📚 A-Level Physics: Formula Derivation from June 2018 Mark Scheme 5 | A-Level 物理：2018年6月评分方案5中的公式推导
Many A-Level Physics papers test the ability to derive key formulas from first principles. A classic example appears in the June 2018 series, where one question (often question 5 in certain boards) required candidates to derive the expression for the radius of curvature of a charged particle moving perpendicularly through a uniform magnetic field. This article breaks down that derivation step by step, linking each stage to the mark scheme points that examiners typically expect. We will also extend the derivation to find the period of circular motion, and discuss common pitfalls.
许多 A-Level 物理试卷都会考查从基本原理推导关键公式的能力。一个经典的例子出现在 2018 年 6 月系列的考试中，其中一道题目（在某些考试局中通常是第五题）要求考生推导带电粒子垂直于匀强磁场运动时的曲率半径表达式。本文将逐步拆解这一推导过程，并将每个阶段与阅卷人通常期望的评分方案得分点联系起来。我们还会进一步推导圆周运动的周期，并讨论常见错误。
1. Setting the Scene: Particle in a Magnetic Field | 情景设定：磁场中的粒子
A charged particle of charge q moves with velocity v at right angles to a uniform magnetic field of flux density B. The particle experiences a force that is always perpendicular to both its velocity and the field.
一个电荷量为 q 的带电粒子以速度 v 垂直于磁感应强度为 B 的匀强磁场运动。粒子会受到一个始终垂直于其速度和磁场方向的力。
The force acting on the particle is the magnetic Lorentz force, given by F = Bqv sinθ. Since the particle enters the field at 90°, sin 90° = 1, so the magnitude of the force is simply F = Bqv.
作用在粒子上的力是洛伦兹磁力，表达式为 F = Bqv sinθ。由于粒子以 90° 进入磁场，sin 90° = 1，因此力的大小简化为 F = Bqv。
This force does no work on the particle because it always acts perpendicular to the instantaneous velocity. Consequently, the speed v remains constant, but the direction changes continuously, forcing the particle into a circular path.
该力对粒子不做功，因为它始终垂直于瞬时速度。因此，速率 v 保持不变，但方向持续改变，迫使粒子进入圆周路径。
2. The Essential Balance of Forces | 力的基本平衡
For circular motion, the net inward force must equal the centripetal force required to keep the particle moving in a circle of radius r. The magnetic force provides this centripetal force.
对于圆周运动，向内的合力必须等于维持粒子在半径为 r 的圆上运动所需的向心力。磁力恰好提供了这个向心力。
The required centripetal force is given by Fc = mv² / r, where m is the mass of the particle.
所需向心力由 Fc = mv² / r 给出，其中 m 是粒子的质量。
Equating the magnetic force and the centripetal force: Bqv = mv² / r. This step is the core of the derivation and a key mark-scheme point.
令磁力与向心力相等：Bqv = mv² / r。这一步是推导的核心，也是评分方案中的一个关键得分点。
3. Deriving the Radius r | 推导半径 r
From the equality Bqv = mv² / r, we can cancel one power of v from each side (assuming v ≠ 0), giving Bq = mv / r. Rearranging to make r the subject yields the familiar form:
由等式 Bqv = mv² / r 出发，我们可以从两边各消去一个 v（假设 v ≠ 0），得到 Bq = mv / r。重新整理，使 r 成为公式的主项，就得到了我们熟悉的形式：
r = mv / (Bq)
It is crucial to present the steps clearly: equating forces, cancelling v, and rearranging. Many mark schemes award marks for each of these algebraic manipulations.
清晰地展示步骤至关重要：让力相等、约去 v 以及重新整理。许多评分方案会对这些代数操作中的每一步分别给分。
One common variation is when the particle is an electron, with charge e. Then the radius becomes r = mv / (Be). Always substitute the appropriate charge symbol as given in the question.
一个常见的变体是当粒子为电子时，其电荷为 e。此时半径变为 r = mv / (Be)。务必根据题目给出的符号代入正确的电荷符号。
4. Checking Units and Proportionalities | 检查单位与比例关系
We can verify the derived formula by examining units: [r] = [m][v] / ([B][q]). Using SI base units: kg × m s⁻¹ / (N A⁻¹ m⁻¹ × A s). Since N = kg m s⁻², N A⁻¹ m⁻¹ simplifies to kg s⁻² A⁻¹. The denominator becomes (kg s⁻² A⁻¹) × (A s) = kg s⁻¹. Thus the whole expression gives m, which matches the unit of radius. Such a unit check can prevent algebraic mistakes.
