗電荷每單位所受到的電力
$$\begin{aligned}&\therefore\vec{F}=q\vec{E}\\&\text{unit: N C}^{-1}\text{or V m}^{-1}\end{aligned}$$
Electric field lines 電場線
- line density: magnitude of E
電場線密度：E 的量值 - arrow: direction of E
方向：E 的方向 - from + to − charges
以正電荷為起點，以負電荷為終點 - never cross, nor branch
不會相交，也不會出現分支
Uniform electric field 勻強電場
- E = constant, everywhere
E = 定值，處處相同 - field lines: straight, parallel and even
電場線：直線、平行、等距
Electric field strength E between two parallel plates
平行板之間的電場強度 E
$$E=\frac{V}{d}$$
Motion in uniform electric field
電荷於勻強電場中的運動
projectile motion 拋體運動
Electric field due to a point source Q
來自點源 Q 的電場
magnitude 量值：
$$E=k\frac{Q}{r^2}\quad\mathrm{where~}k=\frac{1}{4\pi\varepsilon_0}$$
direction 方向：
- +Q =⇒ E radially outwards
+Q =⇒ E 徑向外指 - −Q =⇒ E radially inwards
−Q =⇒ E 徑向內指
Resultant field due to many sources
多個電荷產生的合電場
vector sum 矢量相加
$$\begin{aligned}&\vec{E}=\vec{E}_{1}+\vec{E}_{2}+\cdots\\&\vec{F}=q(\vec{E}_{1}+\vec{E}_{2}+\cdots)\end{aligned}$$
Electric field patterns 電場圖形
Attracting each other 互相吸引
A neutral object can be attracted by a charged object due to induced charges.
電中性物體可被帶電物體影響而感生出電荷，使兩者互相吸引。
Repelling each other 互相排斥
The electric forces on the two balls are action and reaction. The horizontal forces are of the same size.
兩球間的電力為作用力與反作用力。兩力的垂直分量量值相同。
Effect of induction 靜電感應的影響
Induction cannot create net charge; it only separates + and − charges.
靜電感應並不憑空產生淨電荷，只能 分開正、負電荷。
Electric field lines 電場線
The E‑field lines show the direction of force on +q
電場線顯示的是電力作用在正電荷上的方向。
Chapter 21 Circuit and Power
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Current 電流 I
flow of charge (per unit time)
電荷的流動（ 每單位時間 ）
$$I=\frac{Q}{t}\quad\mathrm{or}\quad Q=It$$
$$unit:A(1A=1Cs^{-1})$$
conventional: ϐlow of +q
傳統電流 = 正電荷的流動
Voltage across two points 兩點之間的電壓 V
- voltage = emf of battery or pd (potential difference) across resistor
電壓 = 電池的電動勢 或者電阻器兩端的電勢差 - unit: V
- emf ε = voltage gain = energy supplied per unit charge
電動勢 ε = 電壓增幅 = 每單位電荷獲得的能量 - pd = voltage drop = energy consumed per unit charge
電勢差 = 電壓跌幅 = 每單位電荷損失的能量 - at earthed point, potential = 0 V
接地點的電勢為 0 V
Current direction 電流的方向
in external circuit 在導線內：
- current: from + to −
電流：由正端鈕至負端鈕 - electrons: opposite
自由電子：相反方向
inside battery 在電池內：
- current: from − to +
電流：由正端鈕至負端鈕 - electrons: opposite
電流：由正端鈕至負端鈕
Resistance 電阻值 R
$$\begin{aligned}R&=\rho\frac\ell A\\\text{unit: }\Omega\end{aligned}$$
**注意：考試經常伏一個位，就是圓形橫切面但有時提供radius，有時提供diameter。您一定要睇清楚！
depends on temperature
受温度影響
Resistance, voltage and current 電阻 、電壓與電流
$$V=IR\quad\mathrm{or}\quad R=\frac{V}{I}$$
Ohm’s law 歐姆定律：
- ohmic: R = constant (V ∝ I)
歐姆元件：R = 定值 ( V ∝ I ) - non‑ohmic: R 6= constant
非歐姆元件：R 6= 定值
Equivalent resistance 等效電阻 Req
series combination 串聯組合：
$$\begin{aligned}
&R_{\mathrm{eq}}=R_{1}+R_{2}+\cdots \\
&\mathrm{total}V=V_{1}+V_{2}+\cdots \\
&I={\frac{V_{1}}{R_{1}}}={\frac{V_{2}}{R_{2}}}={\mathrm{constant}} \\
&\implies V_1=V\left(\frac{R_1}{R_{\mathrm{eq}}}\right)
\end{aligned}$$
parallel combination 並聯組合：
$$\begin{aligned}
&\frac{1}{R_{\mathrm{eq}}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\cdots \\
&\text{total }I=I_{1}+I_{2}+\cdots \\
&V=I_{1}R_{1}=I_{2}R_{2}=\mathrm{constant} \\
