Oscillations and Fields: WJEC Physics Unit 3 & 4 Complete Guide
Oscillations and Fields is the cornerstone of A-Level WJEC Physics, spanning both Unit 3 (Oscillations and Nuclei) and Unit 4 (Fields and Options). This module connects foundational mechanics with advanced electromagnetic theory — mastering it is essential for scoring A* and for university-level physics. This guide covers every key concept, formula, and exam technique you need.
振荡与场：WJEC物理单元3和4完整指南
振荡与场是A-Level WJEC物理的核心模块，横跨单元3（振荡与原子核）和单元4（场与选修）。该模块将基础力学与高等电磁理论相衔接——掌握这部分内容对于获得A*以及大学物理学习至关重要。本指南涵盖了你需要的所有关键概念、公式和考试技巧。
Part 1: Simple Harmonic Motion (SHM)
Simple Harmonic Motion is the fundamental oscillatory model. A body exhibits SHM when its acceleration is directly proportional to its displacement from equilibrium and always directed towards the equilibrium position:
a = -ω²x
Where ω = 2πf is the angular frequency, x is displacement, and the negative sign indicates the restoring direction.
Key equations for displacement, velocity, and acceleration:
x = A cos(ωt) | v = ±ω√(A² - x²) | vₘₐₓ = ωA
Energy in SHM: Total energy is conserved and proportional to A². Kinetic energy is maximum at equilibrium (x=0); potential energy is maximum at amplitude (x=±A). The energy continuously transforms between kinetic and potential forms:
Eₖ = ½mω²(A² - x²) | Eₚ = ½mω²x² | Eₜₒₜ = ½mω²A²
Exam tip: WJEC frequently asks you to derive velocity as a function of displacement using energy conservation — practice this derivation until it becomes automatic. Also be ready to sketch displacement-time, velocity-time, and acceleration-time graphs on the same axes, showing correct phase relationships (velocity leads displacement by π/2; acceleration is in antiphase with displacement).
第一部分：简谐运动 (SHM)
简谐运动是最基本的振动模型。当物体的加速度与偏离平衡位置的位移成正比且始终指向平衡位置时，物体做简谐运动。核心方程 a = -ω²x 中的负号表示恢复力的方向。
能量关系是WJEC高频考点：总能量守恒且正比于振幅平方。动能和势能随位置不断转换，在平衡位置动能最大，在最大位移处势能最大。务必熟练掌握 Eₜₒₜ = ½mω²A² 的推导过程。
考试技巧：WJEC常要求用能量守恒推导速度-位移关系，并且在同一坐标轴上绘制位移、速度、加速度的时间图像，展示正确的相位关系——速度超前位移π/2，加速度与位移反相。
Part 2: Damped Oscillations and Resonance
Damping removes energy from an oscillating system. WJEC distinguishes three regimes:
Light damping: Amplitude decreases gradually over many cycles. The period remains approximately constant. This is the most common exam scenario.
Critical damping: The system returns to equilibrium in the shortest possible time without oscillating. Applications include car suspension systems and galvanometer needle damping.
Heavy damping: The system returns to equilibrium slowly with no oscillation. Think of a door closer with thick oil.
Forced oscillations and resonance: When a periodic driving force is applied, the system oscillates at the driving frequency. Resonance occurs when the driving frequency matches the natural frequency f₀ of the system — amplitude becomes maximum. The sharpness of resonance depends on the damping: light damping produces a sharp resonance peak; heavy damping broadens and lowers the peak.
Key applications: Microwave ovens (water molecules resonate at ~2.45 GHz), MRI scanners, musical instruments (air column resonance), and the catastrophic Tacoma Narrows Bridge collapse (wind-induced resonance).
Exam focus: Be able to sketch amplitude-frequency graphs showing the effect of different damping levels. Also be prepared to explain why resonance is both useful (radios, musical instruments) and dangerous (bridges, buildings during earthquakes).
第二部分：阻尼振荡与共振
阻尼从振动系统中移除能量。WJEC区分三种阻尼类型：轻阻尼（振幅在多个周期内逐渐减小，周期近似不变）、临界阻尼（系统在最短时间内回到平衡位置而不发生振荡，应用于汽车悬挂）、重阻尼（系统缓慢回到平衡位置，无振荡）。
受迫振荡与共振：施加周期性驱动力时，系统以驱动频率振动。当驱动频率等于系统固有频率时发生共振，振幅达到最大值。共振的尖锐程度取决于阻尼大小——轻阻尼产生尖锐的共振峰，重阻尼使峰值变宽变低。
重要应用：微波炉（水分子在约2.45GHz处共振）、核磁共振成像、乐器（空气柱共振）以及著名的塔科马海峡大桥坍塌事件（风致共振）。必须能绘制不同阻尼水平下的振幅-频率曲线图。
Part 3: Gravitational Fields
Newton's Law of Gravitation: Every point mass attracts every other point mass with a force proportional to the product of their masses and inversely proportional to the square of their separation:
F = -GMm / r²
The negative sign indicates the force is attractive. G = 6.67 × 10⁻¹¹ N m² kg⁻².
