WJEC物理 振荡 SHM 引力场 电场 电磁感应
WJEC A-Level Unit 3: Oscillations and Fields 是WJEC物理课程中概念密度最高的单元之一。从简谐运动的数学之美到电磁感应的工程应用，Unit 3将经典力学与场论无缝衔接。本期盘点梳理Unit 3十大核心知识点，覆盖SHM、阻尼共振、引力场、电场、电容、磁场和电磁感应，助你系统掌握振荡与场论。
WJEC A-Level Unit 3 is one of the most concept-dense modules in the WJEC Physics syllabus. From the mathematical elegance of simple harmonic motion to the engineering applications of electromagnetic induction, Unit 3 seamlessly bridges classical mechanics and field theory. This checklist breaks down the 10 core knowledge points covering SHM, damping and resonance, gravitational fields, electric fields, capacitors, magnetic fields, and electromagnetic induction ： helping you master oscillations and fields systematically.
1. 简谐运动定义与判据 SHM: Definition and Criteria
简谐运动(SHM)的核心定义是加速度与位移成正比且方向相反：a = -ω²x。任何满足这一关系的运动都是SHM。判别SHM的三个步骤：(1)确定平衡位置；(2)写出回复力/加速度表达式；(3)验证是否满足F ∝ -x或a ∝ -x形式。常见SHM系统包括弹簧振子(T = 2π√(m/k))和单摆(T = 2π√(l/g))。注意：单摆仅在摆角小于约10°时近似为SHM。
The defining equation of SHM is a = -ω²x: acceleration is proportional to displacement and directed toward the equilibrium position. Three steps to identify SHM: (1) locate the equilibrium position; (2) write the restoring force or acceleration expression; (3) verify it matches the form F ∝ -x or a ∝ -x. Common SHM systems include the mass-spring oscillator (T = 2π√(m/k)) and the simple pendulum (T = 2π√(l/g)). Note: the simple pendulum approximates SHM only for angles less than about 10°.
2. SHM位移、速度、加速度方程 SHM Displacement, Velocity and Acceleration
以平衡位置为原点，位移方程x = A cos(ωt)或x = A sin(ωt)，取决于计时起点。速度v = ±ω√(A² - x²)，最大速度v_max = ωA出现在平衡位置。加速度a = -ω²x，最大加速度a_max = ω²A出现在端点。三个量的时间图像：x-t是余弦/正弦曲线，v-t是正弦/余弦曲线(相位超前π/2)，a-t与x-t反相(相位差π)。WJEC考试常要求从图像中读取振幅、周期并验证a-x直线关系。
Taking the equilibrium position as origin, displacement is x = A cos(ωt) or x = A sin(ωt) depending on the timing reference. Velocity is v = ±ω√(A² - x²), with maximum v_max = ωA at equilibrium. Acceleration is a = -ω²x, with maximum a_max = ω²A at the endpoints. Time graphs: x-t is a cosine/sine curve, v-t is a sine/cosine curve (phase lead of π/2), and a-t is in antiphase with x-t (phase difference of π). WJEC exams frequently ask you to read amplitude and period from graphs and verify the a-x linear relationship.
3. SHM中的能量转换 Energy in Simple Harmonic Motion
SHM中动能与势能持续相互转化，总能量守恒：E_total = ½mω²A²。动能KE = ½mω²(A² - x²)，势能PE = ½mω²x²(以平衡位置为零势能点)。在平衡位置(x=0)：KE最大，PE=0。在端点(x=±A)：KE=0，PE最大。能量-位移图：KE是倒抛物线，PE是正抛物线，总能量为水平线。水平弹簧振子中，势能是弹性势能；竖直弹簧振子中需同时考虑弹性势能和重力势能。
In SHM, kinetic and potential energy continuously interchange while total energy is conserved: E_total = ½mω²A². Kinetic energy KE = ½mω²(A² - x²), potential energy PE = ½mω²x² (measured from equilibrium). At equilibrium (x=0): KE is maximum, PE=0. At endpoints (x=±A): KE=0, PE is maximum. Energy-displacement graphs: KE is an inverted parabola, PE is an upright parabola, total energy is a horizontal line. For a horizontal mass-spring system, PE is elastic; for a vertical oscillator, both elastic and gravitational PE must be considered.
