📚 IB Science: Electricity and Magnetism Key Points | IB 科学：电与磁考点精讲
This article covers the essential concepts of electricity and magnetism for IB Science, including electric charge, Coulomb’s law, electric fields, circuits, magnetic effects, electromagnetic induction, and applications. Each section presents key ideas in both English and Chinese to support bilingual learning and revision.
本文涵盖 IB 科学中电与磁的核心概念，包括电荷、库仑定律、电场、电路、磁效应、电磁感应及其应用。每个小节以中英双语呈现重点，帮助双语学习与复习。
1. Electric Charge and Coulomb’s Law | 电荷与库仑定律
Electric charge is a fundamental property of matter. There are two types of charge: positive and negative. Like charges repel, unlike charges attract. Charge is conserved and quantized, with the elementary charge e = 1.6 × 10⁻¹⁹ C.
电荷是物质的基本属性。有正、负两种电荷，同号相斥，异号相吸。电荷守恒且量子化，基本电荷 e = 1.6 × 10⁻¹⁹ C。
Coulomb’s law describes the electrostatic force between two point charges. The magnitude of the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
库仑定律描述两点电荷之间的静电力。力的大小与电荷量的乘积成正比，与它们之间距离的平方成反比。
F = k Q₁ Q₂ / r²
where k = 8.99 × 10⁹ N m² C⁻² (in vacuum) and r is the distance between the charges. This force is central and follows the inverse-square law.
其中 k = 8.99 × 10⁹ N m² C⁻²（真空中），r 为电荷间的距离。该力为有心力且遵从平方反比律。
2. Electric Fields | 电场
An electric field is a region where an electric charge experiences a force. The electric field strength E is defined as the force per unit positive charge.
电场是电荷会受到力的作用的区域。电场强度 E 定义为单位正电荷所受的力。
E = F / q
The direction of the electric field is the direction of the force on a positive test charge. For a point charge Q, the field at a distance r is:
电场方向与正检验电荷的受力方向相同。对于点电荷 Q，距离 r 处的电场为：
E = k Q / r²
In a uniform electric field, such as between two parallel plates, E = V / d, where V is the potential difference and d the plate separation. Field lines show direction and strength; they never cross.
在匀强电场（如两平行板之间）中，E = V / d，V 为电势差，d 为板间距。电场线显示方向与强度；它们永不相交。
3. Electric Current and Resistance | 电流与电阻
Electric current I is the rate of flow of charge. It is measured in amperes (A). Conventional current flows from positive to negative terminals.
电流 I 是电荷流动的速率，单位为安培（A）。传统电流方向是从正极流向负极。
I = ΔQ / Δt
Resistance R measures how much a component opposes the flow of current. Ohm’s law states that for an ohmic conductor at constant temperature, V = I R. Resistance depends on the material’s resistivity ρ, length L, and cross-sectional area A.
电阻 R 衡量元件阻碍电流流动的程度。欧姆定律指出，对于恒温下的欧姆导体，V = I R。电阻取决于材料的电阻率 ρ、长度 L 和横截面积 A。
R = ρ L / A
Resistivity increases with temperature in metal conductors. Non-ohmic components, like filament lamps and diodes, do not have a constant V/I ratio.
金属导体的电阻率随温度升高而增大。非欧姆元件（如灯丝灯泡和二极管）的 V/I 比值不恒定。
4. Ohm’s Law and Circuits | 欧姆定律与电路
Kirchhoff’s laws are essential for circuit analysis. Kirchhoff’s current law (KCL) states that the sum of currents entering a junction equals the sum of currents leaving it. Kirchhoff’s voltage law (KVL) states that the sum of emfs around any closed loop equals the sum of voltage drops.
