📚 Electromagnets 1.2.1 – Series Circuits Formula Derivation | 电磁学 1.2.1 – 串联电路公式推导
In A-level Physics, mastering series circuits is essential for understanding how electromagnets and other electrical components behave when connected in a single loop. This article provides a step-by-step derivation of the core formulas for current, voltage, resistance, power and EMF in series circuits, grounded in conservation laws that govern charge and energy.
在A-level物理中，掌握串联电路对于理解电磁铁和其他电气元件在单一回路中的行为至关重要。本文将逐步推导串联电路中电流、电压、电阻、功率和电动势的核心公式，这些公式扎根于支配电荷与能量的守恒定律。
1. What is a Series Circuit? | 什么是串联电路？
A series circuit is one in which components are connected end-to-end, forming a single continuous path for electric current. There are no junctions or branching nodes – the same current must flow through every component sequentially.
串联电路是指元件首尾相连，形成一条连续的单一电流路径。电路中没有分支节点——同一电流必须依次流过每一个元件。
In an electromagnet setup, coils of wire might be placed in series with a power supply and a variable resistor. Understanding how the total resistance and voltage distribute is critical for designing circuits that produce a desired magnetic field.
在电磁铁装置中，线圈可能与电源和可变电阻串联。理解总电阻和电压如何分配，对于设计产生所需磁场的电路至关重要。
2. Current in a Series Circuit: Conservation of Charge | 串联电路中的电流：电荷守恒
For any series connection, the current is identical everywhere. This stems directly from the principle of conservation of charge: charge cannot be created or destroyed in a closed circuit, nor can it accumulate at any point under steady-state conditions.
对于任何串联连接，各处的电流都相同。这直接源于电荷守恒原理：在稳态条件下，电荷既不能在闭合电路中产生或湮灭，也不能在任一点累积。
Imagine 2 coulombs of charge entering a resistor every second. Since there is only one path, those same 2 coulombs must leave the resistor every second. Hence the rate of flow of charge, which is current, remains constant.
设想每秒钟有2库仑的电荷流入一个电阻器。由于只有一条路径，同样2库仑的电荷必须每秒钟离开电阻器。因此电荷流动的速率，即电流，保持恒定。
I = I₁ = I₂ = I₃ = … = Iₙ
This relationship is used when calculating power dissipation in each component or when determining the appropriate fuse rating for a series branch that powers an electromagnet.
在计算每个元件的功率损耗或确定给电磁铁供电的串联支路所需的熔断器额定值时，会用到这一关系式。
3. Voltage in a Series Circuit: Energy Conservation | 串联电路中的电压：能量守恒
The total voltage supplied by the source is equal to the sum of the individual potential differences (p.d.) across all components. This is a consequence of energy conservation: the energy given to each unit charge by the source must equal the total energy transferred by that charge to the circuit components.
电源提供的总电压等于所有元件两端电势差（p.d.）之和。这是能量守恒的结果：电源给予每单位电荷的能量，必须等于该电荷传递给电路元件的总能量。
If a charge gains 12 J of electrical potential energy from a 12 V battery, then as it flows through two series resistors, it might lose 7 J in the first and 5 J in the second. The voltmeter reading across each component corresponds to these energy transfers per coulomb.
如果一个电荷从12 V电池获得12 J的电势能，那么当它流过两个串联电阻时，可能在第一个电阻中失去7 J，在第二个中失去5 J。每个元件两端的电压表读数对应着每库仑所传递的能量。
Vₛ = V₁ + V₂ + V₃ + … + Vₙ
This is often called Kirchhoff’s voltage law (KVL) and is the foundation for deriving the equivalent resistance and the voltage divider rule.
这通常被称为基尔霍夫电压定律（KVL），是推导等效电阻和分压规则的基础。
4. Deriving Equivalent Resistance | 等效电阻的推导
To find the single resistor that can replace a series combination while drawing the same current from the same source, we combine Ohm’s law with the voltage rule. For each resistor, Vᵣ = Iᵣ Rᵣ. Since current is the same in series, we can write:
为了找出能替代串联组合、且在相同电源下吸取相同电流的单个电阻，我们将欧姆定律与电压规则结合起来。对于每个电阻，Vᵣ = Iᵣ Rᵣ。由于串联中电流相同，我们可以写出：
IRₑ = IR₁ + IR₂ + IR₃ + … + IRₙ
Where Rₑ is the equivalent resistance and I is the common current. Factoring out I on both sides yields:
其中Rₑ为等效电阻，I为公共电流。两边提取出公因子I就得到：
Rₑ = R₁ + R₂ + R₃ + … + Rₙ
Thus the total resistance of a series circuit is simply the arithmetic sum of all individual resistances. Every additional resistor increases the opposition to current flow, which reduces the current for a given supply voltage.
因此，串联电路的总电阻就是各个电阻的算术和。每增加一个电阻都会增大对电流的阻碍，对于给定的电源电压，这将减小电流。
In an electromagnet, the wire itself has resistance, and any added series resistance will lower the current and thus weaken the magnetic flux density unless the supply voltage is adjusted.