Gravitational field strength g is defined as the force per unit mass experienced by a small test mass placed in the field: g = F/m = GM/r². Near Earth's surface, g ≈ 9.81 N kg⁻¹.
Gravitational potential V at a point is the work done per unit mass to bring a test mass from infinity to that point:
V = -GM / r
The negative sign arises because work must be done against the field to move a mass away — gravitational potential is always negative and becomes zero at infinity. The gravitational potential energy of a mass m at a point is U = mV.
Key relationships that WJEC loves to test:
g = -dV/dr (field strength is the negative potential gradient)
ΔW = mΔV (work done = mass × potential difference)
Escape velocity: vₑₛ = √(2GM/R) — derived from equating KE to the work needed to reach infinity
Satellite orbits: For a satellite in circular orbit, centripetal force is provided by gravity: GMm/r² = mv²/r. This yields v = √(GM/r) and T² ∝ r³ (Kepler's Third Law). Geostationary satellites have a period of 24 hours and orbit at approximately 36,000 km above the equator — they appear stationary relative to Earth's surface.
第三部分：引力场
牛顿万有引力定律：F = -GMm/r²，G = 6.67×10⁻¹¹ N·m²·kg⁻²。引力场强度 g = F/m = GM/r²，地表g ≈ 9.81 N/kg。
引力势 V = -GM/r，负号表示将质量从无穷远移到该点需要克服引力场做功。引力势在无穷远处为零，在所有有限距离处为负值。核心关系：g = -dV/dr（场强等于负势梯度），这是WJEC必考公式。
卫星轨道：圆形轨道中引力提供向心力，导出 v = √(GM/r) 和开普勒第三定律 T² ∝ r³。地球同步卫星周期为24小时，轨道高度约36000km，位于赤道平面上方。
Part 4: Electric Fields
Coulomb's Law mirrors Newton's gravitation law in form:
F = kQ₁Q₂ / r² where k = 1/(4πε₀)
The key difference: electric forces can be attractive or repulsive (unlike gravity, which is always attractive). ε₀ = 8.85 × 10⁻¹² F m⁻¹ is the permittivity of free space.
Electric field strength E: Defined as force per unit positive charge: E = F/q. For a point charge: E = kQ/r². For a uniform field between parallel plates: E = V/d (potential difference divided by plate separation).
Electric potential V at a point is the work done per unit charge to bring a small positive test charge from infinity to that point. For a point charge: V = kQ/r. Unlike gravitational potential, electric potential can be positive or negative depending on the sign of Q.
Comparison of fields — WJEC frequently asks you to compare gravitational and electric fields. Key similarities: both obey inverse-square laws, both define field strength as force per unit property, both have potential concepts. Key differences: gravity is always attractive (mass is always positive) while electric forces can repel; gravitational potential is always negative while electric potential can have either sign; G is universal while k is very large (orders of magnitude larger).
Particle motion in electric fields: Charged particles experience a constant force in uniform electric fields, producing parabolic trajectories — analogous to projectile motion in gravitational fields. This is a standard WJEC calculation question involving F = qE, kinematic equations, and sometimes combined electric and magnetic fields (velocity selectors).
第四部分：电场
库仑定律：F = kQ₁Q₂/r²，k = 1/(4πε₀)，ε₀ = 8.85×10⁻¹² F/m。电场力可以是吸引力或排斥力（与始终为吸引的引力不同）。电场强度 E = F/q，点电荷 E = kQ/r²，平行板均匀电场 E = V/d。
电势：V = kQ/r，电势的正负取决于Q的符号。WJEC高频要求比较引力场和电场：相同点是都遵循平方反比定律、都有场强和势的概念；不同点是引力始终为吸引力而电场可推可拉、引力势始终为负而电势可正可负、引力常量G远小于库仑常量k。
带电粒子在电场中的运动产生抛物线轨迹，类似于引力场中的抛体运动，是标准计算题——涉及 F = qE、运动学方程以及电场与磁场结合的速度选择器问题。
Part 5: Capacitors
Capacitance C is defined as the charge stored per unit potential difference: C = Q/V. Unit: farad (F). A capacitor stores energy in the electric field between its plates.
Energy stored: The work done to charge a capacitor equals the area under the Q-V graph, giving:
E = ½QV = ½CV² = ½Q²/C
Capacitor charging and discharging: Through a resistor R, the charge/discharge follows an exponential law. The time constant τ = RC represents the time for charge to fall to 1/e (~37%) of its initial value during discharge, or to reach ~63% of its final value during charging.
Discharge equation: Q = Q₀e^(-t/RC), similarly V = V₀e^(-t/RC) and I = I₀e^(-t/RC).
Graphical analysis: WJEC expects you to use log-linear plots for capacitor discharge — taking natural logs gives ln V = ln V₀ - t/RC, so a graph of ln V vs t yields a straight line with gradient -1/RC. This is a classic data-analysis question.
