欧姆定律是电学中最基本的定律之一，由德国物理学家格奥尔格·欧姆（Georg Ohm）于1826年提出。它描述了在恒温条件下，通过一段导体的电流与导体两端的电压成正比，与导体的电阻成反比。
换句话说，电压越大，电流越大；电阻越大，电流越小。这个简单而深刻的关系，是整个电路分析的基石。
Ohm's Law is one of the most fundamental laws in electricity, proposed by German physicist Georg Ohm in 1826. It states that, at a constant temperature, the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to its resistance.
In simpler terms: higher voltage drives more current; higher resistance restricts more current. This simple yet profound relationship is the cornerstone of all circuit analysis.
电压 (V) 是驱动电荷流动的"推力"，类似于水压。电池和发电机是常见的电压源。
电流 (I) 是电荷的定向移动，表示单位时间内通过导体横截面的电荷量。
电阻 (R) 是导体对电流的阻碍作用。材料、长度、横截面积和温度都会影响电阻的大小。电阻公式为 \(R = \rho\frac{L}{A}\)，其中 \(\rho\) 为电阻率，\(L\) 为长度，\(A\) 为横截面积。
Voltage (V) is the "push" that drives charges to flow, analogous to water pressure. Batteries and generators are common voltage sources.
Current (I) is the directed movement of electric charges, representing the amount of charge passing through a cross-section per unit time.
Resistance (R) is the conductor's opposition to current flow. Material, length, cross-sectional area, and temperature all affect resistance. The formula is \(R = \rho\frac{L}{A}\), where \(\rho\) is resistivity, \(L\) is length, and \(A\) is cross-sectional area.
对于线性电阻（如金属导体在恒温下），电流 I 与电压 V 的关系是一条过原点的直线。这条线被称为伏安特性曲线。
直线的斜率 = 1/R——电阻越大，斜率越小，直线越平缓。用公式表达为 \(I = \frac{1}{R}V\)，斜率为 \(\frac{1}{R}\)。
For a linear resistor (metal conductor at constant temp), current I vs. voltage V is a straight line through the origin — the V-I characteristic curve.
The slope = 1/R — larger resistance means smaller slope and flatter line. Mathematically: \(I = \frac{1}{R}V\), with slope \(\frac{1}{R}\).
欧姆定律的推导基于一个关键的实验观察：对于一段金属导体，在温度不变的情况下，导体两端的电压 V 与通过导体的电流 I 的比值是一个常数。这个常数被定义为电阻 R。
The derivation is based on a key experimental observation: for a metal conductor at constant temperature, the ratio V/I is a constant. This constant is defined as resistance R.
实验观察：搭建电路（电池、可变电阻、电流表、电压表），改变电压，记录多组 \((V, I)\) 数据。
Observation: Set up a circuit (battery, variable resistor, ammeter, voltmeter). Vary voltage, record \((V, I)\) pairs.
数据分析：以 V 为横轴、I 为纵轴绘图。数据点落在一条过原点的直线上 → \(I \propto V\)（正比），即 \(\frac{V}{I} = \text{常数}\)。
Analysis: Plot V (x-axis) vs. I (y-axis). Points form a straight line through origin → \(I \propto V\), i.e., \(\frac{V}{I} = \text{constant}\).
引入比例常数：由 \(I \propto V\) 得 \(I = k \cdot V\)，k 代表导体的导电能力。换不同导体，k 也会改变。
Proportionality constant: From \(I \propto V\), we get \(I = k \cdot V\), where k is the conductor's conductivity. Different conductors → different k.
定义电阻：定义 \(R = \frac{1}{k} = \frac{V}{I}\)。R 越大，阻碍越强。得到欧姆定律：\(I = \dfrac{V}{R}\)。
Define resistance: \(R = \frac{1}{k} = \frac{V}{I}\). Larger R = stronger opposition. Yields Ohm's Law: \(I = \dfrac{V}{R}\).
变形公式：\(V = I \times R\)（求电压）和 \(R = \dfrac{V}{I}\)（求电阻）。三者描述同一规律。
Variants: \(V = I \times R\) and \(R = \dfrac{V}{I}\). All describe the same physical law.
适用于线性电阻——电阻不随电压或电流变化。金属导体在恒温下是典型线性电阻，满足 \(\frac{V}{I} = \text{常数}\)。
不适用：半导体二极管、灯丝（温度升高电阻变大）、气体放电等——属于非线性元件。
Applies to linear resistors — resistance doesn't change with V or I. Metal conductors at constant temp satisfy \(\frac{V}{I} = \text{constant}\).
Not applicable: semiconductor diodes, lamp filaments (R increases with temp), gas discharges — non-linear elements.
拖动滑块改变电压和电阻，观察电流的变化。根据 \(I = \frac{V}{R}\)，试试看：电压加倍时，电流会发生什么变化？
Drag sliders to change V and R, observe the current. Based on \(I = \frac{V}{R}\), try: what happens to current when you double the voltage?
在欧姆定律基础上分析多个电阻组合。两种基本连接：串联（首尾相连）和并联（首首相连、尾尾相连）。
串联：电流处处相等，电压按电阻比例分配；并联：各支路电压相等，电流按电阻反比分配。
Building on Ohm's Law to analyze multiple resistors. Two basic connections: series (end-to-end) and parallel (all starts together, all ends together).
Series: same current everywhere, voltage divides by R ratio; Parallel: same voltage across branches, current divides inversely.
总电阻 Total R: \[R_{\text{总}} = R_1 + R_2 + \cdots + R_n\]
电流 Current: \[I_{\text{总}} = I_1 = I_2 = \cdots = I_n\]
电压 Voltage（分压）: \[U_{\text{总}} = U_1 + U_2 + \cdots + U_n\]
\(U_1 : U_2 = R_1 : R_2\)
总电阻 Total R: \[\frac{1}{R_{\text{总}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}\]
电流 Current（分流）: \[I_{\text{总}} = I_1 + I_2 + \cdots + I_n\]
电压 Voltage: \[U_{\text{总}} = U_1 = U_2 = \cdots = U_n\]
\(I_1 : I_2 = \dfrac{1}{R_1} : \dfrac{1}{R_2} = R_2 : R_1\)
选择连接方式并输入各电阻值（逗号分隔），计算总电阻。试试看：并联电路的总电阻总是小于最小的那个分支电阻，你能验证这个规律吗？
Select connection type and enter resistor values (comma-separated) to calculate total resistance. Try this: in a parallel circuit, the total resistance is always less than the smallest branch resistance — can you verify?
下方是一个由两个电阻串联的电路。观察：总电压等于各电阻电压之和 \(U = U_1 + U_2\)，而电流处处相同 \(I_1 = I_2 = I\)。
Below is a circuit with two resistors in series. Observe: total voltage equals the sum of individual voltages \(U = U_1 + U_2\), while current is the same everywhere \(I_1 = I_2 = I\).
下方是一个由两个电阻并联的电路。观察：各支路电压相等 \(U_1 = U_2 = U\)，总电流等于各支路电流之和 \(I = I_1 + I_2\)。
Below is a circuit with two resistors in parallel. Observe: voltage is equal across branches \(U_1 = U_2 = U\), and total current equals the sum of branch currents \(I = I_1 + I_2\).