Physics Electromagnetism: Core Formula Summary | 物理电磁学核心公式总结

📚 Physics Electromagnetism: Core Formula Summary | 物理电磁学核心公式总结

Electromagnetism is one of the most essential and exam-relevant topics in A-Level and IB Physics. Mastering the core formulas not only helps you solve problems efficiently but also deepens your understanding of how electric and magnetic fields govern the physical world. This article provides a concise yet complete summary of the key equations you need, organised by topic, with explanations for each variable and typical exam applications.

电磁学是 A-Level 和 IB 物理中最重要、最常考的知识板块之一。熟练掌握核心公式不仅帮助你在考试中快速解题，更能加深你对电场和磁场如何支配物理世界的理解。本文按专题整理了你需要掌握的关键方程，逐一解释每个变量的含义，并指出典型的考试应用场景。

1. Coulomb’s Law & Electric Field Strength | 库仑定律与电场强度

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.

库仑定律描述两个点电荷之间的静电力大小。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 = kₑ·|q₁q₂| / r² = (1 / 4πε₀) · |q₁q₂| / r²

F: electrostatic force / 静电力 (N)
q₁, q₂: point charges / 点电荷量 (C)
r: separation distance / 电荷间距离 (m)
kₑ = 8.99 × 10⁹ N·m²·C⁻²: Coulomb’s constant / 库仑常量
ε₀ = 8.85 × 10⁻¹² C²·N⁻¹·m⁻²: permittivity of free space / 真空介电常数

The electric field strength E at a point is defined as the force per unit positive charge acting on a test charge placed at that point.

电场强度 E 定义为单位正电荷在该点所受的电场力。

E = F / q and for a point charge: E = kₑ·Q / r²

For a uniform electric field produced between two parallel plates, the field strength is related to the potential difference and plate separation.

对于平行板之间的匀强电场，场强与电势差和板间距有关。

E = V / d

V: potential difference between plates / 两板间电势差 (V)
d: plate separation / 板间距 (m)

2. Electric Potential & Potential Energy | 电势与电势能

Electric potential V at a point in an electric field is the work done per unit positive charge in bringing a test charge from infinity to that point.

电场中某点的电势 V 等于将单位正电荷从无穷远处移至该点所做的功。

V = W / q and for a point charge: V = kₑ·Q / r

The electric potential energy of a charge q placed at a point where the potential is V is simply:

电荷 q 在电势为 V 的点处所具有的电势能为：

Eₚ = qV

When a charge moves through a potential difference ΔV, the change in electric potential energy equals the work done by the electric field. This concept is crucial for calculations involving charged particles accelerated between plates.

当电荷通过电势差 ΔV 移动时，电势能的变化量等于电场力做的功。这一概念对计算带电粒子在极板间加速的问题至关重要。

W = qΔV = ½mv² (for a charge starting from rest)

3. Capacitance & Energy Stored | 电容与储存能量

Capacitance measures the ability of a conductor or capacitor to store charge per unit potential difference.

电容表示导体或电容器在单位电势差下储存电荷的能力。

C = Q / V

C: capacitance / 电容 (F, farad)
Q: charge stored / 储存的电荷量 (C)
V: potential difference across the capacitor / 电容器两端电势差 (V)

For a parallel-plate capacitor, the capacitance depends on the geometry and the dielectric material between the plates.

对于平行板电容器，电容取决于极板的几何结构以及板间电介质材料。

C = ε₀εᵣA / d

εᵣ: relative permittivity (dielectric constant) / 相对介电常数
A: plate area / 极板面积 (m²)
d: separation / 极板间距 (m)

The energy stored in a charged capacitor can be expressed in three equivalent forms.

充电电容器储存的能量有三种等价表达形式。

E = ½QV = ½CV² = Q² / (2C)

In capacitor discharge problems (e.g., RC circuits), the exponential decay equations are essential. The charge on a capacitor discharging through a resistor decays according to:

在电容器放电问题（如 RC 电路）中，指数衰减方程至关重要。电容器通过电阻放电时，电荷按以下规律衰减：

Q = Q₀·e^(−t/RC)

Q₀: initial charge / 初始电荷量 (C)
t: time elapsed / 经过的时间 (s)
R: resistance in series / 串联电阻 (Ω)

Similarly, the voltage across the capacitor follows V = V₀·e^(−t/RC), and the time constant τ = RC represents the time for the charge to fall to 1/e (about 37%) of its initial value.