基尔霍夫定律对电路分析至关重要。基尔霍夫电流定律：流入节点的电流之和等于流出节点的电流之和。基尔霍夫电压定律：任何闭合回路中电动势之和等于电压降之和。
Resistors in series carry the same current, and the total resistance is R_total = R₁ + R₂ + … Resistors in parallel have the same voltage across them, and the reciprocal of the total resistance is 1/R_total = 1/R₁ + 1/R₂ + …
串联电阻中的电流相同，总电阻 R_total = R₁ + R₂ + …。并联电阻两端电压相同，总电阻的倒数为 1/R_total = 1/R₁ + 1/R₂ + …
| Series | Parallel |
|---|---|
| Current same | Voltage same |
| R_total = sum of R | 1/R_total = sum of 1/R |
An ideal battery provides constant emf ε, but a real battery has internal resistance r. The terminal voltage is V = ε – I r. The emf can be found from the open-circuit voltage.
理想电池提供恒定电动势 ε，但实际电池有内阻 r。路端电压为 V = ε – I r。电动势可通过开路电压测得。
5. Power and Energy in Electric Circuits | 电路中的功率与能量
Electric power P is the rate at which energy is converted or transferred in a circuit. It is measured in watts (W).
电功率 P 是电路中能量转化或传递的速率，单位为瓦特（W）。
P = I V = I² R = V² / R
The energy dissipated as heat in a resistor over time t is E = P t = I V t. Electrical energy is often measured in kilowatt-hours (kWh), where 1 kWh = 3.6 × 10⁶ J.
电阻中随时间 t 耗散的热能为 E = P t = I V t。电能常用千瓦时（kWh）衡量，1 kWh = 3.6 × 10⁶ J。
In a circuit, the power supplied by the battery (ε I) is distributed among the external load and the internal resistance (I² r). Maximum power is transferred to the load when R_load = r.
电路中，电池提供的功率（ε I）分配给外负载和内阻（I² r）。当负载电阻 R_load = r 时，负载获得的最大功率。
6. Magnetic Fields from Currents | 电流的磁效应
A moving charge or a current-carrying conductor creates a magnetic field around it. The direction of the magnetic field can be determined by the right-hand grip rule: point the thumb in the direction of conventional current, and the fingers curl in the direction of the magnetic field.
运动电荷或载流导体在其周围产生磁场。磁场方向可用右手螺旋定则判定：拇指指向传统电流方向，弯曲的四指即表示磁场方向。
For a long straight wire, the magnetic field at a distance r is given by B = μ₀ I / (2π r), where μ₀ = 4π × 10⁻⁷ T m A⁻¹ is the permeability of free space. The field lines form concentric circles around the wire.
对于长直导线，距离 r 处的磁感应强度为 B = μ₀ I / (2π r)，其中 μ₀ = 4π × 10⁻⁷ T m A⁻¹ 为真空磁导率。磁感线为围绕导线的同心圆。
For a solenoid, the internal field is strong and uniform: B = μ₀ n I, where n is the number of turns per unit length. The right-hand grip rule also applies: if the fingers follow the current, the thumb points to the north pole of the electromagnet.
螺线管内部磁场强而均匀：B = μ₀ n I，n 为单位长度匝数。同样适用右手螺旋定则：四指沿电流方向弯曲，拇指指向电磁铁 N 极。
7. Magnetic Force on Moving Charges and Conductors | 运动电荷与通电导体在磁场中所受的力
A charge q moving with velocity v in a magnetic field B experiences a magnetic force (Lorentz force). The magnitude is F = q v B sin θ, where θ is the angle between v and B. The direction is given by the left-hand rule (for positive charge) or right-hand rule (for negative charge).
以速度 v 运动的电荷 q 在磁场 B 中会受到磁力（洛伦兹力）。其大小为 F = q v B sin θ，θ 为 v 与 B 之间的夹角。方向可用左手定则（正电荷）或右手定则（负电荷）判定。
F = q v B sin θ
If the velocity is perpendicular to a uniform magnetic field, the particle moves in a circular path. The magnetic force provides the centripetal force: q v B = m v² / r, leading to the radius r = m v / (q B).
若速度垂直于匀强磁场，粒子将做圆周运动。磁力提供向心力：q v B = m v² / r，得半径 r = m v / (q B)。
A current-carrying wire of length L in a magnetic field experiences a force: F = B I L sin θ. The direction is perpendicular to both B and I, determined by Fleming’s left-hand rule.
