📚 Exponential Change Formula Derivation | 指数变化公式推导
In A-Level Physics, many processes follow an exponential law where the rate of change of a quantity is proportional to the quantity itself. Capacitor charge and discharge, radioactive decay, and Newton's law of cooling all exhibit this behaviour. A solid grasp of the underlying differential equation and its solution is essential for tackling related problems and understanding the unifying nature of exponential change across different topics.
在 A-Level 物理中，许多过程遵循指数规律：量的变化率与量本身成正比。电容的充放电、放射性衰变以及牛顿冷却定律都表现出这种行为。扎实掌握其背后的微分方程及其解法，对于解决相关问题和理解指数变化在不同专题中的统一本质至关重要。
1. What is Exponential Change? | 什么是指数变化？
Exponential change occurs when a variable y changes at a rate directly proportional to its current value. If the proportionality constant is negative, y decreases exponentially; if positive, y increases exponentially. The rate is often written as dy/dt ∝ y.
当一个变量 y 的变化率与其当前值成正比时，就发生指数变化。如果比例常数为负，y 指数衰减；如果为正，y 指数增长。变化率常写作 dy/dt ∝ y。
2. The General Differential Equation | 一般微分方程
The fundamental differential equation for exponential change is:
指数变化的基本微分方程为：
dy/dt = ± k y
where k is a positive constant. Choosing the minus sign gives exponential decay; the plus sign gives exponential growth. This simple first-order equation is the starting point for all derivations in this topic.
其中 k 为正的常数。选择负号得出指数衰减；选择正号得出指数增长。这个简单的一阶方程是本主题所有推导的起点。
3. Solving the Differential Equation for Decay | 求解衰减的微分方程
he decay equation dy/dt = −k y. The solution is obtained by separation of variables.
我们从衰减方程 dy/dt = −k y 开始。通过分离变量法求解。
Step 1: Separate the variables so that all y terms are on one side and all t terms on the other: (1/y) dy = −k dt.
步骤一：分离变量，使所有含 y 的项在一边，含 t 的项在另一边：(1/y) dy = −k dt。
n integration constant C: ∫ (1/y) dy = ∫ −k dt → ln y = −k t + C.
步骤二：两边积分。左边积分得 ln y，右边得 −k t 加上积分常数 C：∫ (1/y) dy = ∫ −k dt → ln y = −k t + C。
Step 3: Apply the initial condition. At t = 0, let y = y0. Then ln y0 = C, giving ln(y/y0) = −k t. Exponentiating yields the well-known exponential decay law:
步骤三：应用初始条件。设 t = 0 时 y = y0，则 ln y0 = C，得到 ln(y/y0) = −k t。取指数后得到著名的指数衰减律：
y = y0 e−k t
4. Exponential Growth Derivation | 指数增长推导
and integration steps, but starting with a positive constant, leads to the solution y = y0 ek t.
增长方程为 dy/dt = +k y。遵循完全相同的分离和积分步骤，但使用正常数，得到解 y = y0 ek t。
capacitor initially grows exponentially but approaches a maximum value, described by V = V0(1 − e−t/RC), which is a 'growth towards a limit' derived from a related decay in the difference.
在许多物理情景中，纯指数增长最终会受外部因素限制。例如，充电电容两端的电压初期指数增长，但趋近一个最大值，可描述为 V = V0(1 − e−t/RC)，这是一种由差值衰减导出的“趋向极限”的增长。
5. Capacitor Discharge Equation | 电容放电公式
Consider a simple RC circuit with a charged capacitor, resistor R and capacitor C in a closed loop. Kirchhoff's voltage law states that the sum of potential differences around the loop is zero: VR + VC = 0.
考虑一个简单的 RC 回路，包含已充电的电容 C 和电阻 R。基尔霍夫电压定律指出，回路中电位差之和为零：VR + VC = 0。
Using VR = I R, VC acitor), we substitute to get R dQ/dt + Q/C = 0. Rearranging gives dQ/dt = − Q/(RC).
代入 VR = I R，VC = Q/C 以及 I = dQ/dt（电荷离开电容的速率），得到 R dQ/dt + Q/C = 0。整理得 dQ/dt = − Q/(RC)。
dy/dt = −k y, we identify the decay constant k = 1/(RC). The solution is therefore:
与标准衰减方程 dy/dt = −k y 比较，可得衰减常数 k = 1/(RC)。因此解为：
Q = Q0 e−t / RC
Since V = Q/C, the voltage across the capacitor also decays exponentially: V = V0 e−t/RC, and the current I = dQ/dt = −(Q0/RC) e−t/RC, with magnitude I = I0 e−t/RC.
因为 V = Q/C，电容两端电压也呈指数衰减：V = V0 e−t/RC，而电流 I = dQ/dt = −(Q0/RC) e−t/RC，其大小为 I = I0 e−t/RC。
6. Time Constant and Half-Life | 时间常数与半衰期
; = 1/k. For an RC circuit, τ = RC. After one time constant, the charge falls to Q = Q0 e−1 ≈ 0.37 Q0. The time constant represents the time for the quantity to reduce to 1/e of its initial value.
时间常数 τ 定义为 τ = 1/k。对于 RC 电路
Published by TutorHao | A-Level Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图