在电磁铁中，导线本身具有电阻，任何增加的串联电阻都会降低电流，从而减弱磁通密度，除非调整电源电压。
5. Voltage Divider Rule | 分压规则
Using the equivalent resistance expression, we can determine how the supply voltage is divided among series resistors without explicitly calculating current. From Ohm’s law, I = Vₛ / Rₑ. The p.d. across a particular resistor Rₓ is:
利用等效电阻表达式，我们可以确定电源电压如何在串联电阻之间分配，而不必显式计算出电流。由欧姆定律，I = Vₛ / Rₑ。某一特定电阻Rₓ两端的电势差为：
Vₓ = I × Rₓ = (Vₛ / Rₑ) × Rₓ
This simplifies to the voltage divider formula:
Vₓ = (Rₓ / Rₑ) × Vₛ
In a simple two-resistor series network, V₁ = (R₁/(R₁+R₂))×Vₛ and V₂ = (R₂/(R₁+R₂))×Vₛ. This is invaluable when designing sensing circuits, such as using a thermistor and fixed resistor in series to create a temperature-dependent voltage output for a control system.
在一个简单的两个电阻串联网络中，V₁ = (R₁/(R₁+R₂))×Vₛ，V₂ = (R₂/(R₁+R₂))×Vₛ。这在设计传感电路时非常有用，例如使用热敏电阻和固定电阻串联来为控制系统产生一个依赖于温度的电压输出。
For an electromagnet coil with a known resistance, the voltage divider rule helps in selecting a series dropping resistor to obtain the exact current required for a target magnetic field strength.
对于已知电阻的电磁铁线圈，分压规则有助于选择一个串联降压电阻，以获得目标磁场强度所需的精确电流。
6. Power Distribution in Series | 串联中的功率分配
The total power delivered by the source equals the sum of the power dissipated by all components. This follows from P = IV. The source supplies power at a rate Pₛ = I × Vₛ. Using the voltage sum rule:
电源提供的总功率等于所有元件消耗的功率之和。这由P = IV推出。电源以Pₛ = I × Vₛ的速率提供功率。利用电压求和规则：
Pₛ = I×(V₁ + V₂ + … ) = IV₁ + IV₂ + … = P₁ + P₂ + …
Since I is common, the power dissipated by each resistor is also proportional to its resistance: P = I²R. A higher resistance in a series network will dissipate more heat.
由于电流I相同，每个电阻消耗的功率也与其电阻成正比：P = I²R。串联网络中电阻值较高的元件会耗散更多热量。
For electromagnets, this thermal behaviour is important: excessive current can overheat the coil, increasing resistance further and potentially damaging insulation. Designers often calculate the I²R loss to determine cooling requirements.
对于电磁铁，这种热行为很重要：过大的电流会使线圈过热，进一步提高电阻并可能损坏绝缘层。设计者通常计算I²R损耗来确定冷却需求。
7. EMF Sources in Series | 串联的电动势源
When cells or batteries are connected in series, their electromotive forces (EMFs) add algebraically. If the polarities are aligned (positive to negative), the total EMF is the sum:
εₜₒₜₐₗ = ε₁ + ε₂ + … + εₙ
If one cell is reversed, its EMF subtracts. The internal resistances of the cells also add in series, just like resistors, because the same current passes through each cell’s internal resistance r.
当电池或电芯串联连接时，它们的电动势（EMF）按照代数相加。如果极性一致（正极接负极），总电动势就是各电动势之和：εₜₒₜₐₗ = ε₁ + ε₂ + … + εₙ。如果某节电池反接，其电动势则相减。各电池的内阻也像电阻一样串联相加，因为相同的电流流过每个电池的内阻r。
The total internal resistance of the series combination is rₜₒₜₐₗ = r₁ + r₂ + … . This affects the terminal voltage available to the external circuit: V_terminal = εₜₒₜₐₗ – I × rₜₒₜₐₗ.
串联组合的总内阻为rₜₒₜₐₗ = r₁ + r₂ + … 。这会影响外电路可用的端电压：V_terminal = εₜₒₜₐₗ – I × rₜₒₜₐₗ。
In an electromagnet circuit powered by a battery pack, the internal resistances are often small compared to the coil resistance, but they cannot be ignored when the coil resistance is low and a high current is drawn.
在由电池组供电的电磁铁电路中，内阻通常比线圈电阻小，但当线圈电阻较低且吸取大电流时，它们不可忽略。
8. Worked Example and Measurement Considerations | 例题与测量注意事项
Consider a 9.0 V battery connected in series with a 120 Ω resistor and an electromagnet coil of 80 Ω. Calculate the current, the p.d. across the coil, and the power dissipated in the coil.
假设一个9.0 V电池与一个120 Ω电阻和一个80 Ω的电磁铁线圈串联。计算电流、线圈两端的电压以及线圈消耗的功率。
Total resistance Rₑ = 120 + 80 = 200 Ω. Current I = V / Rₑ = 9.0 / 200 = 0.045 A (45 mA). Voltage across coil V_coil = (R_coil / Rₑ) × V = (80 / 200) × 9.0 = 3.6 V. Power in coil P = I²R_coil = (0.045)² × 80 = 0.162 W.
总电阻Rₑ = 120 + 80 = 200 Ω。电流I = V / Rₑ = 9.0 / 200 = 0.045 A (45 mA)。线圈两端电压V_coil = (R_coil / Rₑ) × V = (80 / 200) × 9.0 = 3.6 V。线圈功率P = I²R_coil = (0.045)² × 80 = 0.162 W。
When measuring these quantities, ammeters must be placed in series to experience the full current, while voltmeters are placed in parallel across the component. This preserves the series relationships and ensures accurate readings.
在测量这些量时，电流表必须串联接入以通过全部电流，而电压表则并联在元件两端。这保持了串联关系并确保读数准确。
The derivations above are not merely abstract; they are the essential toolkit for analysing any series arrangement, from simple electromagnet circuits to complex sensor networks.
上述推导不仅仅是抽象的，它们是分析所有串联布置——从简单的电磁铁电路到复杂的传感器网络——的基本工具。
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