📚 Electromagnets 1.1.1: Circuits and Symbols – Key Formula Derivations | 电磁学 1.1.1：电路与符号——关键公式推导
Understanding circuit symbols and their arrangement is the bedrock of electromagnetism. This section introduces standard symbols, explains how to draw and interpret circuit diagrams, and walks through the essential formula derivations linking current, voltage, resistance, power and energy – all vital for analysing even the most complex electromagnetic systems.
理解电路符号及其布局是电磁学的基础。本节介绍标准符号，讲解如何绘制和解读电路图，并细致推导电流、电压、电阻、功率和能量之间的核心公式——这些对于分析任何复杂的电磁系统都至关重要。
1. Common Circuit Symbols | 常见电路符号
Standardised symbols are used to represent components such as cells, batteries, fixed and variable resistors, lamps, switches (open and closed), ammeters, voltmeters, fuses and diodes. Knowing these symbols allows anyone to read a schematic without ambiguity.
标准化符号用于表示电池、固定和可变电阻、灯泡、开关（断与通）、电流表、电压表、熔断器和二极管等元件。掌握这些符号能让任何人毫无歧义地读懂原理图。
A single cell is shown by a long thin line (positive) and a short thick line (negative). A battery is a combination of several cells. A fixed resistor is a rectangle, while a variable resistor (rheostat) has an arrow across it. An ammeter is drawn as a circle with an A, a voltmeter as a circle with a V, connected in series and parallel respectively.
单个电池用一长一短两条线表示（长正短负），电池组是多个电池的组合。固定电阻为矩形，可变电阻（变阻器）带有一个箭头。电流表画成带 A 的圆圈，电压表画成带 V 的圆圈，分别串联和并联接入。
2. Drawing Circuit Diagrams | 绘制电路图
Diagrams must be drawn with straight lines and right-angled corners. Components should be evenly spaced and labelled. The circuit should be closed for current to flow, unless you wish to show an open switch. Correct placement of measuring devices is critical: ammeters in series, voltmeters in parallel.
电路图必须用直线和直角绘制。元件应均匀分布并加以标注。电路应闭合以使电流流通，除非需要显示断开的开关。测量仪器的正确放置至关重要：电流表串联，电压表并联。
Always start from the positive terminal of the power source and trace the path back to the negative terminal. Use a ruler for neat diagrams. Label values of resistance, voltage or current if known, and indicate the direction of conventional current flow from positive to negative.
始终从电源正极出发，沿着路径追踪回负极。画图时使用直尺保持整洁。如果知道电阻、电压或电流的值，应标注出来，并标明从正极到负极的常规电流方向。
3. Electric Current and Charge | 电流与电荷
ere I is current in amperes (A), Q is charge in coulombs (C) and t is time in seconds (s). This definition underpins all circuit calculations.
电流是电荷流动的速率。用公式表示为 I = Q / t，其中 I 为电流（安培），Q 为电荷（库仑），t 为时间（秒）。这一定义是所有电路计算的基础。
Charge carriers in metallic conductors are free electrons, moving from negative to positive, but conventional current points in the opposite direction. In a series circuit, the current is the same at all points. In parallel, the total current splits across different branches.
金属导体中的载流子是自由电子，从负极移向正极，但常规电流方向与此相反。串联电路中各点电流相同。并联电路中总电流会在各支路中分配。
4. Voltage, Energy and EMF | 电压、能量与电动势
V), W is work done or energy in joules (J). The electromotive force (EMF) of a source is the total energy supplied per coulomb of charge.
电压（电势差）是每单位电荷转移的能量。V = W / Q，其中 V 为电压（伏特），W 为所做的功或能量（焦耳）。电源的电动势（EMF）是每库仑电荷提供的总能量。
When current passes through a component, energy is transferred to that component. A voltmeter measures the potential difference across two points. In a circuit diagram the voltmeter is always drawn in parallel with the component being measured, and has a very high resistance so it draws negligible current.