类似地，电容器两端电压满足 V = V₀·e^(−t/RC)，时间常数 τ = RC 表示电荷降至初始值 1/e（约 37%）所需的时间。

4. Ohm’s Law & Electrical Power | 欧姆定律与电功率

Ohm’s Law is fundamental to DC circuit analysis. It states that the current through a conductor is directly proportional to the potential difference across it, provided temperature and other physical conditions remain constant.

欧姆定律是直流电路分析的基础。它表明在温度和物理条件保持恒定的情况下，通过导体的电流与导体两端的电势差成正比。

V = IR

V: potential difference / 电势差 (V)
I: current / 电流 (A)
R: resistance / 电阻 (Ω)

The resistance of a uniform wire depends on its resistivity, length, and cross-sectional area.

均匀导线的电阻取决于其电阻率、长度和横截面积。

R = ρ·L / A

ρ: resistivity / 电阻率 (Ω·m)
L: length / 长度 (m)
A: cross-sectional area / 横截面积 (m²)

Electrical power dissipated in a resistor can be calculated using any of the following equivalent formulas.

电阻上消耗的电功率可用以下等价公式计算。

P = VI = I²R = V² / R

When a battery with emf E and internal resistance r drives current through an external load R, the terminal voltage is less than the emf. Kirchhoff’s voltage law gives:

当电动势为 E、内阻为 r 的电池驱动电流通过外部负载 R 时，端电压小于电动势。基尔霍夫电压定律给出：

E = I(R + r) = V + Ir

Maximum power transfer occurs when the external resistance equals the internal resistance (R = r), a classic exam question.

当外部电阻等于内阻（R = r）时，负载获得最大功率，这是一个经典考题。

5. Kirchhoff’s Laws & Circuit Analysis | 基尔霍夫定律与电路分析

Kirchhoff’s Current Law (KCL) states that the sum of currents entering a junction equals the sum of currents leaving the junction. This is a consequence of charge conservation.

基尔霍夫电流定律（KCL）指出：流入节点（结点）的电流之和等于流出该节点的电流之和。这是电荷守恒的必然结果。

ΣIᵢₙ = ΣIₒᵤₜ

Kirchhoff’s Voltage Law (KVL) states that the sum of the electromotive forces (emfs) around any closed loop in a circuit equals the sum of the potential drops across all components in that loop.

基尔霍夫电压定律（KVL）指出：沿电路中任一闭合回路，电动势之和等于回路中所有元件上电势降落之和。

Σ emf = Σ IR (around any closed loop)

For resistors in series, the total resistance is the sum of individual resistances. For resistors in parallel, the reciprocal of the total resistance equals the sum of the reciprocals.

对于串联电阻，总电阻等于各电阻之和；对于并联电阻，总电阻的倒数等于各电阻倒数之和。

Rₛ = R₁ + R₂ + R₃ + …

1/Rₚ = 1/R₁ + 1/R₂ + 1/R₃ + …

For capacitors, the rules are reversed: capacitors in parallel add directly, while capacitors in series combine as reciprocals.

对于电容器，规则恰好相反：并联电容直接相加，串联电容按倒数方式组合。

Cₚ = C₁ + C₂ + C₃ + …

1/Cₛ = 1/C₁ + 1/C₂ + 1/C₃ + …

6. Magnetic Force on a Moving Charge | 运动电荷在磁场中的受力

A charged particle moving perpendicular to a uniform magnetic field experiences a force perpendicular to both its velocity and the magnetic field direction. This is described by the Lorentz force law for a point charge.

带电粒子垂直于匀强磁场运动时，会受到一个同时垂直于速度方向和磁场方向的力。这就是点电荷的洛伦兹力定律。

F = qvB·sinθ

F: magnetic force / 磁场力 (N)
q: charge / 电荷量 (C)
v: speed of the particle / 粒子速度 (m/s)
B: magnetic flux density / 磁感应强度 (T, tesla)
θ: angle between v and B / v 与 B 之间的夹角

When θ = 90°, the particle moves in a circular path because the magnetic force acts as the centripetal force. Equating the two forces gives:

当 θ = 90° 时，粒子做匀速圆周运动，因为磁场力充当向心力。两种力相等可得：

qvB = mv² / r

Rearranging, the radius of the circular path is:

整理后，圆周运动的半径为：

r = mv / (qB)

The cyclotron frequency (angular speed) is given by ω = qB/m, and the period of revolution is T = 2πm/(qB). These formulas are commonly tested in particle accelerator and mass spectrometry problems.