&\implies I_1:I_2=\frac{1}{R_1}:\frac{1}{R_2} \\
&\implies I_1=I\left(\frac{R_{\mathrm{eq}}}{R_1}\right)
\end{aligned}$$
two resistors in parallel 兩個並聯電阻器
$$R_{\mathrm{eq}}=\left(\frac1{R_1}+\frac1{R_2}\right)^{-1}=\frac{R_1R_2}{R_1+R_2}$$
如果有兩個resistor in parallel，最好的方法是直接用這條公式。（相乘除相加）
如果有兩個resistor in parallel 而且resistance R 一樣，最後的方式是直接除2，即是R/2
Equipotential point 等勢點
- points with equal potential (voltage)
處於相同電勢之地方 - the circuit will not be altered, when equipotential points are connected
將等勢點互相連接 並不影響整個電路 ⇒ 簡化電路
Conservation 守恆
energy 能量守恆：
total emf = total pd
總電動勢 = 總電勢差
charge 電荷守恆：
total current into a point = total current out of that point
流入的總電流 = 流出的總電流
Internal resistance 內電阻 r
| Device | Ideal | Practical |
| Battery | r = 0 | r ≠ 0 |
| Vout = ε | Vout = ε – I * r | |
| Ammeter | r = 0 | r very small |
| Across Ammeter V = 0 | V ≠ 0 | |
| Voltmeter | r → ∞ | r very large |
| Through Voltmeter I = 0 | I ≠ 0 |
Connecting ammeters and voltmeters
安培計和伏特計的接駁方法
voltmeter 伏特計
ammeter 安培計
Electric power 電功率 P
$$\begin{aligned}&\mathrm{energy}=P\times t\\&P=\frac{\mathrm{energy}}{t}=\underbrace{\frac{\mathrm{energy}}{Q}}_{V}\cdot\underbrace{\frac{Q}{t}}_{I}=VI=\frac{V^{2}}{R}=I^{2}R\end{aligned}$$
Conservation of energy 能量守恆
ideal 理想狀況：
total Pin = total Pout (ideal wires and battery)
總輸入功率 = 總輸出功率（ 理想導線和電源 ）
in practice 實際狀況：
total Pin = total Pout + power loss in wires and internal r
總輸入功率 = 總輸出功率 + 導線電阻和電池內阻的功率損耗
Efficiency 效率 η%
- useful energy out = total energy in × η%
有用輸出能量 = 總輸入能量 × η% - useful Pout = total Pin ×η%
有用輸出功率 = 總輸入功率 × η%
$$\eta\%=\frac{\text{useful }P_{\mathrm{out}}}{\text{total }P_{\mathrm{in}}}\times100\%$$
- perfect = 100% efϐicient = no loss
理想 = 效率為 100% = 沒有功率損
經典問題：Voltmeter‑ammeter method 伏安法
Ideal的情況大家遇到非常多，但如果是practical situation大家又是否處理到呢？
| small R (≈ Rammeter) | voltage | current | |
| circuit 1 | exact reading | ≈ reading | 應使用 |
| circuit 2 | ≈ 50%× reading | exact reading | 不應使用 |
| small R (≈ Rvoltmeter) | voltage | current | |
| circuit 1 | exact reading | ≈ 50%× reading | 不應使用 |
| circuit 2 | ≈ reading | exact reading | 應使用 |
Chapter 22 AC and Domestic Electricity
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AC 交流電是什麽
AC 代表交流電，全稱為交流電源（Alternating Current）。交流電是一種電流類型，其特點是電流方向和大小以固定的頻率進行周期性變化。在交流電中，電流週期性地從正向流動到負向，再從負向流回正向，這種變化稱為振盪。交流電廣泛用於家庭、工業和商業場所的電力供應。
Domestic electricity 家居用電
mains in HK: 220 V ac, 50 Hz
香港市電規格：220 V ac, 50 Hz
unit of electricity (energy): 1 kW h = 1 kW×1 h = 3.6 MJ
用電單位（ 能量 ）：1 kW h = 1 kW×1 h = 3.6 MJ
Appliance rating (e.g. ‘220 V, 100 W’) = values designed for normal operation
電器的額定值（ 例如「 220 V, 100 W 」）= 正常運作時預設的功率與電壓
Three‑pin plugs and sockets 三腳插頭和插座
小編建議您記死一邊。小編會選擇插座，然後，當問及插頭，小編會幻想將插頭插進插座，從而分辨出哪一條是 earth，neutral，live。
| Wire | Color | Function | Voltage |
| Live wire | Brown | Normal path for current | Varies between +220 V and -220 V |
| Neutral wire | Blue | Normal path for current | Fixed at 0 V |
| Earth wire | Yellow & Green | Safety device | Connected to metal cases, Earthed (fixed at 0 V) |
Switch and fuse 開關和保險絲
- installed in live wire, not neutral wire
安裝在活線上 而非中線上 - otherwise, the appliance circuit would still be live even if it is switched off or broken.