当电流通过元件时，能量会转移到该元件。电压表测量两点之间的电势差。在电路图中，电压表始终与被测元件并联，并且具有极高的电阻，因此汲取的电流可忽略不计。
5. Ohm's Law: The Fundamental Relationship | 欧姆定律：基本关系式
Ohm' potential difference across it, provided temperature remains constant. This gives the well‑known equation V = I × R, where R is resistance in ohms (Ω).
欧姆定律指出，在温度保持不变的条件下，通过金属导体的电流与其两端的电势差成正比。由此得出著名的公式 V = I × R，其中 R 为电阻（欧姆）。
From this, resistance can be defined as R = V / I. A component that obeys Ohm's law is called an ohmic conductor, and its I–V graph is a straight line through the origin. Non‑ not follow this linear relationship due to temperature changes or semiconductor behaviour.
由此可将电阻定义为 R = V / I。遵守欧姆定律的元件称为欧姆导体，其 I–V 图是一条通过原点的直线。非欧姆导体（如灯丝灯泡或二极管）由于温度变化或半导体特性，不遵循这种线性关系。
6. Resistance in Series Circuits | 串联电路中的电阻
When resistors are connected end‑to‑end, the total resistance Rtotal is simply the sum of individual resistances. This can be derived from the fact that the same current I flows through each resistor, and the total voltage is the sum of individual voltages: Vtotal = V₁ + V₂ + V₃ + ...
当电阻器首尾相连时，总电阻 Rtotal 等于各个电阻之和。这一结论可推导如下：每个电阻器中流过相同的电流 I，总电压等于各电压之和：Vtotal = V₁ + V₂ + V₃ + ...
Using V = I R for each: V₁ = I R₁, V₂ = I R₂. So Vtotal = I R₁ + I R₂ = I (R₁ + R₂). Since Vtotal = I Rtotal, we obtain:
Rtotal = R₁ + R₂ + R₃ + ...
对每个电阻使用 V = I R：V₁ = I R₁，V₂ = I R₂。因此 Vtotal = I R₁ + I R₂ = I (R₁ + R₂)。由于 Vtotal = I Rtotal，可得：
Rtotal = R₁ + R₂ + R₃ + ...
Adding more resistors in series increases the total resistance, reducing the current for a given supply voltage.
串联更多电阻会增加总电阻，从而在给定电源电压下减小电流。
7. Resistance in Parallel Circuits | 并联电路中的电阻
For resistors in parallel, the potential difference across each branch is the same. The total current entering the parallel network splits among the branches: Itotal = I₁ + I₂ + I₃ + ...
对于并联的电阻，各支路两端的电势差相同。流入并联网络的总电流在各支路中分配：Itotal = I₁ + I₂ + I₃ + ...
Applying Ohm's law to each branch: I₁ = V / R₁, I₂ = V / R₂, and Itotal = V / Rtotal. Substituting gives V / Rtotal = V / R₁ + V / R₂ + ... Cancelling the common voltage V yields the reciprocal formula:
1 / Rtotal = 1 / R₁ + 1 / R₂ + 1 / R₃ + ...
对每一支路应用欧姆定律：I₁ = V / R₁，I₂ = V / R₂，且 Itotal = V / Rtotal。代入得 V / Rtotal = V / R₁ + V / R₂ + ... 消去公因子 V 得到倒数公式：
1 / Rtotal = 1 / R₁ + 1 / R₂ + 1 / R₃ + ...
For two resistors in parallel, this simplifies to Rtotal ance in parallel is always less than the smallest individual resistance, which is why household circuits are wired in parallel.
对于两个并联电阻，该式简化为 Rtotal = (R₁ × R₂) / (R₁ + R₂)。并联总电阻总是小于最小的单个电阻，这正是家用电路采用并联布线的原因。
8. Electrical Power Derivation | 电功率公式推导
eliminate Q: from W = V Q and Q = I t, we get W = V I t. Therefore power P = W / t = V I.
功率是能量转移的速率。在电学中，结合定义 V = W / Q 和 I = Q / t，消去 Q：由 W = V Q 和 Q = I t 得 W = V I t。因此功率 P = W / t = V I。
P = V × I
This is the fundamental power equation. Substituting V = I R (Ohm's law) gives two other useful forms:
P = I² × R
P = V² / R
这是基本的功率方程。代入 V = I R（欧姆定律）可得到另外两种有用的形式：
P = I² × R
P = V² / R
These three formulas are used depending on which quantities are known, and are essential for selecting components with adequate power ratings.
