docs(notes): add 2025-05-25-exponential-en.md

Author: lzwjava

notes/2025-05-25-exponential-en.md

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+---
+title: Exponential Formulas
+lang: en
+layout: post
+audio: false
+translated: false
+generated: true
+---
+
+In science, several formulas describe transient or exponential growth/decay processes similar to the RL circuit current equation, where the behavior starts at a specific value (often zero) at \\( t = 0 \\) and approaches a steady-state value (like \\( V_0 \\) or \\( I_0 \\)) as time progresses, often involving the exponential term \\( e \\). Below is a list of such formulas from various domains of physics and engineering, each exhibiting similar exponential behavior.
+
+### 1. **RC Circuit Charging (Capacitor Voltage)**
+   - **Context**: In an RC circuit (resistor and capacitor in series), when a voltage is applied, the capacitor charges over time.
+   - **Formula**:
+     \\[
+     V_C(t) = V_0 \left( 1 - e^{-\frac{t}{RC}} \right)
+     \\]
+   - **Variables**:
+     - \\( V_C(t) \\): Voltage across the capacitor at time \\( t \\).
+     - \\( V_0 \\): Maximum voltage (source voltage).
+     - \\( R \\): Resistance (ohms).
+     - \\( C \\): Capacitance (farads).
+     - \\( RC \\): Time constant (\\( \tau \\)).
+   - **Behavior**: At \\( t = 0 \\), \\( V_C = 0 \\). As \\( t \to \infty \\), \\( V_C \to V_0 \\).
+   - **Similarity**: Like the RL circuit, it starts at 0 and approaches a maximum value exponentially.
+
+### 2. **RC Circuit Discharging (Capacitor Voltage)**
+   - **Context**: When a charged capacitor in an RC circuit is allowed to discharge through a resistor.
+   - **Formula**:
+     \\[
+     V_C(t) = V_0 e^{-\frac{t}{RC}}
+     \\]
+   - **Variables**:
+     - \\( V_0 \\): Initial voltage across the capacitor.
+     - Others same as above.
+   - **Behavior**: At \\( t = 0 \\), \\( V_C = V_0 \\). As \\( t \to \infty \\), \\( V_C \to 0 \\).
+   - **Similarity**: Involves \\( e \\), but decays from a maximum to zero, complementary to the RL charging case.
+
+### 3. **Radioactive Decay**
+   - **Context**: In nuclear physics, the number of radioactive atoms decreases over time.
+   - **Formula**:
+     \\[
+     N(t) = N_0 e^{-\lambda t}
+     \\]
+   - **Variables**:
+     - \\( N(t) \\): Number of radioactive atoms at time \\( t \\).
+     - \\( N_0 \\): Initial number of atoms.
+     - \\( \lambda \\): Decay constant (s⁻¹).
+     - \\( \tau = \frac{1}{\lambda} \\): Mean lifetime.
+   - **Behavior**: At \\( t = 0 \\), \\( N = N_0 \\). As \\( t \to \infty \\), \\( N \to 0 \\).
+   - **Similarity**: Uses \\( e \\) for exponential decay, analogous to RC discharging or RL circuit current decay when the voltage is removed.
+
+### 4. **Newton’s Law of Cooling**
+   - **Context**: Describes the cooling of an object in a cooler environment.
+   - **Formula**:
+     \\[
+     T(t) = T_{\text{env}} + (T_0 - T_{\text{env}}) e^{-kt}
+     \\]
+   - **Variables**:
+     - \\( T(t) \\): Temperature of the object at time \\( t \\).
+     - \\( T_0 \\): Initial temperature of the object.
+     - \\( T_{\text{env}} \\): Ambient temperature.
+     - \\( k \\): Cooling constant (s⁻¹).
+   - **Behavior**: At \\( t = 0 \\), \\( T = T_0 \\). As \\( t \to \infty \\), \\( T \to T_{\text{env}} \\).
+   - **Similarity**: Exponential approach from an initial value to a steady-state value, using \\( e \\).
+
+### 5. **Population Growth (Exponential Model)**
+   - **Context**: In biology, models unrestricted population growth.
+   - **Formula**:
+     \\[
+     P(t) = P_0 e^{rt}
+     \\]
+   - **Variables**:
+     - \\( P(t) \\): Population at time \\( t \\).
+     - \\( P_0 \\): Initial population.
+     - \\( r \\): Growth rate (s⁻¹ or other time units).
+   - **Behavior**: At \\( t = 0 \\), \\( P = P_0 \\). As \\( t \to \infty \\), \\( P \to \infty \\) (unbounded growth).
+   - **Similarity**: Uses \\( e \\), but grows exponentially rather than approaching a finite limit (unlike RL/RC circuits).
+
+### 6. **RL Circuit Current Decay (After Voltage Removal)**
+   - **Context**: When the voltage source is removed from an RL circuit, the current decays.
+   - **Formula**:
+     \\[
+     I(t) = I_0 e^{-\frac{R}{L}t}
+     \\]
+   - **Variables**:
+     - Same as in the RL circuit charging formula.
+   - **Behavior**: At \\( t = 0 \\), \\( I = I_0 \\). As \\( t \to \infty \\), \\( I \to 0 \\).
+   - **Similarity**: Complementary to the RL charging case, showing exponential decay with \\( e \\).
+
+### 7. **Damped Harmonic Oscillator (Under-Damped)**
+   - **Context**: In mechanics, describes a system (e.g., spring-mass with friction) with damping.
+   - **Formula**:
+     \\[
+     x(t) = A e^{-\gamma t} \cos(\omega t + \phi)
+     \\]
+   - **Variables**:
+     - \\( x(t) \\): Displacement at time \\( t \\).
+     - \\( A \\): Initial amplitude.
+     - \\( \gamma \\): Damping constant.
+     - \\( \omega \\): Angular frequency of oscillation.
+     - \\( \phi \\): Phase angle.
+   - **Behavior**: At \\( t = 0 \\), oscillation starts with amplitude \\( A \\). As \\( t \to \infty \\), \\( x \to 0 \\).
+   - **Similarity**: The amplitude decays exponentially with \\( e^{-\gamma t} \\), similar to RL/RC decay.
+
+### Common Characteristics
+- **Exponential Term**: All formulas use \\( e \\) to describe growth or decay.
+- **Initial and Final States**: They start at an initial value (often 0 or a maximum) and approach a steady-state value (0, \\( V_0 \\), or another constant).
+- **Time Constant**: Most involve a time constant (e.g., \\( L/R \\), \\( RC \\), \\( 1/\lambda \\)) that governs the rate of change.
+- **Applications**: These appear in electrical circuits, thermodynamics, nuclear physics, biology, and mechanics.
+
+If you’d like, I can provide a chart comparing the behavior of a few of these (e.g., RL charging, RC charging, and Newton’s cooling) over time to visualize their exponential nature. Would you like me to generate such a chart? Alternatively, I can dive deeper into any of these formulas or explore others in specific fields!