Electromagnets 1.1.2: Deriving Current and Potential Difference Formulas | 电磁铁 1.1.2：电流与电势差公式推导

📚 Electromagnets 1.1.2: Deriving Current and Potential Difference Formulas | 电磁铁 1.1.2：电流与电势差公式推导

Electromagnets transform electrical energy into magnetic force, and their behaviour is governed by the interplay between current and potential difference. In this section, we unpack the core formulas V = IR and P = IV, deriving them from first principles and linking every step to how a coiled conductor creates a controllable magnetic field.

电磁铁将电能转化为磁力，其行为由电流与电势差之间的相互作用决定。在本节中，我们将从基本原理出发推导核心公式 V = IR 和 P = IV，并将每一步推导与线圈导体如何产生可控磁场联系起来。

1. Understanding Electromagnets and Circuit Basics | 理解电磁铁与电路基础

An electromagnet consists of a coil of wire, often wrapped around a soft iron core. When a potential difference is applied across the coil, current flows, generating a magnetic field whose strength depends directly on the current and the number of turns.

电磁铁由一个线圈组成，通常绕在软铁芯上。当在线圈两端施加电势差时，电流流过，产生磁场，其强度直接取决于电流大小和线圈匝数。

The circuit containing an electromagnet behaves like any resistive load, which means the fundamental rules connecting voltage, current, resistance, and power are essential for designing and analysing electromagnets in relays, motors, and lifting devices.

含有电磁铁的电路与任何电阻性负载一样，因此连接电压、电流、电阻和功率的基本规则对于设计及分析继电器、电机和起重设备中的电磁铁至关重要。

2. Defining Current: Flow of Charge | 定义电流：电荷的流动

Electric current I is the rate of flow of charge. If an amount of charge ΔQ passes through a cross‑section of the coil in time Δt, the current is given by:

电流 I 是电荷流动的速率。若在时间 Δt 内有电荷量 ΔQ 通过线圈的横截面，则电流由下式给出：

I = ΔQ / Δt

In an electromagnet, mobile electrons in the copper wire drift along the circuit. The larger the current, the more charge carriers pass a point per second, and consequently the stronger the magnetic field around each turn of the coil.

在电磁铁中，铜导线中的自由电子沿电路漂移。电流越大，每秒钟通过某一点的载流子越多，因此线圈每一匝周围的磁场也越强。

The unit of current is the ampere (A), where 1 A = 1 C s⁻¹. Using this definition, we can later connect current to the work done per unit charge, which is the potential difference.

电流的单位是安培（A），1 A = 1 C s⁻¹。利用这一定义，我们随后可以将电流与单位电荷所做的功（即电势差）联系起来。

3. Potential Difference: The Driving Force | 电势差：驱动力

Potential difference V (often called voltage) is the energy transferred per unit charge as charge moves between two points in a circuit. If a component transfers energy E when charge Q passes through it, then:

电势差 V（常称作电压）是单位电荷在电路中两点间移动时所转移的能量。若某元件在电荷 Q 通过时转移能量 E，则有：

V = E / Q

For an electromagnet, the power supply does work on each coulomb of charge, delivering energy that is converted partly into the magnetic field and partly into heat due to the coil’s resistance.

对于电磁铁，电源对每库仑电荷做功，提供的能量一部分转化为磁场，另一部分由于线圈电阻而转化为热量。

This definition allows us to see that 1 V = 1 J C⁻¹. A 12 V battery, for example, supplies 12 joules of energy to each coulomb of charge that circulates through the electromagnet’s windings.

这一定义让我们明白 1 V = 1 J C⁻¹。例如，一个 12 V 的电池为流过电磁铁绕组的每库仑电荷提供 12 焦耳的能量。

4. Ohm’s Law and Resistance in Electromagnet Coils | 欧姆定律与电磁铁线圈中的电阻

Most electromagnet coils are made from copper wire which obeys Ohm’s law at constant temperature. The resistance R of the wire opposes the flow of charge, and for a metallic conductor the current is proportional to the applied potential difference:

大多数电磁铁线圈由铜线制成，在恒温下遵循欧姆定律。导线的电阻 R 阻碍电荷流动，对于金属导体，电流与施加的电势差成正比：

V ∝ I

Introducing the constant of proportionality, which is the resistance, we obtain the familiar equation. This relationship is the foundation for designing an electromagnet that draws a predictable current for a given supply voltage.

引入比例常数，即电阻，我们便得到熟悉的方程。这一关系是设计电磁铁的基础，使其在给定电源电压下消耗可预测的电流。

5. Deriving V = IR from First Principles | 从基本原理推导 V = IR

Start with the definitions of current and potential difference. Current is I = Q / t, and potential difference is V = E / Q.

