📚 3 Physical Models of a Differential Equation | 微分方程的三个物理模型
In physics, many natural processes exhibit a behaviour where a quantity decreases at a rate proportional to its current value. This simple rule translates into the first-order differential equation dy/dt = -k y, where y represents the quantity of interest and k is a positive constant. While the mathematical structure remains the same, the physical interpretation of y and k varies across different phenomena, giving us multiple models rooted in one differential equation.
在物理学中，许多自然过程表现为某个量以其当前值的恒定比例减少。这一简单规则转化为一阶微分方程 dy/dt = -k y，其中 y 代表所关注的物理量，k 是一个正常数。尽管数学结构保持不变，但 y 和 k 的物理解释在不同的现象中各不相同，从而为我们提供了基于同一个微分方程的多重模型。
This article explores three classic physical models described by this exponential decay equation: radioactive decay, the discharge of a capacitor through a resistor, and Newton’s law of cooling. Each model illustrates how identifying the underlying differential equation unlocks powerful predictive tools, a key skill in IB Physics.
本文探讨由这一指数衰减方程描述的三种经典物理模型：放射性衰变、电容器通过电阻放电以及牛顿冷却定律。每个模型都展示了识别底层微分方程如何开启强大的预测工具，这是 IB 物理中的一项关键技能。
1. Introduction: The Common Differential Equation | 引言：共同的微分方程
The equation dy/dt = -k y states that the rate of change of y is negative and directly proportional to y itself. The larger the value of y, the more rapidly it decreases. Such behaviour is known as exponential decay, and its solution is y(t) = y₀ e^(-kt), where y₀ is the initial value at time t = 0.
方程 dy/dt = -k y 表明 y 的变化率为负，且与 y 本身成正比。y 的值越大，减少得越快。这种行为被称为指数衰减，其解为 y(t) = y₀ e^(-kt)，其中 y₀ 是 t = 0 时的初始值。
In every case, the constant k carries units of inverse time and determines the speed of the decay. A large k means a rapid drop, while a small k corresponds to a slow process. The exponential function e^(-kt) ensures that the quantity never reaches zero in finite time, but approaches it asymptotically.
在每种情况下，常数 k 的单位为时间的倒数，决定了衰减的快慢。大的 k 意味着快速下降，而小的 k 则对应缓慢的过程。指数函数 e^(-kt) 确保了该量在有限时间内绝不会达到零，而是渐近地趋近于零。
2. Radioactive Decay: The Decay Law | 放射性衰变：衰变定律
Radioactive nuclei transform into other nuclei spontaneously. For a given isotope, the probability that a nucleus decays in a short time interval Δt is λ Δt, where λ is the decay constant. If N is the number of undecayed nuclei present, the expected change dN in an infinitesimal time dt is dN = -λ N dt, giving dN/dt = -λ N.
放射性原子核会自发地转变为其他原子核。对于给定的同位素，原子核在很短的时间间隔 Δt 内衰变的概率为 λ Δt，其中 λ 为衰变常数。如果 N 是现存未衰变原子核的数量，在无限小时间 dt 内的预期变化为 dN = -λ N dt，从而得到 dN/dt = -λ N。
This is our prototype differential equation with y ≡ N and k ≡ λ. The activity A, defined as the number of decays per unit time, is A = -dN/dt = λ N, which itself decays exponentially. The SI unit of activity is the becquerel (Bq), equal to one decay per second.
这正是我们的原型微分方程，其中 y ≡ N，k ≡ λ。活度 A 定义为单位时间内的衰变次数，A = -dN/dt = λ N，本身也呈指数衰减。活度的 SI 单位是贝克勒尔 (Bq)，等于每秒一次衰变。
3. Half-life and Activity | 半衰期与活度
Integrating the decay equation yields the familiar exponential law N(t) = N₀ e^(-λt). The half-life T₁/₂ is defined as the time after which half of the original nuclei have decayed. Setting N(T₁/₂) = N₀/2 gives e^(-λ T₁/₂) = 1/2, so T₁/₂ = ln2 / λ ≈ 0.693/λ.
对衰变方程积分得到我们熟悉的指数规律 N(t) = N₀ e^(-λt)。半衰期 T₁/₂ 定义为初始原子核衰变一半所需的时间。令 N(T₁/₂) = N₀/2，得到 e^(-λ T₁/₂) = 1/2，因此 T₁/₂ = ln2 / λ ≈ 0.693/λ。
The constant half-life is a hallmark of exponential decay: no matter how many nuclei are present initially, the time for the population to halve remains fixed. This property is exploited in radiocarbon dating, where the decay of carbon-14 (half-life 5730 years) reveals the age of archaeological samples.
恒定的半衰期是指数衰减的标志：无论初始有多少原子核，其数量减半所需的时间始终不变。这一性质被应用于放射性碳定年法，其中碳-14（半衰期 5730 年）的衰变揭示了考古样品的年龄。
4. RC Circuit Discharge | RC 电路放电
A capacitor of capacitance C charged to an initial potential difference V₀ stores charge Q₀ = C V₀. When the capacitor discharges through a resistor R, the current I in the circuit is related to the rate of charge loss by I = -dQ/dt. Kirchhoff’s voltage law requires that the potential difference across the capacitor, V = Q/C, equals the voltage drop across the resistor, IR.