磁场中长度为 L 的载流导线所受的力为 F = B I L sin θ。其方向垂直于 B 和 I，可用弗莱明左手定则判定。
8. Electromagnetic Induction | 电磁感应
Electromagnetic induction is the production of an emf (voltage) across a conductor when it is exposed to a varying magnetic field. The induced emf is given by Faraday’s law.
电磁感应是当导体处于变化的磁场中时，在其两端产生电动势（电压）的现象。感应电动势由法拉第定律给出。
ε = -N ΔΦ / Δt
where N is the number of turns in a coil, and Φ is the magnetic flux. Magnetic flux through an area A is Φ = B A cos θ, where θ is the angle between the magnetic field and the normal to the area.
其中 N 为线圈匝数，Φ 为磁通量。通过面积 A 的磁通量为 Φ = B A cos θ，θ 为磁场方向与面积法线之间的夹角。
Lenz’s law states that the direction of the induced current is such that it opposes the change in magnetic flux that produced it. This is represented by the minus sign in Faraday’s law.
楞次定律指出，感应电流的方向总是阻碍引起感应电流的磁通量变化。这体现在法拉第定律的负号中。
For a straight conductor moving with speed v perpendicular to a magnetic field, the induced emf between its ends is ε = B L v.
对于垂直于磁场、以速度 v 运动的直导体，其两端产生的感应电动势为 ε = B L v。
9. Transformers and Alternating Current | 变压器与交流电
A transformer consists of two coils wound on a common iron core. It changes the voltage of an alternating current (AC) supply. For an ideal transformer, the ratio of voltages equals the ratio of turns:
变压器由绕在同一铁心上的两个线圈组成，用于改变交流电电压。对于理想变压器，电压比等于匝数比：
Vp / Vs = Np / Ns
where Vp, Np refer to the primary coil and Vs, Ns to the secondary coil. Power is conserved: Vp Ip = Vs Is. Step-up transformers increase voltage and decrease current, reducing power loss in long-distance transmission lines.
其中 Vp、Np 指初级线圈，Vs、Ns 指次级线圈。功率守恒：Vp Ip = Vs Is。升压变压器升高电压、降低电流，从而减少长距离输电的功率损耗。
AC voltage and current vary sinusoidally. The root-mean-square (rms) value relates to the peak value: V_rms = V₀ / √2 and I_rms = I₀ / √2. These rms values are used to calculate average power.
交流电压和电流呈正弦变化。有效值（rms）与峰值的关系为：V_rms = V₀ / √2，I_rms = I₀ / √2。有效值用于计算平均功率。
10. Problem-Solving Strategies for Electricity and Magnetism | 电与磁问题解题策略
Draw clear circuit diagrams and label all known quantities, including internal resistance. Apply Kirchhoff’s laws systematically to solve for unknown currents and voltages. When working with resistors, simplify networks using series and parallel combinations stepwise.
绘制清晰的电路图并标注所有已知量，包括内阻。系统地应用基尔霍夫定律求解未知电流和电压。处理电阻时，逐步用串并联公式简化网络。
In magnetic problems, first determine the field direction using the right-hand rule, then apply the appropriate force equation. Use the left-hand rule carefully for currents in magnetic fields. For electromagnetic induction, identify the change in flux and apply Lenz’s law to find the direction of induced emf.
在磁学问题中，先用右手定则确定磁场方向，然后应用相应的力公式。载流导线在磁场中的受力须谨慎使用左手定则。对于电磁感应，找出磁通量变化并运用楞次定律判断感应电动势方向。
Always check units: charge in C, field in N/C or V/m, potential difference in V, resistance in Ω, magnetic field in T, flux in Wb. For numerical answers, express results with appropriate significant figures. Practice problems that combine electric and magnetic concepts, such as the motion of a charged particle in crossed E and B fields.
始终检查单位：电荷用 C，电场用 N/C 或 V/m，电势差用 V，电阻用 Ω，磁感应强度用 T，磁通量用 Wb。数值答案应以合适的有效数字表示。多练习结合电、磁概念的题目，例如带电粒子在正交电场和磁场中的运动。
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