题型概述：热学与近代物理长题Question types: thermal and modern physics

本组精选 Section 2 三道各 25 分的长题。题目常从熟悉的能量或动量守恒起步，再要求建立微分方程、做几何投影或积分，最后联系真实实验。失分通常来自投影次数不足、温差与温度混淆、概率密度误用，以及在符号推导中漏写假设。先完成能直接得分的估算，再把模型连接起来。This set contains three Section 2 long questions worth 25 marks each. They start with familiar energy or momentum conservation, develop differential equations, projections or integrals, and connect the model to experiments. Common losses come from missing projections, confusing temperature with excess temperature, misusing probability densities and omitting assumptions. Secure the direct estimates first, then connect the models.

1储存量与流率Stored quantities and flow rates

水的升温和电容充电都可写成“输入−漏出”。恒定输入与正比于储存量的损失竞争，会产生指数趋近稳态的曲线。Heating water and charging a capacitor both obey input minus leakage. Constant input competing with loss proportional to the stored quantity produces exponential approach to equilibrium.

mc dT/dt = P−kT; T(t) = (P/k)(1−e^−t/τ), τ=mc/k

T 是相对环境的温差；时间常数后走完稳态变化量的 63.2%。T is excess above ambient; after one time constant, 63.2% of the equilibrium change is complete.

2动量传递与几何投影Momentum transfer and projection

光子带有动量。镜面倾斜既减少截获光子数，也减少每个光子传递的法向动量；曲面合力还要沿指定方向再投影。Photons carry momentum. Tilting a mirror reduces both the number intercepted and each photon’s normal momentum transfer; a curved-surface resultant needs a further directional projection.

p_γ=E/c; P_mirror=2I cos²α/c; dF_axis=P_mirror cosα dA

每一个余弦因子都要能说清对应哪一次投影。Be able to explain the physical origin of every cosine factor.

3指数规律与首次碰撞Exponential laws and first collisions

每一小段路径有相同的单位长度碰撞概率时，未碰撞比例指数衰减。它与热学中的偏离稳态量服从相同的数学形式，但物理量和单位不同。When each small path segment has the same collision probability per unit length, the uncollided fraction decays exponentially. This shares the mathematical form of a thermal departure from equilibrium, but the quantities and units differ.

S(x)=e^−x/μ; P(x)=1−S(x); f(x)=S(x)/μ; μ=1/(nσ)

拖动同一个无量纲变量，看“剩余量”和“累计变化量”如何互补。Move the dimensionless variable to compare the remaining and accumulated fractions.

无量纲时间或距离 uDimensionless time or distance u

子题型Subtype	识别特征Recognise	核心步骤Key steps	真题Question
热损失与类比Heat loss and analogy	升温、冷却、漏电容Heating, cooling, leaky capacitor	投影→守恒→稳态Projection → balance → equilibrium	2024 Q2
光压与曲面Radiation on curves	反射、倾斜、半球Reflection, tilt, hemisphere	动量→分量→环积分Momentum → components → rings	2025 Q4
微观碰撞与电离Collisions and ionisation	密度、自由程、变电场Density, mean free path, varying field	分子数→概率→能量Molecule count → probability → energy	2025 Q5