欧姆定律 · Ohm's Law — 互动学习平台

什么是欧姆定律？ / What is Ohm's Law?

🇨🇳 中文

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。

🇬🇧 English

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.

I = \frac{V}{R}

V = I \times R

R = \frac{V}{I}

I

— 电流 Current（A） |

V

— 电压 Voltage（V） |

R

— 电阻 Resistance（Ω）

三个物理量 / The Three Quantities

V

电压 / Voltage

单位：伏特 volt (V)

I

电流 / Current

单位：安培 ampere (A)

R

电阻 / Resistance

单位：欧姆 ohm (Ω)

🇨🇳 中文

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。

这三个量的关系可以用一个简单的记忆三角形来表示：V 在顶部，I 和 R 在底部——遮住你想求的量，剩下的就是计算公式。

🇬🇧 English

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 resistance formula is $R = \rho\frac{L}{A}$ , where $\rho$ is resistivity, $L$ is length, and $A$ is cross-sectional area.

The relationship can be remembered using a memory triangle: V on top, I and R on the bottom — cover the quantity you want to find, and the remaining expression is your formula.

伏安特性曲线 / V-I Characteristic Curve

🇨🇳 中文

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。用数学公式表达即为 $I = \frac{1}{R}V$ ，在 V-I 坐标系中斜率为 $\frac{1}{R}$ 。

点击下方按钮切换不同电阻值，观察曲线斜率如何变化。

🇬🇧 English

For a linear resistor (such as a metal conductor at constant temperature), current I versus voltage V is a straight line through the origin. This line is called the V-I characteristic curve.

The slope = 1/R — the larger the resistance, the smaller the slope, and the flatter the line. Mathematically: $I = \frac{1}{R}V$ , where the slope in V-I coordinates is $\frac{1}{R}$ .

Click the buttons below to switch between different resistance values and observe how the slope changes.

公式推导 / Formula Derivation

🇨🇳 中文

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。

下面是详细的推导步骤。

🇬🇧 English

The derivation of Ohm's Law is based on a key experimental observation: for a metal conductor at constant temperature, the ratio of voltage V across the conductor to the current I through it is a constant. This constant is defined as resistance R.

Below are the detailed derivation steps.

1

实验观察：搭建一个简单电路，包含电池、可变电阻、电流表和电压表。改变电池电压，记录多组 $(V, I)$ 数据。

Experimental observation: Set up a simple circuit with a battery, variable resistor, ammeter, and voltmeter. Vary the battery voltage and record multiple $(V, I)$ data pairs.

2

数据分析：以 V 为横轴，I 为纵轴绘制数据点。发现所有数据点都落在一条过原点的直线上，说明 $I \propto V$ （正比关系），即 $\frac{V}{I} = \text{常数}$ 。

Data analysis: Plot the data with V as the x-axis and I as the y-axis. All data points fall on a straight line through the origin, indicating $I \propto V$ (direct proportionality), i.e. $\frac{V}{I} = \text{constant}$ .

3

引入比例常数：因为 $I \propto V$ ，所以 $I = k \cdot V$ ，其中 k 是比例常数，代表导体的导电能力。实验发现，换个电阻不同的导体，k 的值也会改变。

Introduce proportionality constant: Since $I \propto V$ , we have $I = k \cdot V$ , where k is a proportionality constant representing the conductor's ability to conduct. Experiments show that changing the conductor changes k.

4

定义电阻：定义电阻 $R = \frac{1}{k} = \frac{V}{I}$ 。R 越大，导体对电流的阻碍越强。于是得到欧姆定律的标准形式： $I = \dfrac{V}{R}$ 。

Define resistance: Define resistance $R = \frac{1}{k} = \frac{V}{I}$ . The larger R is, the more the conductor opposes current flow. This yields the standard form of Ohm's Law: $I = \dfrac{V}{R}$ .

5

变形公式：通过代数变换，得到两种等价形式： $V = I \times R$ （求电压）和 $R = \dfrac{V}{I}$ （求电阻）。这三个公式描述的是同一个物理规律。

Variant formulas: Through algebraic manipulation: $V = I \times R$ (solve for voltage) and $R = \dfrac{V}{I}$ (solve for resistance). All three formulas describe the same physical law.

适用范围与条件 / Scope and Conditions

🇨🇳 中文

欧姆定律适用于线性电阻——即电阻值不随电压或电流变化的导体。金属导体在温度恒定的条件下是典型的线性电阻，满足 $\frac{V}{I} = \text{常数}$ 。

不适用的情况包括：半导体二极管（非对称导电）、灯丝（温度升高导致电阻增大）、气体放电等。这些元件的 V-I 曲线不是直线，属于非线性元件。

🇬🇧 English

Ohm's Law applies to linear resistors — conductors whose resistance does not change with voltage or current. Metal conductors at constant temperature are typical linear resistors, satisfying $\frac{V}{I} = \text{constant}$ .