4. 阻尼振动 Damped Oscillations
阻尼指振动系统能量逐渐耗散、振幅随时间衰减的现象。三种阻尼类型：(1)轻阻尼(underdamping)：振幅指数衰减，系统仍完成多次振动，周期略大于无阻尼周期；(2)临界阻尼(critical damping)：系统以最快速度返回平衡位置而不发生振荡：：汽车减震器、电流计设计即利用此特性；(3)过阻尼(overdamping)：系统缓慢返回平衡位置，不振荡。WJEC常考阻尼曲线识别：通过半衰期比较阻尼强度，或判断某系统属于哪种阻尼类型。
Damping refers to the gradual dissipation of energy in an oscillating system, causing amplitude to decay with time. Three damping regimes: (1) Underdamping: amplitude decays exponentially while the system completes many oscillations ： period is slightly longer than the undamped case; (2) Critical damping: the system returns to equilibrium in the shortest possible time without oscillating ： used in car shock absorbers and galvanometer design; (3) Overdamping: the system returns slowly to equilibrium without oscillation. WJEC exams often test damping curve identification: comparing damping strength via half-life or determining which damping regime a system falls into.
5. 受迫振动与共振 Forced Oscillations and Resonance
当周期性外力作用于振动系统时，系统以驱动频率振动，而非固有频率。共振发生在驱动频率等于系统固有频率f₀时：此时振幅达到最大值，因为能量转移效率最高。共振曲线的关键特征：(1)轻阻尼时共振峰尖锐、振幅极大；(2)阻尼增大时共振峰变宽变矮，峰值频率略低于f₀。现实中的共振：塔科马海峡大桥(风致共振)、微波炉(水分子介电共振)、MRI(核磁共振)。WJEC要求能从共振曲线中读取固有频率并比较不同阻尼下的振幅响应。
When a periodic external force acts on an oscillating system, the system vibrates at the driving frequency, not its natural frequency. Resonance occurs when the driving frequency equals the natural frequency f₀: amplitude reaches its maximum because energy transfer is most efficient. Key features of resonance curves: (1) Light damping produces a sharp, tall resonance peak; (2) Increased damping broadens and lowers the peak, with the peak frequency shifting slightly below f₀. Real-world resonance: Tacoma Narrows Bridge (wind-induced), microwave ovens (dielectric resonance of water molecules), MRI (nuclear magnetic resonance). WJEC expects you to read natural frequency from resonance curves and compare amplitude responses under different damping conditions.
6. 引力场 Gravitational Fields
牛顿万有引力定律：F = GMm/r²。引力场强度g = F/m = GM/r²，在均匀场(地表附近)中g ≈ 9.81 N/kg，在径向场中g ∝ 1/r²。引力势V = -GM/r(取无穷远处V=0)，引力势能U = -GMm/r。关键概念：g是矢量(指向质量中心)，V是标量。场线从质量向外辐射(或指向质量)，等势面为同心球面且垂直于场线。WJEC强调g和V的图像对比：g-r图(线性下降于地表内，1/r²衰减于地表外)，V-r图(连续曲线，梯度= -g)。
Newton's law of gravitation: F = GMm/r². Gravitational field strength g = F/m = GM/r², which is approximately 9.81 N/kg in a uniform field near Earth's surface and follows g ∝ 1/r² in a radial field. Gravitational potential V = -GM/r (taking V=0 at infinity), gravitational potential energy U = -GMm/r. Key concept: g is a vector (pointing toward the mass center), V is a scalar. Field lines radiate outward from a mass (or point inward), and equipotential surfaces are concentric spheres perpendicular to field lines. WJEC emphasizes comparing g-r and V-r graphs: g-r shows a linear drop inside the Earth and 1/r² decay outside; V-r is a continuous curve whose gradient equals -g.
7. 轨道运动与开普勒定律 Orbital Motion and Kepler's Laws
对于圆形轨道，引力提供向心力：GMm/r² = mv²/r = mrω²。由此推导：v = √(GM/r)，T² = (4π²/GM)r³(开普勒第三定律)。重要应用：(1)地球同步卫星(r ≈ 4.23×10⁷ m, T=24h)；(2)从轨道周期推算中心天体质量M = 4π²r³/GT²；(3)逃逸速度v_esc = √(2GM/R)。注意：轨道速度v_orb与逃逸速度v_esc相差√2倍。WJEC计算题常要求运用T² ∝ r³比较不同半径的轨道周期。
For circular orbits, gravitational force provides centripetal force: GMm/r² = mv²/r = mrω². This yields: v = √(GM/r) and T² = (4π²/GM)r³ (Kepler's third law). Key applications: (1) geostationary satellites (r ≈ 4.23×10⁷ m, T=24h); (2) deriving central body mass from orbital period: M = 4π²r³/GT²; (3) escape velocity v_esc = √(2GM/R). Note: orbital velocity and escape velocity differ by a factor of √2. WJEC calculation questions frequently require applying T² ∝ r³ to compare orbital periods at different radii.