否則，在電路斷開或保險絲熔斷後，電路依然處於高電勢
Household circuits 家居電路
ring circuits for wall sockets 插座均連接至環形電路
- reduces chance of overloading
避免超負荷 - still functions if broken at one point
若其中一點損壞仍可照常運作
wall sockets in parallel 插座以並聯方式連接
- do not affect each other
互不影響
separate circuits for lightings and high‑power appliances
照明裝置及大功率電器各自連接用戶電箱
- reduces chance of overloading
避免超負荷
Possible consequences of short circuit
短路可能引致的結果
- huge current =⇒ overloading =⇒ overheating
巨大電流 =⇒ 超負荷 =⇒ 過熱 - metal case becomes live =⇒ user gets electric shock
金屬外殼處於高電勢 =⇒ 使用者觸電
Safety devices: overloading protection
安全裝置（ 防止超負荷 ）
fuse 保險絲
- if overloaded, blows and thus breaks the circuit (replace to resume)
- 若電流過大，便會熔斷，以切斷電路（ 須更換以恢復供電 ）
circuit breaker 斷路器
- if overloaded, switches off the circuit (reset to resume)
- 若電流過大，便會切斷電路（ 可重設以恢復供電）
earth wire 地線
- earths the metal case, preventing it from being live
- 將金屬外殼接地，避免外殼意外處於高電壓
double insulation 雙重絕緣
- plastic case + insulates the live parts with extra plastic cases
- 將金屬外殼接地，避免外殼意外處於高電壓
earth leakage circuit breaker (ELCB) 漏電斷路器 (ELCB)
if IL ≠ IN ,switches off the circuit (reset to resume)
若 IL ≠ IN ，便會切斷電路（ 可重設以恢復供電 ）
DC 直流電
one‑direction 單一電流方向
無論是constant，還是varying，只要沒有sign change就是DC
AC 交流電
reversing direction over time 方向隨時間改變
無論有多端正，還是irregular，只要有sign change就是AC
Average power for DC and AC 電功率的平均值
general 一般情況
在這裏，我們會引入一套新的理論，就是root mean square。其實道理非常簡單，只要照字面意思解就可以。將數字square，再取平均數，然後take square root。我們就可以得出有代表性的胡數字。
$$V=IR\quad\Longrightarrow\quad V_{\mathrm{rms}}=I_{\mathrm{rms}}R$$
$$P=I^2R\quad\Longrightarrow\quad\langle P\rangle=I_{\mathrm{rms}}^2R=\frac{V_{\mathrm{rms}}^2}R=V_{\mathrm{rms}}I_{\mathrm{rms}}$$
battery (constant dc) 電池（ 恆定直流電 ）
$$\begin{aligned}&V_{\mathrm{rms}}=V_0\\&I_{\mathrm{rms}}=I_0\end{aligned}$$
mains (sinusoidal ac) 家居電路（ 正弦交流電 ）
$$V_{\mathrm{rms}}=\frac{V_0}{\sqrt{2}}\\I_{\mathrm{rms}}=\frac{I_0}{\sqrt{2}}$$
常見錯誤：applied 輸入與額定rated的分別
rated只是聲稱而已，考生要時時刻刻檢查applied是否達到rated的數值。
Chapter 23 Electromagnetism
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Permanent magnet 永久磁鐵
- poles always in pair (N, S), isolated pole does not exist
磁極總是成對出現（ N 和 S ），並不單獨存在 - like poles repel; unlike poles attract
同性磁極互相排斥；異性磁極則互相吸引
Magnetic field 磁場 B
vector 向量
unit：T
**記住記住！！
direction: can be shown by plotting compass
方向：指南針方向 = 磁場方向
Magnetic field lines 磁場線
always form closed loops
磁場線必然形成 閉合迴路
outside magnet: N → S
磁鐵 之外：N → S
inside magnet: S → N
磁鐵 之內：S → N
line density is a measure of magnitude of the field
磁場線密度反映磁場量值 B
B = constant×(line density)
所以睇B-field要睇field lines的密度！
必背答案 ：如何visualize整個B-field Pattern
Put a board on top of a magnet. Sprinkle iron fillings on the board and tap the board gently. The B-field pattern is shown by the pattern of the fillings.