根据已知量可选择这三个公式之一，它们对于选择具有足够额定功率的元件至关重要。
9. Energy Transferred in Circuits | 电路中的能量转化
transferred W in a circuit can be calculated from W = V I t. Using substitutions from Ohm's law, we also obtain:
W = I² R t
W = (V² / R) t
电路中的能量转移 W 可通过 W = V I t 计算。利用欧姆定律代换，同样可得：
W = I² R t
W = (V² / R) t
The form W = I² R t is often called Joule' n carrying large currents and forms the basis of heating elements in kettles and toasters.
W = I² R t 这一形式常被称为焦耳热定律，它描述电能如何在电阻器中转化为热能。这一原理解释了为何大电流通过导线时会发热，也是电热水壶和烤面包机中加热元件的工作基础。
In the context of electromagnets, minimising unwanted heat is important; using thick copper wire with low resistance reduces the I² R loss, making the magnet more efficient.
在电磁铁的应用中，减少不必要的发热很重要；使用低电阻的粗铜线可降低 I² R 损耗，使电磁铁效率更高。
10. Short Circuits and Fuses | 短路与保险丝
A short circuit occurs when a low‑ comes extremely small, current surges. The huge I² R heating can melt wires and cause fires.
短路是指低电阻路径绕过了预定负载，导致电流过大。根据 I = V / R，若 R 变得极小，电流就会激增。巨大的 I² R 热量可能熔化导线并引发火灾。
Fuses are safety devices containing a thin wire that melts when current exceeds a rated value. A fuse must be connected in series with the live wire. Choosing the correct fuse rating slightly above the normal operating current protects the circuit without unnecessary interruptions.
保险丝是含有细金属线的安全装置，当电流超过额定值时金属线会熔断。保险丝必须与火线串联。选择额定值略高于正常工作电流的保险丝，可在保护电路的同时避免不必要的断电。
11. Circuits in Electromagnets and Relays | 电磁铁与继电器中的电路
An electromagnet typically consists of a coil of insulated copper wire wound around a soft iron core. The circuit controlling it is simple: a power source, a switch and the coil (often shown as an inductor symbol). When the switch is closed, current flows, creating a magnetic field that magnetises the core.
电磁铁通常由缠绕在软铁芯上的绝缘铜线圈构成。其控制电路很简单：一个电源、一个开关和线圈（常用电感符号表示）。闭合开关，电流流通，产生的磁场将铁芯磁化。
Relays use an electromagnet to operate a separate switch. A small current in the coil circuit closes or opens contacts in a higher‑power circuit. This allows safe isolation and remote control. The circuit symbols for a relay coil and its contacts are distinct and must be drawn correctly in diagrams.
继电器利用电磁铁来操作一个独立的开关。线圈电路中的小电流能使更高功率电路中的触点闭合或断开，从而实现安全隔离和远程控制。继电器线圈及其触点的电路符号是截然不同的，在图中必须正确绘制。
12. Practical Measurement Tips | 实验测量技巧
When measuring current, break the circuit at the point of interest and insert the ammeter in series. Voltmeters are connected directly across the component. Always use the correct range to avoid damaging the meter or blowing internal fuses.
测量电流时，应在待测点断开电路并将电流表串联接入。电压表则直接并联在元件两端。务必选择正确量程，以免损坏仪表或烧断内部保险丝。
For resistance, use the voltmeter‑ meter can also measure resistance directly, but the component must be removed from the circuit to avoid parallel paths affecting the reading. These practices reinforce the theoretical derivations and help verify Ohm's law experimentally.
测量电阻可采用伏安法：同时测量 V 和 I，然后应用 R = V / I。万用表也能直接测量电阻，但必须将元件从电路中断开，以免并联路径影响读数。这些操作可巩固理论推导，并有助于通过实验验证欧姆定律。
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