8. 电场 Electric Fields
库仑定律：F = kQq/r² = Qq/(4πε₀r²)。电场强度E = F/q。点电荷的径向场：E = kQ/r²，方向由正电荷向外。均匀电场(平行板间)：E = V/d，方向由高电势指向低电势。电场线从正电荷出发、终止于负电荷。电势能变化ΔU = qΔV，电子伏特1 eV = 1.60×10⁻¹⁹ J。带电粒子在均匀电场中的运动：(1)平行于场方向：匀加速直线运动，a = qE/m；(2)垂直于场方向：类平抛运动，用于示波器和速度选择器。WJEC喜欢考电场与引力场的类比：两者都服从平方反比律，E类比g，V类比φ。
Coulomb's law: F = kQq/r² = Qq/(4πε₀r²). Electric field strength E = F/q. For a point charge (radial field): E = kQ/r², direction radially outward from a positive charge. Uniform field (between parallel plates): E = V/d, direction from high to low potential. Field lines originate on positive charges and terminate on negative charges. Potential energy change ΔU = qΔV; 1 eV = 1.60×10⁻¹⁹ J. Charged particle motion in a uniform field: (1) parallel to field: uniformly accelerated linear motion, a = qE/m; (2) perpendicular to field: parabolic trajectory, used in oscilloscopes and velocity selectors. WJEC favours gravitational-electric field analogies: both obey inverse-square laws, E corresponds to g, V corresponds to φ.
9. 电容器 Capacitors
电容定义：C = Q/V，单位法拉(F)。平行板电容：C = ε₀A/d，插入介质后C = εᵣε₀A/d。电容器储能：E = ½QV = ½CV² = ½Q²/C。RC电路的充放电：充电时Q = Q₀(1 - e^{-t/RC})，放电时Q = Q₀ e^{-t/RC}。时间常数τ = RC，在t=τ时放电至37%，充电至63%。WJEC实验常要求通过ln V-t图线性化确定时间常数(梯度 = -1/RC)。电容器的串并联：串联1/C_total = Σ1/C_i，并联C_total = ΣC_i(注意与电阻规则相反)。
Capacitance definition: C = Q/V, unit farad (F). Parallel-plate capacitor: C = ε₀A/d; with dielectric: C = εᵣε₀A/d. Energy stored: E = ½QV = ½CV² = ½Q²/C. RC circuit charging and discharging: charging Q = Q₀(1 - e^{-t/RC}), discharging Q = Q₀ e^{-t/RC}. Time constant τ = RC; at t=τ, charge drops to 37% (discharging) or rises to 63% (charging). WJEC practicals frequently require linearizing ln V-t graphs to determine time constant (gradient = -1/RC). Capacitors in series and parallel: series 1/C_total = Σ1/C_i, parallel C_total = ΣC_i (note: opposite to resistor rules).
10. 磁场与电磁感应 Magnetic Fields and Electromagnetic Induction
磁通量密度B：F = BIl sinθ(F垂直于B和I)。弗莱明左手定则确定力的方向。带电粒子在磁场中做圆周运动：半径r = mv/(Bq)，周期T = 2πm/(Bq)(与速度无关：：这是回旋加速器的原理)。电磁感应：法拉第定律ε = -N dΦ/dt，磁通量Φ = BA cosθ。楞次定律：感应电流的方向总是阻碍引起感应的磁通量变化。应用：发电机(机械能→电能)、变压器(V_s/V_p = N_s/N_p)、涡流制动。WJEC重点考查Φ-t图到ε-t图的转换(ε是Φ-t的负梯度)以及楞次定律的定性解释。
Magnetic flux density B: F = BIl sinθ (F perpendicular to both B and I). Fleming's left-hand rule determines force direction. Charged particles in a magnetic field move in circular paths: radius r = mv/(Bq), period T = 2πm/(Bq) ： independent of speed, which is the operating principle of the cyclotron. Electromagnetic induction: Faraday's law ε = -N dΦ/dt, magnetic flux Φ = BA cosθ. Lenz's law: the direction of induced current always opposes the change in magnetic flux causing it. Applications: generators (mechanical to electrical energy), transformers (V_s/V_p = N_s/N_p), eddy-current braking. WJEC focuses on converting Φ-t graphs to ε-t graphs (ε is the negative gradient of Φ-t) and qualitative explanations using Lenz's law.