Non-applicable cases include: semiconductor diodes (asymmetric conduction), lamp filaments (resistance increases with temperature), gas discharges, etc. These are non-linear elements.

互动电路模拟 / Interactive Circuit Simulation

🇨🇳 中文

拖动下方滑块来改变电压和电阻，观察电路中电流的变化。右侧的电压表和电流表会实时显示读数。根据 $I = \frac{V}{R}$ ，试试看：当你把电压加倍时，电流会发生什么变化？

🇬🇧 English

Drag the sliders below to change the voltage and resistance, and observe how the current changes. The voltmeter and ammeter display real-time readings. Based on $I = \frac{V}{R}$ , try this: what happens to the current when you double the voltage?

🔋 电压 Voltage (V) 6.0 V

⚡ 电阻 Resistance (R) 3.0 Ω

📐 电流 Current

2.00 A

I = \frac{V}{R}

串联与并联电路 / Series and Parallel Circuits

🇨🇳 中文

在欧姆定律的基础上，我们可以进一步分析多个电阻组合的电路。电阻有两种基本连接方式：串联（首尾相连）和并联（首首相连、尾尾相连）。

串联电路的电流处处相等，电压按电阻比例分配；并联电路的各支路电压相等，电流按电阻反比分配。

🇬🇧 English

Building on Ohm's Law, we can analyze circuits with multiple resistors. Resistors can be connected in two basic ways: series (end-to-end) and parallel (all starts together, all ends together).

In series circuits, current is the same throughout, and voltage divides in proportion to resistance. In parallel circuits, voltage is the same across each branch, and current divides inversely proportional to resistance.

🔗 串联电路 / Series Circuit

总电阻 / Total Resistance:

R_{\text{总}} = R_1 + R_2 + R_3 + \cdots + R_n

电流特性 / Current:

I_{\text{总}} = I_1 = I_2 = I_3 = \cdots = I_n

电压特性 / Voltage (分压公式):

U_{\text{总}} = U_1 + U_2 + U_3 + \cdots + U_n

U_1 : U_2 = R_1 : R_2

推导思路 / Derivation:

串联电路中电流处处相等，由欧姆定律 $U=IR$ 得： $U_1=IR_1$ ， $U_2=IR_2$ ，总电压 $U=U_1+U_2=I(R_1+R_2)=IR_{\text{总}}$ ，故 $R_{\text{总}}=R_1+R_2$ 。

In series, current is equal everywhere. From Ohm's Law $U=IR$ : $U_1=IR_1$ , $U_2=IR_2$ , total $U=U_1+U_2=I(R_1+R_2)=IR_{\text{total}}$ , so $R_{\text{total}}=R_1+R_2$ .

🔀 并联电路 / Parallel Circuit

总电阻 / Total Resistance:

\frac{1}{R_{\text{总}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots + \frac{1}{R_n}

电流特性 / Current (分流公式):

I_{\text{总}} = I_1 + I_2 + I_3 + \cdots + I_n

I_1 : I_2 = \frac{1}{R_1} : \frac{1}{R_2} = R_2 : R_1

电压特性 / Voltage:

U_{\text{总}} = U_1 = U_2 = U_3 = \cdots = U_n

推导思路 / Derivation:

并联电路各支路电压相等，由欧姆定律： $I_1=\frac{U}{R_1}$ ， $I_2=\frac{U}{R_2}$ ，总电流 $I=I_1+I_2=U\left(\frac{1}{R_1}+\frac{1}{R_2}\right)=\frac{U}{R_{\text{总}}}$ ，故 $\frac{1}{R_{\text{总}}}=\frac{1}{R_1}+\frac{1}{R_2}$ 。

In parallel, voltage is equal across branches. From Ohm's Law: $I_1=\frac{U}{R_1}$ , $I_2=\frac{U}{R_2}$ , total $I=I_1+I_2=U\left(\frac{1}{R_1}+\frac{1}{R_2}\right)=\frac{U}{R_{\text{total}}}$ , so $\frac{1}{R_{\text{total}}}=\frac{1}{R_1}+\frac{1}{R_2}$ .

串并联计算器 / Series & Parallel Calculator

选择连接方式并输入各电阻值（用逗号分隔），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 the 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 this?

连接方式 / Connection: 电阻值 / Resistor values (Ω):

串联电路模拟 / Series Circuit Simulation

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 $I_1 = I_2 = I$ .

🔋 电压 Voltage (V) 12.0 V

电阻 R₁ 4.0 Ω

电阻 R₂ 8.0 Ω

📐 电流 I (处处相等)

1.00 A

U₁ = 4.0 V U₂ = 8.0 V

R_总 = 12.0 Ω U₁:U₂ = 1:2

并联电路模拟 / Parallel Circuit Simulation

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 $I = I_1 + I_2$ .

