📚 Exponential Change | 指数变化
Many processes in physics do not change at a constant rate. Instead, the rate of change is proportional to the current value.
物理学中的许多过程并非以恒定速率变化，而是变化率与当前值成正比。
This topic focuses on exponential growth and decay, with key examples from capacitor circuits and radioactivity in the AQA International A-level specification.
本专题聚焦指数增长与指数衰减，重点例子来自 AQA 国际 A-level 考纲中的电容器电路与放射性。
1. What is Exponential Change? | 什么是指数变化？
Exponential change occurs when a quantity changes at a rate proportional to its instantaneous value.
当一个量的变化率与其瞬时值成正比时，就发生指数变化。
If the quantity decreases, we call it exponential decay; if it increases towards a limiting value, it is often modelled as exponential growth towards an asymptote.
如果该量减少，称为指数衰减；如果它增大并趋近某个极限值，通常可建模为趋向渐近线的指数增长。
In A-level Physics, the most common examples are capacitor discharge, capacitor charging, and radioactive decay.
在 A-level 物理中，最常见的例子是电容器放电、电容器充电和放射性衰变。
2. The Mathematics of Exponential Decay | 指数衰减的数学
The general decay equation is written as X = X₀e−t/τ, where X₀ is the initial value and τ is the time constant.
一般衰减方程写为 X = X₀e−t/τ，其中 X₀ 是初始值，τ 是时间常数。
After one time constant t = τ, the value falls to X₀/e ≈ 0.37X₀, meaning about 63% of the change has occurred.
经过一个时间常数 t = τ 后，数值降至 X₀/e ≈ 0.37X₀，意味着大约 63% 的变化已经发生。
X = X₀ e−t/τ
capacitor discharge, τ = RC; in radioactive decay, τ = 1/λ, where λ is the decay constant.
在电容器放电中，τ = RC；在放射性衰变中，τ = 1/λ，其中 λ 是衰变常量。
3. Time Constant τ | 时间常数 τ
The time constant τ is the time taken for a decaying quantity to fall to 1/e of its initial value.
时间常数 τ 是衰减量降到初始值 1/e 所需的时间。
It is also the time at which the initial tangent reaches the horizontal axis on a graph of X versus t.
它也是在 X-t 图上初始切线与横轴相交所对应的时间。
τ is the second. In an RC circuit, τ = RC, so an ohm multiplied by a farad gives a second.
τ 的单位是秒。在 RC 电路中，τ = RC，因此欧姆乘以法拉得到秒。
This is useful for checking that an expression for time is dimensionally correct.
这有助于检查时间表达式在量纲上是否正确。
4. Half-Life t½ | 半衰期 t½
s the time taken for the quantity to halve. For exponential decay, t½ = ln 2 × τ ≈ 0.693τ.
半衰期是量减半所需的时间。对于指数衰减，t½ = ln 2 × τ ≈ 0.693τ。
In radioactivity, t½ = ln 2/λ; in capacitor discharge, t½ = RC ln 2.
在放射性中，t½ = ln 2/λ；在电容器放电中，t½ = RC ln 2。
The half-life is constant: whatever the starting value, the same time interval is needed for it to halve.
半衰期是恒定的：无论起始值如何，减半所需的时间间隔相同。
This property is often used to identify an exponential change from experimental data.
这一特性常用于通过实验数据识别指数变化。
5. Capacitor Discharge | 电容器放电
When a charged capacitor discharges through a fixed resistor, the charge, voltage and current all decay exponentially.
当已充电的电容器通过固定电阻放电时，电荷、电压和电流均按指数规律衰减。
Q = Q₀ e−t/RC, V = V₀ e−t/RC, I = I₀ e−t/RC
Here Q is the charge on the capacitor plates, V is the potential difference across the capacitor, and I is the discharge current.
其中 Q 是电容器极板上的电荷，V 是电容器两端的电势差，I 是放电电流。
e initial current is I₀ = V₀/R, and energy is dissipated in the resistor during discharge.
初始电流为 I₀ = V₀/R，放电过程中能量在电阻中耗散。
All three quantities decrease by the same exponential factor because they are proportional to each other.
这三个量都按相同的指数因子减小，因为它们彼此成正比。
6. Charging a Capacitor | 电容器充电
During charging, the voltage across the capacitor rises according to V = V₀(1 − e−t/RC).
充电过程中，电容器两端电压按 V = V₀(1 − e−t/RC) 上升。
Here V₀ is the final steady voltage, equal to the supply emf if the internal resistance is negligible.
这里的 V₀ 是最终稳态电压，如果内阻可忽略，则等于电源电动势。
The current starts at a maximum value and decays exponentially to zero as the capacitor approaches full charge.
电流从最大值开始，并随电容器接近充满而指数衰减至零。
The quantity V₀ − V decays exponentially with time constant RC, so the same time constant applies to both charging and discharging.