Worked Example 典型例题: Capacitor Discharge
A 470 μF capacitor is charged to 6.0 V and discharged through a 10 kΩ resistor. (a) Calculate the time constant. (b) Find the p.d. across the capacitor after 9.4 s. (c) Determine the time taken for the p.d. to fall to 1.5 V.
Solution: (a) τ = RC = (10×10³)(470×10⁻⁶) = 4.7 s. (b) At t=9.4 s = 2τ: V = V₀ e^{-t/RC} = 6.0 × e^{-2} = 6.0 × 0.135 = 0.81 V. (c) 1.5 = 6.0 e^{-t/4.7}, so e^{-t/4.7} = 0.25, t/4.7 = ln 4, t = 4.7 × ln 4 = 6.5 s. The "rule of thumb": after 1τ, V drops to 37%; after 2τ to 13.5%; after 5τ to <1%.
Key Bilingual Terms 关键双语术语
Simple Harmonic Motion 简谐运动 | Angular Frequency 角频率 | Amplitude 振幅 | Restoring Force 回复力 | Natural Frequency 固有频率 | Damping 阻尼 | Critical Damping 临界阻尼 | Resonance 共振 | Driving Frequency 驱动频率 | Gravitational Field 引力场 | Gravitational Potential 引力势 | Escape Velocity 逃逸速度 | Kepler's Laws 开普勒定律 | Geostationary Orbit 地球同步轨道 | Electric Field 电场 | Coulomb's Law 库仑定律 | Permittivity 介电常数 | Equipotential Surface 等势面 | Capacitance 电容 | Time Constant 时间常数 | Dielectric 电介质 | Exponential Decay 指数衰减 | Magnetic Flux Density 磁通量密度 | Fleming's Left-Hand Rule 弗莱明左手定则 | Faraday's Law 法拉第定律 | Lenz's Law 楞次定律 | Electromagnetic Induction 电磁感应 | Eddy Current 涡流 | Transformer 变压器 | Cyclotron 回旋加速器
WJEC Unit 3 Exam Tips 考试技巧
WJEC Unit 3(振荡与场)考试时间为2小时，满分90分，含Section A(选择题和简答题)和Section B(长答题和数据分析题)。以下为高分策略：(1)SHM题目中，务必明确给出a = -ω²x并标出负号：：缺少负号直接失分；(2)共振曲线题中，注意区分自然频率和驱动频率，振幅最大处对应f = f₀而非f=0；(3)引力场与电场的平行计算：两者公式结构完全相同(G↔k, M↔Q, g↔E)，利用类比快速检查答案合理性；(4)电容器放电实验中，取自然对数将指数曲线线性化：：gradient = -1/RC，截距 = ln V₀，这是WJEC高频考点；(5)电磁感应中，ε = -N dΦ/dt的负号代表楞次定律：：解释时必须强调"opposes the change"而不仅仅是"opposes the flux"。
The WJEC Unit 3 (Oscillations and Fields) exam is 2 hours, 90 marks total, split into Section A (multiple-choice and short-answer) and Section B (long-answer and data analysis). Top-scoring strategies: (1) In SHM questions, always write a = -ω²x with the negative sign explicit ： omitting it loses the mark; (2) For resonance curve questions, distinguish between natural and driving frequencies: maximum amplitude corresponds to f = f₀, not f=0; (3) Gravitational and electric field calculations are structurally identical (G↔k, M↔Q, g↔E) ： use analogy to quickly sanity-check your answers; (4) In capacitor discharge experiments, take natural logs to linearise the exponential curve ： gradient = -1/RC, y-intercept = ln V₀, a WJEC examiner favourite; (5) In electromagnetic induction, the negative sign in ε = -N dΦ/dt encodes Lenz's law ： your explanation must say "opposes the change in flux", not just "opposes the flux".
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