🔋 电压 Voltage (V) 12.0 V

电阻 R₁ 4.0 Ω

电阻 R₂ 12.0 Ω

📐 总电流 I_总

4.00 A

I₁ = 3.00 A I₂ = 1.00 A

R_总 = 3.00 Ω I₁:I₂ = 3:1

习题测试 / Quiz

第 1 题 / Question 1

一段导体的电阻为

4\ \Omega

，两端电压为

12\ \text{V}

，通过的电流是多少？A conductor has resistance

4\ \Omega

and voltage

12\ \text{V}

across it. What is the current?

A. 2 A

B. 3 A

C. 48 A

D. 0.33 A

第 2 题 / Question 2

根据欧姆定律，当电阻不变时，电压增大为原来的 3 倍，电流会如何变化？According to Ohm's Law, if resistance stays constant and voltage triples, what happens to the current?

A. 不变 / Stays the same

B. 变为原来的 1/3 / Becomes 1/3

C. 变为原来的 3 倍 / Triples

D. 变为原来的 9 倍 / Becomes 9×

第 3 题 / Question 3

以下哪个元件的 V-I 特性曲线不是一条直线？Which component does NOT have a straight-line V-I characteristic?

A. 恒温铜导线 / Copper wire at constant temp

B. 碳膜电阻 / Carbon film resistor

C. 半导体二极管 / Semiconductor diode

D. 镍铬合金丝 / Nichrome wire

第 4 题 / Question 4

两个电阻

R_1=6\ \Omega

、

R_2=3\ \Omega

串联，总电阻是多少？Two resistors

R_1=6\ \Omega

and

R_2=3\ \Omega

are connected in series. What is the total resistance?

A. 2 Ω

B. 9 Ω

C. 18 Ω

D. 3 Ω

第 5 题 / Question 5

两个电阻

R_1=6\ \Omega

、

R_2=3\ \Omega

并联，总电阻最接近哪个值？Two resistors

R_1=6\ \Omega

and

R_2=3\ \Omega

are in parallel. Which value is closest to the total resistance?

A. 9 Ω

B. 4.5 Ω

C. 2 Ω

D. 1.5 Ω

第 6 题 / Question 6

在并联电路中，下列说法正确的是？Which statement about parallel circuits is correct?

A. 各支路电流相等 / Branch currents are equal

B. 各支路电压相等 / Branch voltages are equal

C. 总电阻等于各电阻之和 / Total R = sum of resistances

D. 总电阻大于任何一个支路电阻 / Total R > any branch R

在下方填入数值（保留两位小数），然后点击"检查答案"。

Enter numerical values below (two decimal places), then click "Check Answers".

第 1 题 / Question 1（基础欧姆定律）

一个

8\ \Omega

的电阻接在

24\ \text{V}

的电源两端，电流 I = A An

8\ \Omega

resistor is connected to a

24\ \text{V}

source. I = ? A

第 2 题 / Question 2（求电阻）

某段电路中电流为

0.5\ \text{A}

，电压为

6\ \text{V}

，则电阻 R = Ω A circuit has

0.5\ \text{A}

and

6\ \text{V}

. R = ? Ω

第 3 题 / Question 3（求电压）

一个

15\ \Omega

电阻中通过

3\ \text{A}

的电流，则两端电压 V = V A

15\ \Omega

resistor has

3\ \text{A}

flowing through it. V = ? V

第 4 题 / Question 4（串联总电阻）

三个电阻

5\ \Omega

、

10\ \Omega

、

15\ \Omega

串联，总电阻 R_总 = Ω Three resistors

5\ \Omega

,

10\ \Omega

,

15\ \Omega

in series. R_total = ? Ω

第 5 题 / Question 5（并联总电阻）

两个电阻

6\ \Omega

和

3\ \Omega

并联，总电阻 R_总 = Ω Two resistors

6\ \Omega

and

3\ \Omega

in parallel. R_total = ? Ω

使用下面的电路模拟器，调整电压和电阻来达成目标，然后点击"提交"按钮检查你的操作是否正确！

Use the circuit simulator below to adjust voltage and resistance to meet the target, then click "Submit" to check your work!

🎯 当前目标 / Current Target:

🔋 电压 Voltage (V) 6.0 V

⚡ 电阻 Resistance (R) 3.0 Ω

📐 你的电流 Your Current

2.00 A

⚡ 欧姆定律 · Ohm's Law

什么是欧姆定律？ / What is Ohm's Law?

三个物理量 / The Three Quantities

伏安特性曲线 / V-I Characteristic Curve

公式推导 / Formula Derivation

适用范围与条件 / Scope and Conditions

互动电路模拟 / Interactive Circuit Simulation

串联与并联电路 / Series and Parallel Circuits

🔗 串联电路 / Series Circuit

🔀 并联电路 / Parallel Circuit

串并联计算器 / Series & Parallel Calculator

串联电路模拟 / Series Circuit Simulation

并联电路模拟 / Parallel Circuit Simulation

习题测试 / Quiz