量 V₀ − V 以时间常数 RC 指数衰减，因此充电和放电具有相同的时间常数。
7. Radioactive Decay | 放射性衰变
The number of undecayed nuclei N follows N = N₀ e−λt, where λ is the decay constant in s⁻¹.
未衰变原子核数 N 遵循 N = N₀ e−λt，其中 λ 是衰变常量，单位为 s⁻¹。
Activity A is proportional to N, so A = A₀ e−λt.
活度 A 与 N 成正比，因此 A = A₀ e−λt。
The decay constant λ is the probability per unit time that a given nucleus will decay.
衰变常量 λ 是给定原子核在单位时间内发生衰变的概率。
The half-life is related to the decay constant by t½ = ln 2/λ.
半衰期与衰变常量的关系为 t½ = ln 2/λ。
Radioactive decay is random and spontaneous, but the large number of nuclei gives a predictable exponential pattern.
放射性衰变是随机且自发的，但大量原子核给出了可预测的指数规律。
8. Logarithmic Linearisation | 对数线性化
Taking natural logs of an exponential decay equation gives ln X = ln X₀ − t/τ.
对指数衰减方程取自然对数得到 ln X = ln X₀ − t/τ。
A graph of ln X against t is a straight line with gradient −1/τ and intercept ln X₀.
ln X 对 t 作图是一条直线，斜率为 −1/τ，截距为 ln X₀。
This technique allows the time constant and the initial value to be determined from experimental data.
这一技术使得可以从实验数据确定时间常数和初始值。
For radioactive decay, plotting ln A against t gives gradient −λ and intercept ln A₀.
对于放射性衰变，绘制 ln A 对 t 的图，斜率为 −λ，截距为 ln A₀。
9. Graphical Analysis | 图像分析
On an X-t graph, exponential decay is a curve that never reaches zero but approaches the time axis asymptotically.
在 X-t 图上，指数衰减是一条永不为零但渐近地趋近时间轴的曲线。
Successive intervals of one half-life reduce the quantity by half, giving a characteristic stepped pattern if marked on the graph.
每经过一个半衰期，量减少一半，如果在图上标出，会形成特征性的阶梯模式。
For growth towards a limit, a graph of the difference (X₀ − X) decays exponentially to zero.
对于趋近极限的增长，差值 (X₀ − X) 按指数衰减至零。
A curved graph alone is not sufficient evidence of exponential change; a straight-line log graph provides much stronger evidence.
单凭曲线图不足以证明指数变化；对数图的直线提供了更强有力的证据。
10. Experimental Determination of RC | RC 的实验测定
You can measure capacitor discharge by charging a capacitor, then recording the voltage at regular time intervals during discharge.
可通过给电容器充电，然后在放电过程中每隔一定时间记录电压来测量电容放电。
Plotting ln V against t gives a gradient equal to −1/RC, from which RC can be calculated.
绘制 ln V 对 t 的图，斜率等于 −1/RC，由此可计算出 RC。
Alternatively, find the time for V to halve and use t½ = RC ln 2.
或者，找出电压减半的时间并使用 t½ = RC ln 2。
In an experiment, a high-impedance voltmeter or data logger should be used to minimise additional discharge paths.
实验中应使用高阻抗电压表或数据记录仪，以尽量减少额外的放电回路。
11. Worked Example | 例题解析
A 100 μF capacitor is charged to 12 V and discharged through a 50 kΩ resistor. Find the time constant and the voltage after 5 s.
一个 100 μF 电容器充电至 12 V，通过 50 kΩ 电阻放电。求时间常数和 5 s 后的电压。
The time constant is τ = RC = 50 × 10³ Ω × 100 × 10⁻⁶ F = 5.0 s.
时间常数为 τ = RC = 50 × 10³ Ω × 100 × 10⁻⁶ F = 5.0 s。
Using V = V₀ e−t/τ, we get V = 12 e−5/5 = 12 e⁻¹ = 4.4 V.
使用 V = V₀ e−t/τ，得到 V = 12 e−5/5 = 12 e⁻¹ = 4.4 V。
So the voltage after 5 seconds is approximately 4.4 V.
因此 5 秒后的电压约为 4.4 V。
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
Always check that RC has units of seconds: Ω × F = s.
务必检查 RC 的单位为秒：Ω × F = s。
Do not confuse time constant τ with half-life t½: t½ = τ ln 2.
不要混淆时间常数 τ 与半衰期 t½：t½ = τ ln 2。
When using logs, plot ln X on the y-axis, not log₁₀ X unless a different base is specified in the question.
使用对数时，y 轴应绘制 ln X，除非题目指定使用 log₁₀ X。
For charging curves, remember that the exponential part describes the difference between the final value and the current value.
对于充电曲线，记住指数部分描述的是最终值与当前值之间的差值。
Always quote the correct unit for decay constant λ as s⁻¹, and for RC as s.
务必正确写出衰变常量 λ 的单位为 s⁻¹，RC 的单位为 s。
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