回旋频率（角速度）为 ω = qB/m，回旋周期为 T = 2πm/(qB)。这些公式常见于粒子加速器和质谱仪相关考题中。

7. Magnetic Force on a Current-Carrying Wire | 载流导线在磁场中的受力

A wire carrying a current in a magnetic field experiences a force given by the equation below. The direction is determined by Fleming’s left-hand rule.

载流导线在磁场中受到的作用力由以下方程给出，方向用弗莱明左手定则判断。

F = BIL·sinθ

F: magnetic force / 磁场力 (N)
B: magnetic flux density / 磁感应强度 (T)
I: current / 电流 (A)
L: length of wire in the field / 导线处于磁场中的长度 (m)
θ: angle between the wire and B / 导线与 B 的夹角

When the wire is perpendicular to the magnetic field (θ = 90°), the force is maximised: F = BIL.

当导线垂直于磁场时（θ = 90°），力最大：F = BIL。

This principle underlies the operation of electric motors, galvanometers, and loudspeakers. In exam questions, you may be asked to calculate the force on a rectangular coil in a uniform magnetic field, where the torque is τ = BINA·cosθ.

这一原理是电动机、电流计和扬声器工作的基础。在考试中，可能会要求你计算匀强磁场中矩形线圈所受的力矩，其公式为 τ = BINA·cosθ。

8. Magnetic Flux & Faraday’s Law | 磁通量与法拉第定律

Magnetic flux Φ through a surface is the product of the magnetic flux density and the area perpendicular to the field. It measures the total magnetic field passing through a given area.

通过某一表面的磁通量 Φ 等于磁感应强度与垂直于磁场的面积的乘积，它度量穿过给定面积的总磁感线数量。

Φ = BA·cosθ

Φ: magnetic flux / 磁通量 (Wb, weber)
B: magnetic flux density / 磁感应强度 (T)
A: area / 面积 (m²)
θ: angle between the normal to the area and B / 面积法线与 B 的夹角

Faraday’s Law of electromagnetic induction states that the induced electromotive force (emf) in a coil equals the negative rate of change of magnetic flux linkage through the coil.

法拉第电磁感应定律指出：线圈中产生的感应电动势等于通过线圈的磁通链变化率的负值。

ε = −N·(ΔΦ / Δt)

ε: induced emf / 感应电动势 (V)
N: number of turns / 线圈匝数
ΔΦ / Δt: rate of change of flux / 磁通量变化率 (Wb/s)

Lenz’s Law, which is a consequence of energy conservation, determines the direction of the induced current: it always opposes the change in flux that produced it. The negative sign in Faraday’s Law embodies this principle.

楞次定律是能量守恒的推论，它决定了感应电流的方向：感应电流总是阻碍引起它的磁通量变化。法拉第定律中的负号正体现了这一原理。

For a rod of length L moving perpendicular to a magnetic field with speed v, the motional emf is simply:

对于以速度 v 垂直于磁场运动的长度为 L 的导体棒，动生电动势为：

ε = BLv

9. Transformers & Alternating Current | 变压器与交流电

A transformer operates on the principle of mutual induction. For an ideal transformer, the ratio of the voltages across the primary and secondary coils equals the ratio of the number of turns.

变压器基于互感原理工作。对于理想变压器，原副线圈电压之比等于匝数之比。

Vₛ / Vₚ = Nₛ / Nₚ

Vₛ, Vₚ: secondary and primary voltages / 副线圈与原线圈电压 (V)
Nₛ, Nₚ: number of turns in secondary and primary / 副线圈与原线圈匝数

For an ideal transformer, power is conserved, so the current ratio is inversely proportional to the turns ratio.

对于理想变压器，功率守恒，因此电流之比与匝数之比成反比。

VₚIₚ = VₛIₛ

The root-mean-square (rms) values for a sinusoidal alternating current are essential for power calculations. The rms voltage and current are given by:

正弦交流电的均方根（rms）值对功率计算至关重要。rms 电压和电流为：

V_rms = V₀ / √2 and I_rms = I₀ / √2

The average power dissipated in an AC circuit is P = V_rms·I_rms, which is equal to the power that would be dissipated by the equivalent DC values.