我们可以通过检查单位来验证推导出的公式：[r] = [m][v] / ([B][q])。采用国际单位制基本单位：kg × m s⁻¹ / (N A⁻¹ m⁻¹ × A s)。因为 N = kg m s⁻²，N A⁻¹ m⁻¹ 可简化为 kg s⁻² A⁻¹。分母变为 (kg s⁻² A⁻¹) × (A s) = kg s⁻¹。因此整个表达式得出 m，与半径的单位一致。这样的单位检查有助于避免代数错误。
The equation also shows that r ∝ v (directly proportional to speed) and r ∝ 1/(Bq) (inversely proportional to both magnetic flux density and charge). Understanding these proportionalities helps in qualitative questions.
该方程还表明 r ∝ v（与速度成正比）以及 r ∝ 1/(Bq)（与磁感应强度和电荷量均成反比）。理解这些比例关系有助于解答定性问题。
5. Deriving the Period of Circular Motion | 推导圆周运动周期
Once the radius is known, we can find the time taken for one complete revolution, i.e. the period T. The circumference of the circular path is 2πr, and the particle moves at constant speed v, so the period is T = 2πr / v.
一旦知道了半径，我们就可以求出完成一整圈所需的时间，即周期 T。圆形路径的周长为 2πr，而粒子以恒定速率 v 运动，因此周期为 T = 2πr / v。
Substituting r = mv/(Bq) into T = 2πr/v gives:
将 r = mv/(Bq) 代入 T = 2πr/v，得到：
T = 2πm / (Bq)
Notice that v cancels out, meaning the period is independent of the particle’s speed. This counter-intuitive result is often tested in multiple-choice questions.
注意 v 被消去了，这意味着周期与粒子的速度无关。这一违反直觉的结果常在选择题中被考查。
The frequency of revolution (cyclotron frequency) is f = 1/T = Bq/(2πm). The derivation of T is a natural extension and is frequently part of the same question.
回旋频率（cyclotron frequency）为 f = 1/T = Bq/(2πm)。T 的推导是自然的延伸，且通常是同一道题的一部分。
6. Linking to the Mark Scheme – Typical Scoring Points | 关联评分方案 – 典型得分点
In the June 2018 mark scheme for a typical awarding body, the derivation question (often Q5) had the following allocation of marks:
在 2018 年 6 月一个典型考试局的评分方案中，推导题（通常是第5题）的分数分配如下：
- Identifying magnetic force equation F = Bqv (1 mark)
- Stating or using centripetal force equation F = mv²/r (1 mark)
- Equating the two forces correctly (1 mark)
- Algebraic manipulation to r = mv/(Bq) (1 mark)
- Optional substitution and derivation of T = 2πm/(Bq) (1 additional mark)
- 写出磁力方程 F = Bqv（1 分）
- 给出或使用向心力方程 F = mv²/r（1 分）
- 正确令两个力相等（1 分）
- 通过代数运算得到 r = mv/(Bq)（1 分）
- 可选代入并推导 T = 2πm/(Bq)（额外 1 分）
Examiners’ reports often highlight that candidates lose marks by forgetting to state that the magnetic force is the centripetal force, or by not justifying the use of mv²/r. Explicit verbal explanation is rewarded.
考官报告通常会强调，考生因忘记说明磁力就是向心力，或者没有解释为何使用 mv²/r 而失分。明确的文字说明会得到加分。
7. The Role of ‘Perpendicular’ in the Derivation | 推导中“垂直”的作用
The original step F = Bqv relies on the velocity being perpendicular to the magnetic field. If the particle enters at an angle θ to the field, the perpendicular component v sinθ must be used. The mark scheme often requires stating ‘for perpendicular entry’ or ‘since v ⊥ B’.
最初的步骤 F = Bqv 依赖于速度与磁场垂直。如果粒子以与磁场成 θ 角的方向进入，则必须使用垂直分量 v sinθ。评分方案常要求说明“适用于垂直入射”或“由于 v ⊥ B”。
In the June 2018 question, the scenario typically specified that the particle enters a region of magnetic field at right angles. Make sure you read the question carefully to pick up these details.