从电流和电势差的定义出发。电流为 I = Q / t，电势差为 V = E / Q。

For a resistive component such as a coil, the energy E dissipated as heat when charge Q moves through a potential difference V is E = VQ. The power P dissipated is the rate of energy transfer, so:

对于线圈这样的电阻性元件，当电荷 Q 通过电势差 V 时，以热量形式耗散的能量 E 为 E = VQ。耗散的功率 P 是能量转移的速率，因此：

P = E / t = (VQ) / t = V × (Q / t) = V I

In a purely resistive component, Ohm’s empirical law tells us the ratio V/I is constant and equal to R. Thus we can write V/I = R, which rearranges to V = IR.

在纯电阻元件中，欧姆经验定律告诉我们 V/I 为常数且等于 R。因此我们可以写成 V/I = R，重新整理得到 V = IR。

Equivalently, we can derive V = IR by considering the microscopic drift of electrons: the electric field (potential gradient) drives current, and the resistance quantifies the collisions that limit drift velocity. For a uniform wire of length L and cross‑sectional area A, R = ρL/A, which directly links the geometry of an electromagnet’s coil to its resistance.

同样，我们也可以通过考虑电子微观漂移来推导 V = IR：电场（电势梯度）驱动电流，而电阻量化了限制漂移速度的碰撞。对于长度为 L、横截面积为 A 的均匀导线，R = ρL/A，这就将电磁铁线圈的几何形状与其电阻直接联系起来。

Therefore, the current flowing through the electromagnet can be predicted as I = V / R, an essential calculation for selecting the correct wire gauge and number of turns.

因此，流经电磁铁的电流可以预测为 I = V / R，这是选择正确线径和匝数时必不可少的计算。

6. Power in Electromagnet Circuits: P = IV | 电磁铁电路中的功率：P = IV

We have already seen that P = V I from the definition of potential difference. For an electromagnet, the total electrical power supplied is the product of the terminal voltage and the current.

我们已经从电势差的定义推得 P = V I。对于电磁铁，输入的总电功率是端电压与电流的乘积。

This power is split between the useful magnetic field energy and the resistive heating of the coil. The efficiency of an electromagnet is therefore a crucial design factor, and it depends on minimising the coil resistance while maintaining the required ampere‑turns (NI).

这部分功率分为有用的磁场能量和线圈的电阻发热。因此，电磁铁的效率是一个关键的设计因素，它取决于在维持所需安匝数（NI）的同时最小化线圈电阻。

We can also write P = E / t, so the energy consumed in a time interval Δt is ΔE = V I Δt. This is used to calculate the temperature rise in the windings.

我们还可以写为 P = E / t，因此在时间间隔 Δt 内消耗的能量为 ΔE = V I Δt，这用于计算绕组中的温升。

7. Combining with Resistance: P = I²R and P = V²/R | 结合电阻：P = I²R 与 P = V²/R

Substituting V = IR into P = V I gives another useful form:

将 V = IR 代入 P = V I 可得到另一种有用的形式：

P = I × (I R) = I² R

This formula explains why doubling the current through an electromagnet quadruples the heat generated in the coil, which is a major consideration for high‑power electromagnets.

该公式解释了为什么通过电磁铁的电流加倍会使线圈产生的热量增加四倍，这是设计大功率电磁铁时的一个主要考虑因素。

Similarly, rearranging V = IR to I = V/R and substituting into P = V I yields:

类似地，将 V = IR 整理为 I = V/R 并代入 P = V I，得到：

P = V × (V / R) = V² / R

This version is convenient when the supply voltage is fixed, allowing a quick assessment of how changes in coil resistance affect power consumption and heat dissipation.

当电源电压固定时，这种形式很方便，可以快速评估线圈电阻变化如何影响功耗和散热。

All three power expressions are mathematically equivalent; choosing the right one depends on the known quantities in an electromagnet design problem.

这三种功率表达式在数学上是等效的；选择哪一个取决于电磁铁设计问题中的已知量。

8. Practical Implications for Electromagnet Design | 电磁铁设计的实际意义

Using V = IR, a designer can determine the required supply voltage for a target magnetic field strength. A higher current increases the magnetomotive force (MMF = N I), but also increases I²R losses.

利用 V = IR，设计者可以确定达到目标磁场强度所需的电源电压。较大的电流会增大磁动势（MMF = N I），但也会增加 I²R 损耗。

To reduce heating, a thicker wire (lower resistance) is preferred, though this adds bulk and cost. Alternatively, increasing the number of turns N allows the same MMF with a smaller current, reducing heat while keeping the magnetic field strong.

为了减少发热，更粗的导线（低电阻）是首选，但这会增加体积和成本。或者，增加匝数 N 可以在较小电流下获得相同的磁动势，从而减少发热并保持强磁场。

The potential difference across the coil must be controlled to avoid insulation breakdown; hence the insulation rating of the wire and the voltage source are selected together using V = IR and P = V²/R.