一个电容为 C 的电容器被充电至初始电势差 V₀，存储的电荷为 Q₀ = C V₀。当电容器通过电阻 R 放电时，电路中的电流 I 与电荷损失率的关系为 I = -dQ/dt。基尔霍夫电压定律要求电容器两端的电势差 V = Q/C 等于电阻两端的电压降 IR。
Substituting gives Q/C = -R (dQ/dt), which rearranges to dQ/dt = -Q/(RC). Comparing with dy/dt = -k y shows that the charge Q plays the role of y, and the constant k = 1/(RC). The product RC has dimensions of time and is called the time constant τ.
代入后得到 Q/C = -R (dQ/dt)，整理得 dQ/dt = -Q/(RC)。与 dy/dt = -k y 比较可知，电荷 Q 扮演了 y 的角色，常数 k = 1/(RC)。乘积 RC 具有时间的量纲，称为时间常数 τ。
5. Time Constant and Exponential Decay | 时间常数与指数衰减
Solving the differential equation gives Q(t) = Q₀ e^(-t/RC). After one time constant τ = RC, the charge falls to Q₀ e⁻¹ ≈ 0.37 Q₀, or 37% of its initial value. After 5τ, the charge drops to less than 1% of Q₀, and for most practical purposes the capacitor can be considered discharged.
求解微分方程得到 Q(t) = Q₀ e^(-t/RC)。经过一个时间常数 τ = RC 后，电荷降至 Q₀ e⁻¹ ≈ 0.37 Q₀，即初始值的 37%。经过 5τ 后，电荷降至 Q₀ 的 1% 以下，对大多数实际应用而言，电容可视为已放电。
Since the potential difference V = Q/C and the current I = V/R, both also decay exponentially: V(t) = V₀ e^(-t/RC) and I(t) = (V₀/R) e^(-t/RC). The exponential behaviour is identical to radioactive decay, with the half-life given by T₁/₂ = RC ln2.
由于电势差 V = Q/C 且电流 I = V/R，因此两者也呈指数衰减：V(t) = V₀ e^(-t/RC)，I(t) = (V₀/R) e^(-t/RC)。这一指数行为与放射性衰变完全相同，半衰期为 T₁/₂ = RC ln2。
6. Newton’s Law of Cooling | 牛顿冷却定律
Isaac Newton observed that the rate of heat loss from a body is proportional to the difference between its own temperature and that of its surroundings. Mathematically, dT/dt = -k (T – T_s), where T is the object’s temperature, T_s is the constant ambient temperature, and k is a positive cooling constant depending on the object’s properties.
艾萨克·牛顿观察到，物体散失热量的速率与其自身温度和环境温度之差成正比。数学表达为 dT/dt = -k (T – T_s)，其中 T 是物体的温度，T_s 是恒定的环境温度，k 是一个正的冷却常数，取决于物体的性质。
This equation is not immediately in the form dy/dt = -k y. However, by defining the temperature excess θ = T – T_s, we obtain dθ/dt = -k θ, which is exactly the same differential equation as before. The physical variable θ decays exponentially towards zero, meaning T approaches T_s.
该方程并非直接是 dy/dt = -k y 的形式。然而，通过定义温度过剩量 θ = T – T_s，我们得到 dθ/dt = -k θ，这与之前的微分方程完全相同。物理量 θ 指数衰减至零，意味着 T 趋近于 T_s。
7. Cooling Curve and Applications | 冷却曲线与应用
The solution for the temperature excess is θ(t) = θ₀ e^(-kt), where θ₀ = T₀ – T_s. Consequently, the temperature at any time is T(t) = T_s + (T₀ – T_s) e^(-kt). A graph of temperature against time shows a smooth exponential approach to the ambient temperature.
温度过剩量的解为 θ(t) = θ₀ e^(-kt)，其中 θ₀ = T₀ – T_s。因此，任意时刻的温度为 T(t) = T_s + (T₀ – T_s) e^(-kt)。温度对时间的图像显示出平滑指数趋近于环境温度。
This model is used in forensic science to estimate the time of death by measuring the core body temperature of a deceased person. It also appears in engineering contexts such as predicting the cool-down of electronic components and in meteorology for understanding temperature changes of land and sea.
该模型在法医学中被用来通过测量死者的核心体温来估算死亡时间。它也出现在工程领域中，例如预测电子元器件的冷却过程，以及在气象学中用于理解陆地和海洋的温度变化。
8. Unified Mathematics: The Exponential Function | 统一数学：指数函数
All three models reduce to the same fundamental equation dy/dt = -k y. The solution y = y₀ e^(-kt) implies that in any fixed time interval Δt, the quantity is multiplied by the same factor e^(-k Δt). This is the source of the term ‘exponential decay’: equal time steps produce equal fractional reductions.
这三个模型都归结为同一个基本方程 dy/dt = -k y。其解 y = y₀ e^(-kt) 意味着在任何固定的时间间隔 Δt 内，该量都会乘以相同的因子 e^(-k Δt)。这正是“指数衰减”一词的由来：相等的时间步长产生相等的分数减少。
Taking natural logarithms linearises the model: ln y = ln y₀ – k t. A plot of ln y versus t yields a straight line with gradient -k and intercept ln y₀. This linear relationship is an essential tool for verifying exponential behaviour in experimental data.
取自然对数可将模型线性化：ln y = ln y₀ – k t。绘制 ln y 对 t 的曲线可得到一条斜率为 -k、截距为 ln y₀ 的直线。这一线性关系是验证实验数据中指数行为的基本工具。
9. Comparison Table of the Three Models | 三个模型的比较表
The following table summarises the key quantities, constants, and characteristic times for each physical model governed by dy/dt = -k y.
下表总结了由 dy/dt = -k y 支配的每个物理模型的关键量、常数和特征时间。
| Physical model | Variable y |
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