交流电路中消耗的平均功率为 P = V_rms·I_rms，它等于等效直流值所消耗的功率。

10. Charge & Electric Field Relationships | 电荷与电场关系综合补充

In addition to the standard equations above, several key relationships among electric field, force, and potential are frequently examined in short-answer questions. The electric field is the negative gradient of potential in one dimension.

除上述标准方程外，电场、力与电势之间还有几个关键关系经常在简答题中出现。在一维情况下，电场是电势的负梯度。

E = −dV / dx

The work done moving a charge q through a potential difference ΔV is equal to the change in kinetic energy, which is the basis for many acceleration problems.

将电荷 q 移动通过电势差 ΔV 所做的功等于动能的变化量，这是很多加速问题的解题基础。

For a system of two point charges, the electric potential energy is:

对于两个点电荷组成的系统，电势能为：

U = kₑ·q₁q₂ / r

This energy is positive for like charges (repulsive) and negative for opposite charges (attractive), a detail that is often tested conceptually.

同种电荷（排斥）时能量为正，异种电荷（吸引）时能量为负，这一概念细节经常在选择题中出现。

11. Key Units & Conversion Table | 关键单位与换算表

The following table summarises the SI units associated with the formulas covered in this article. Avoid common unit conversion mistakes by memorising these fundamental definitions.

下表总结了本文涉及公式的 SI 单位。牢记这些基本定义可以避免常见的单位换算错误。

Quantity / 物理量	Symbol / 符号	SI Unit / 单位	Equivalent / 等价形式
Electric Charge / 电荷量	Q	coulomb (C)	A·s
Electric Field / 电场强度	E	N/C	V/m
Electric Potential / 电势	V	volt (V)	J/C
Capacitance / 电容	C	farad (F)	C/V
Resistance / 电阻	R	ohm (Ω)	V/A
Magnetic Flux Density / 磁感应强度	B	tesla (T)	N/(A·m)
Magnetic Flux / 磁通量	Φ	weber (Wb)	T·m²

12. Exam Tips & Common Misconceptions | 考试技巧与常见误区

Many students lose marks on electromagnetism questions not because they cannot recall formulas, but because they apply them incorrectly. Here are the most common pitfalls to avoid.

很多学生在电磁学题目上丢分，不是因为他们记不住公式，而是因为用错了方法。以下是最常见的一些陷阱，需要特别注意。

Confusing E = V/d with E = kₑQ/r². The former applies only to uniform fields (parallel plates); the latter applies to point charges or radially symmetric fields.

混淆 E = V/d 与 E = kₑQ/r²。前者仅适用于匀强电场（平行板）；后者适用于点电荷或球对称电场。
Forgetting the sinθ term in magnetic force equations. The maximum force occurs when velocity/current is perpendicular to the magnetic field; if not stated explicitly, check the angle.

在磁场力公式中忘记 sinθ。当速度/电流垂直于磁场时力最大；如果题目没有明确说明角度，一定要检查。
Signs in Faraday’s Law. The negative sign indicates Lenz’s Law; remember that the induced current opposes the change in flux, not necessarily the field itself.

法拉第定律中的正负号。负号表示楞次定律；要记住感应电流阻碍的是磁通量的变化，不一定与磁场本身方向相反。
Using peak values instead of rms values in AC power calculations. Always use V_rms and I_rms unless the question specifically asks for peak values.

在交流功率计算中使用峰值而非 rms 值。除非题目明确要求峰值，否则始终使用 V_rms 和 I_rms。
Mixing up series and parallel combinations for capacitors versus resistors. Capacitors in parallel combine like resistors in series, and vice versa.

混淆电容器与电阻的串并联组合规则。电容并联的组合方式相当于电阻串联，反之亦然。
Ignoring units when using ε₀. Always convert distances to metres and areas to square metres before substituting into formulas.

使用 ε₀ 时忽略单位换算。代入公式前，务必将距离换算为米、面积换算为平方米。

Finally, draw a clear diagram for every problem involving magnetic fields or circuits. Label the direction of B, v, I, and F using Fleming’s rules. This will save you from careless directional errors and help you earn partial credit in written answer questions.

最后，遇到磁场或电路问题时，务必画出清晰的示意图。用弗莱明定则标出 B、v、I 和 F 的方向。这能避免因方向判断失误而丢分，也有助于在解答题中获得过程分。

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Physics Electromagnetism: Core Formula Summary | 物理电磁学核心公式总结