將一塊板放在磁鐵上。 將鐵粉撒在板上，然後輕輕敲打板子。 磁場圖案由鐵粉的圖案顯示。
Disadvantage 壞處
看不到方向
Magnetic field due to current 電流產生的磁場
- moving charges (current) set up magnetic field
運動中的電荷（ 電流 ）會產生磁場 - direction: by right hand grip rule
方向：以右手握拳定則來判斷 - magnitude: larger I =⇒ stronger B
量值：I 越大則 B 越強
Magnetic field of wire, coil and long solenoid
Magnetic field of wire
$$B=\frac{\mu_{0}I}{2\pi r}$$
B drops as r increases
Magnetic field of coil
B = constant× I (middle)
Magnetic field of long solenoid
$$\text{middle: }B=\mu_0nI\quad\text{(very long)}$$
$$\Text{two ends:}B\approx\frac{1}{2}\mu_0nI$$
（好多考生都不記得頭尾是除2）
Electromagnet 電流產生的磁場
- Advantage：both strength and polarity can be controlled
優點：磁場量值及方向皆可控制 - factors increasing its strength 增強磁場的方法：
- using a soft‑iron core (µ > µ0)
在螺線管內放入軟鐵心 (µ > µ0) - more number of turns per unit length (n)
增加每單位長度的線圈匝數 (n) - larger current (I)
增強電流 (I)
| Core Material | Magnetic Material? | Can Strengthen Electromagnet? | Easy to be Demagnetized? |
| Soft Iron | Yes | Yes | Easy |
| Steel | Yes | Yes | Difficult |
| Copper | No | No | N/A |
Magnetic force on current in B‑field
作用在電流上的磁力
- magnetic ϐield acts on moving charges (current) =⇒ magnetic force F
磁場對移動電荷（ 電流 ）作用，產生磁力 - direction: by Fleming’s left hand rule
方向：以弗林明左手定則判斷 - magnitude: F = ℓ(I ×B⊥) = ℓIB sinθ
量值：F = ℓ(I ×B⊥) = ℓIB sinθ
Force between parallel currents
一對平行電流的相互作用
direction 方向
parallel currents in the same direction attract; opposite currents repel.
同向相吸，反向相斥
magnitude 量值
$$F=\ell IB’\implies\text{force per unit length}F/\ell=IB’=\frac{\mu_0II’}{2\pi r}$$
definition of ampere 安培的定義：
For I = I‘ and r = 1 m apart, if F/ℓ = 2×10−7 N m−1，then I = 1 A
Turning of coil in magnetic Field 線圈在磁場下的轉動
- direction of force: by Fleming’s left hand rule
力的方向：以弗林明左手定則判斷 - max. torque τ = NBAI (when coil plane aligns with B, i.e. 1)
最大力矩 τ = NB AI（ 當磁場跟線圈面平行之時，即 1 ）
simple DC motor 簡單直流電動機
commutator (split/half rings) 換向器（ 一對半環 ）：
enables the coil to turn continuously (reverses current in coil every half cycle)
每當半環之間的縫隙經過碳刷，電流方向便會逆轉，使線圈不停轉動
必背答案：commutator的作用
換向器用於每半圈反轉流過線圈的電流方向。 因此，線圈將繼續沿著同一方向旋轉
factors increasing motor power
增加電動機功率的因素：
- more number of turns of coil (N)
增加線圈匝數 (N) - with stronger magnet (B)
更換較強的磁鐵 (B) - with soft‑iron core (B)
把線圈繞在軟鐵心上 (B) - larger coil area in ϐield (A)
增加磁場內的線圈面積 (A) - larger current (I)
增強電流 (I)
soft‑iron core 軟鐵心
collects and guides magnetic field lines
集中及引導磁場線
force on moving charge in B‑field
- current = a flow of charge
電流 = 帶電粒子的集體運動 - direction: by Fleming’s left hand
方向：以弗林明左手定則判斷
$$F=qv\times B_{\perp}=qvB\sin\theta $$
Magnetic force never does work 磁力不會作功
- FB ⊥ v, always
- moving charge turns in B‑field
帶電粒子在磁場內偏轉 - speed remains unchanged =⇒ KE remains unchanged
速率不變 =⇒ 動能不變
Uniform circular motion of charge in uniform B‑Field
帶電粒子在勻強磁場中的運動
radius r = mv/qB, depending on v and charge‑to‑mass ratio
period T = 2π/ω = 2πm/(qB), depending on charge‑to‑mass ratio
Physics 必修課題
- Temperature and Thermometers
- Heat Capacity
- Change of State
- Gas laws and Kinetic Theory
- Motion
- Force
- More about Force
- Work Energy and Power
- Momentum
- Projectile Motion
- Uniform Circular Motion
- Gravitation
- Wave Motion
- Reflection Refraction and Diffraction
- Interference and Stationary Waves
- Light and Sound
- Reflection of Light
- Refraction of Light
- Lenses
- Electrostatics
- Circuit and Power
- AC and Domestic Electricity
- Electromagnetism
- Electromagnetic Induction
- Radiation and Radioactivity
- Rate of Decay and Uses of Radionuclides
- Nuclear Energy