Dielectrics: Inserting a dielectric material between plates increases capacitance (C = εᵣε₀A/d where εᵣ > 1 is the relative permittivity). The dielectric becomes polarised, reducing the effective electric field and allowing more charge to be stored for the same voltage.
第五部分：电容器
电容 C = Q/V，单位法拉(F)。储存能量 E = ½QV = ½CV² = ½Q²/C。充放电遵循指数规律，时间常数 τ = RC，是电荷降至初始值1/e（约37%）所需的时间。放电方程 Q = Q₀e^(-t/RC)。
对数图分析：取自然对数得 ln V = ln V₀ - t/RC，ln V对t的图是直线，斜率 -1/RC。这是WJEC经典的数据分析题。电介质通过极化效应增加电容，C = εᵣε₀A/d，相对介电常数 εᵣ > 1。
Part 6: Magnetic Fields and Electromagnetic Induction
Magnetic force on a moving charge: A charge q moving with velocity v in a magnetic field B experiences a force:
F = qvB sinθ
The direction is given by Fleming's left-hand rule (for conventional current) or the right-hand palm rule for positive charges. The force is always perpendicular to both v and B, so it does no work — it changes direction but not speed.
Circular motion in magnetic fields: For a charged particle moving perpendicular to a uniform B field, the magnetic force provides centripetal force: qvB = mv²/r, giving radius r = mv/(qB). The period T = 2πm/(qB) is independent of speed — this is the principle behind the cyclotron.
Force on a current-carrying conductor: F = BIL sinθ, where L is the length of conductor in the field. This is the foundation of electric motors and loudspeakers.
Electromagnetic induction (Faraday's Law): The induced emf in a circuit equals the rate of change of magnetic flux linkage:
ε = -N dΦ/dt
The negative sign represents Lenz's Law: the induced current flows in a direction that opposes the change causing it — a manifestation of energy conservation.
Magnetic flux Φ = BA cosθ and flux linkage = NΦ. Key scenarios: a conductor moving through a field (ε = BLv), a coil rotating in a field (generator — producing AC), and a changing current in one coil inducing emf in another (transformer principle).
Transformers: Vₛ/Vₚ = Nₛ/Nₚ. For an ideal transformer (100% efficient), IₚVₚ = IₛVₛ. Real transformers have losses due to eddy currents, hysteresis, and copper losses.
第六部分：磁场与电磁感应
洛伦兹力：带电粒子在磁场中受力 F = qvB sinθ，方向由左手定则判断。粒子垂直射入均匀磁场时做圆周运动，半径 r = mv/(qB)，周期 T = 2πm/(qB) 与速度无关——这是回旋加速器的原理。
通电导体受力：F = BIL sinθ，是电动机和扬声器的工作原理基础。
法拉第电磁感应定律：ε = -N dΦ/dt，负号代表楞次定律——感应电流的方向总是阻碍引起感应的变化，这是能量守恒的体现。磁通量 Φ = BA cosθ，磁链 = NΦ。关键场景：导体切割磁感线（ε = BLv）、线圈在磁场中旋转（交流发电机）、变压器原理（Vₛ/Vₚ = Nₛ/Nₚ）。
WJEC Exam Strategy and Common Pitfalls
1. Definitions are marks in the bank: Memorise precise definitions for SHM, damping types, field strength, potential, capacitance, magnetic flux, and Lenz's Law. WJEC mark schemes are strict — paraphrasing often loses marks.
2. Graph sketching: Practise sketching every graph mentioned above. Always label axes with quantities AND units. Show key features: amplitude, period, asymptotes, intercepts.
3. Derivations: Master these derivations — they appear every year: v = ±ω√(A² - x²) from energy, T² ∝ r³ for satellites, E = ½CV² for capacitor energy, r = mv/(qB) for circular motion, and the transformer equation.
4. Exponential processes: Capacitor charge/discharge, radioactive decay (from Unit 3) — recognise the common mathematical structure. Time constant, half-life, and logarithmic analysis apply to both.
5. Comparison questions: When asked to compare gravitational and electric fields, use a structured approach: list similarities, then differences, giving specific equations for each point. Use a table format in your revision notes.
6. Units and significant figures: WJEC penalises missing units and incorrect significant figures. Always give final answers to 3 significant figures unless the data suggests otherwise.
WJEC考试策略与常见失分点
1. 定义题送分：精确记忆SHM、阻尼类型、场强、电势、电容、磁通量和楞次定律的定义。WJEC评分标准严格——换说法往往丢分。
2. 图像绘制：练习上面提到的每一类图像，轴标清物理量和单位，展示振幅、周期、渐近线和截距等关键特征。
3. 推导题：每年必考——能量法推导v-x关系、卫星周期-半径关系、电容器能量公式、磁场中圆周运动半径、变压器方程。
4. 指数过程：电容器充放电与放射性衰变（单元3）共享相同的数学结构——时间常数、半衰期和对数分析法互通。
5. 对比题：比较引力场和电场时采用结构化方法——先列相同点再列不同点，每点配具体方程。
6. 单位和有效数字：WJEC对遗漏单位和错误有效数字扣分，最终答案通常保留三位有效数字。
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