在 2018 年 6 月的题目中，通常设定粒子以直角进入磁场区域。务必仔细审题，以抓住这些细节。
8. Common Errors and How to Avoid Them | 常见错误及如何避免
One mistake is incorrectly writing the centripetal force as mv²r or as mvr. Always remember it is mv²/r. Another is failing to cancel v correctly, resulting in r = mv²/(Bq). Show each algebraic step to minimise slip-ups.
一个错误是将向心力误写为 mv²r 或 mvr。务必记住它是 mv²/r。另一个错误是未能正确约去 v，从而导致 r = mv²/(Bq)。逐步展示每个代数步骤可以减少笔误。
Some candidates mistakenly use the formula for electric force or confuse B with E. Keep the forces distinct: magnetic force is Bqv (for a moving charge), electric force is Eq.
有些考生错误地使用了电场力公式，或混淆了 B 与 E。要区分不同的力：磁力是 Bqv（对于运动电荷），电场力是 Eq。
In the derivation of period, a common slip is to write T = 2πmv/(Bq) rather than substituting r correctly. Always substitute the derived radius formula as a whole.
在推导周期时，一个常见的笔误是写出 T = 2πmv/(Bq) 而不是正确地代入 r。务必整体代入推导出的半径公式。
9. Practical Applications of the Formula | 公式的实际应用
The formula r = mv/(Bq) is not just a textbook exercise. It explains how mass spectrometers separate ions of different mass-to-charge ratio. Ions with the same speed but different masses will have different radii, enabling identification.
公式 r = mv/(Bq) 不只是一个课本练习。它解释了质谱仪如何分离不同质荷比的离子。速度相同但质量不同的离子会有不同的半径，从而实现识别。
In a cyclotron, the period T = 2πm/(Bq) determines the frequency of the alternating voltage needed to accelerate particles. Because T is independent of v, the synchronisation remains constant even as the particles gain energy.
在回旋加速器中，周期 T = 2πm/(Bq) 决定了加速粒子所需的交变电压的频率。由于 T 与 v 无关，即使粒子能量增加，同步性仍然保持恒定。
10. Extension: Relativistic Correction? | 拓展：相对论修正？
At speeds approaching the speed of light, the mass m increases according to relativistic principles. However, at A-Level, we assume non-relativistic speeds, so m is constant and the derivation holds. Relativistic corrections are beyond the scope of the June 2018 paper.
当速度接近光速时，质量 m 会根据相对论原理增大。然而，在 A-Level 阶段，我们假设速度非相对论性，因此 m 是常数，推导成立。相对论修正超出了 2018 年 6 月试卷的范围。
11. Summary of the Derivation Flow | 推导流程总结
To recap:
总结一下：
| 1. Magnetic force: F = Bqv | 磁力：F = Bqv |
| 2. Centripetal force: F = mv²/r | 向心力：F = mv²/r |
| 3. Equate: Bqv = mv²/r | 令其相等：Bqv = mv²/r |
| 4. Cancel v: Bq = mv/r | 约去 v：Bq = mv/r |
| 5. Rearrange: r = mv/(Bq) | 整理：r = mv/(Bq) |
| 6. Period: T = 2πr/v → T = 2πm/(Bq) | 周期：T = 2πr/v → T = 2πm/(Bq) |
Rehearse this sequence until you can reproduce it without hesitation, and you will be well prepared for similar questions.
反复演练这个顺序，直到你能毫不犹豫地复述它，这样你就能为类似的题目做好充分准备。
12. Final Exam Tips | 最后的备考建议
When tackling a formula derivation in the exam, always state the relevant physical principles in words before writing equations. This not only shows understanding but also often earns a mark even if the algebra later goes wrong. Keep your working logical and well-spaced, and double-check your cancellations.
在考试中处理公式推导时，始终先用文字陈述相关的物理原理，再写出方程。这不仅能展示你的理解，而且即使后续代数运算出错，也往往能拿到分数。保持你的运算逻辑清晰、步骤间隔合理，并再次检查你的约分。
Finally, practise with past papers, paying close attention to the wording of mark schemes. The June 2018 mark scheme 5 is an excellent resource to see exactly how marks are awarded for derivations.
最后，利用历年真题进行练习，尤其注意评分方案的措辞。2018 年 6 月的评分方案 5 是了解推导题如何给分的一个极佳资源。
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