线圈两端的电势差必须加以控制，以避免绝缘击穿；因此，导线的绝缘等级和电压源需要结合 V = IR 和 P = V²/R 共同选择。

9. Measuring Current and Potential Difference in Experiments | 实验中电流和电势差的测量

In the laboratory, an ammeter is connected in series with the electromagnet coil to measure the current, while a voltmeter is connected in parallel across its terminals to measure the potential difference.

在实验室中，将电流表与电磁铁线圈串联以测量电流，同时将电压表并联在其两端以测量电势差。

These measurements allow students to verify Ohm’s law by plotting V against I and observing a straight line through the origin, confirming that the coil’s resistance is constant under low‑current conditions.

通过这些测量，学生可以绘制 V-I 图并观察一条通过原点的直线，从而验证欧姆定律，确认线圈在低电流条件下的电阻是恒定的。

For an electromagnet, the current is typically limited by a rheostat, and the potential difference is varied step by step. Recording data for V and I enables calculation of R using R = V/I and power using the derived formulas.

对于电磁铁，电流通常通过变阻器限制，电势差则逐步改变。记录 V 和 I 的数据后，即可使用 R = V/I 计算电阻，并使用推导出的公式计算功率。

10. Sample Calculations with an Electromagnet | 电磁铁示例计算

A 24 V DC supply powers a coil of resistance 8 Ω. Using V = IR, the current drawn is I = 24 V / 8 Ω = 3 A.

一个 24 V 直流电源为一个电阻为 8 Ω 的线圈供电。利用 V = IR，流过的电流为 I = 24 V / 8 Ω = 3 A。

The power consumption is P = V I = 24 V × 3 A = 72 W. This can be cross‑checked with P = I²R = (3 A)² × 8 Ω = 72 W, and P = V²/R = (24 V)² / 8 Ω = 72 W.

功耗为 P = V I = 24 V × 3 A = 72 W。这也可以分别通过 P = I²R = (3 A)² × 8 Ω = 72 W 以及 P = V²/R = (24 V)² / 8 Ω = 72 W 进行交叉校验。

If the coil has 500 turns, the magnetomotive force is MMF = N I = 500 × 3 A = 1500 A‑turns. If we wished to reduce heat while keeping the same MMF, we could use 1000 turns with a current of 1.5 A, then recalculate the new required resistance and supply voltage using V = IR and P = I²R.

若线圈有 500 匝，则磁动势为 MMF = N I = 500 × 3 A = 1500 安匝。如果希望在保持相同磁动势的同时减少发热，可以使用 1000 匝、电流 1.5 A，然后利用 V = IR 和 P = I²R 重新计算所需的新电阻和电源电压。

11. Common Misconceptions Clarified | 常见误解澄清

Misconception 1: “A bigger electromagnet automatically draws more current.” In reality, current is determined by V and R. A large electromagnet with high resistance may draw less current than a small one with low resistance.

误解1：“更大的电磁铁会自动消耗更大的电流。”实际上，电流由 V 和 R 决定。一个电阻很大的大型电磁铁可能比电阻很小的小型电磁铁消耗更小的电流。

Misconception 2: “Potential difference and current are the same thing.” Potential difference drives current, but they are distinct quantities linked by resistance. A high voltage can exist with negligible current if resistance is extremely high.

误解2：“电势差和电流是一回事。”电势差驱动电流，但它们是不同的物理量，通过电阻相联系。若电阻极高，可以存在高电压而仅有可忽略的电流。

Misconception 3: “Power formula P = V I is separate from Ohm’s law.” Actually, P = V I is derived directly from energy definitions, and it connects seamlessly with V = IR to produce the other two forms, as shown above.

误解3：“功率公式 P = V I 与欧姆定律是分开的。”事实上，P = V I 直接由能量定义推导而来，并与 V = IR 无缝衔接，产生上文中另外两种形式。

12. Summary of Key Formulas | 关键公式总结

The essential relationships for current and potential difference as they apply to electromagnets are:

电流和电势差在电磁铁中的基本关系如下：

Formula	Meaning
I = ΔQ / Δt	Current as charge flow rate
V = E / Q	Potential difference as energy per unit charge
V = I R	Ohm’s law for a resistive coil
P = V I	Electrical power
P = I² R	Power in terms of current and resistance
P = V² / R	Power in terms of voltage and resistance

These equations form the backbone of electromagnet circuit analysis, allowing you to balance magnetic force, heat generation, and energy efficiency in any design.

这些方程式构成了电磁铁电路分析的支柱，让你能在任何设计中平衡磁力、发热和能效。

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