Ohm's Law | 欧姆定律

📚 Ohm’s Law | 欧姆定律

Georg Simon Ohm published his landmark work in 1827, establishing a simple yet profound relationship between voltage, current, and resistance in electrical circuits. This relationship, now known as Ohm’s Law, forms the bedrock of circuit analysis in A-Level Physics and underpins everything from simple resistor networks to complex electronic devices.

格奥尔格·西蒙·欧姆于1827年发表了具有里程碑意义的研究成果，确立了电压、电流和电阻之间简单而深刻的关系。这一关系如今被称为欧姆定律，是A-Level物理中电路分析的基石，支撑着从简单电阻网络到复杂电子器件的全部内容。

1. Statement of Ohm’s Law | 欧姆定律的表述

Ohm’s Law states that the current flowing through a metallic conductor at constant temperature is directly proportional to the potential difference (voltage) across it, provided physical conditions such as temperature, pressure, and strain remain constant.

欧姆定律指出：在温度恒定的条件下，通过金属导体的电流与导体两端的电势差（电压）成正比，前提是温度、压力和应变等物理条件保持不变。

Mathematically, this is expressed as V ∝ I at constant temperature, which leads to the familiar equation:

数学上，这表示为恒定温度下 V ∝ I，由此得出熟悉的方程：

V = IR

where V is the potential difference in volts (V), I is the current in amperes (A), and R is the resistance in ohms (Ω).

其中 V 是电势差，单位为伏特（V）；I 是电流，单位为安培（A）；R 是电阻，单位为欧姆（Ω）。

A conductor that obeys Ohm’s Law is called an ohmic conductor, meaning its resistance R is constant regardless of the applied voltage — the ratio V/I remains fixed.

遵守欧姆定律的导体称为欧姆导体（ohmic conductor），即其电阻 R 不随外加电压变化——比值 V/I 保持恒定。

2. Physical Meaning of Resistance | 电阻的物理意义

Resistance is the opposition that a material offers to the flow of electric charge. In microscopic terms, it arises from collisions between conduction electrons and the vibrating lattice ions of the conductor. A higher resistance means that for a given voltage, fewer charges flow per second.

电阻是材料对电荷流动的阻碍作用。从微观角度看，电阻源于传导电子与导体晶格离子振动之间的碰撞。电阻越大，意味着在给定电压下，每秒流过的电荷越少。

The unit of resistance, the ohm (Ω), is defined as: 1 Ω = 1 V A⁻¹. This means a resistor has a resistance of 1 ohm if a potential difference of 1 volt across it produces a current of 1 ampere.

电阻的单位——欧姆（Ω）的定义为：1 Ω = 1 V A⁻¹。这意味着，如果电阻两端施加1伏特的电势差能产生1安培的电流，则该电阻的阻值为1欧姆。

It is crucial to understand that resistance is an inherent property of the conductor itself—it depends on the material, its geometry, and its temperature—while current and voltage are quantities that describe the state of the circuit.

必须理解，电阻是导体本身固有的属性——取决于材料、几何形状和温度——而电流和电压则是描述电路状态的物理量。

3. Conditions for Validity | 适用条件

Ohm’s Law is not a universal law of physics; it holds only under specific conditions. The most important condition is constant temperature. When current flows through a conductor, Joule heating occurs, which raises the temperature and, in turn, increases the resistance of most metallic conductors.

欧姆定律并非普适物理定律，仅在特定条件下成立。最重要的条件是温度恒定。当电流通过导体时会产生焦耳热，使温度升高，进而增大大多数金属导体的电阻。

The law also assumes that the conductor is isotropic and homogeneous—meaning its properties are uniform in all directions and throughout its volume. Additionally, the applied voltage must be steady (direct current or low-frequency AC); at very high frequencies, capacitive and inductive effects begin to dominate, and the simple V = IR relationship breaks down.

该定律还假设导体是各向同性和均匀的——即其性质在各个方向及其整个体积内都一致。此外，所施加的电压必须是稳定的（直流或低频交流）；在极高频率下，电容和电感效应开始占据主导，简单的 V = IR 关系不再成立。

It is also worth noting that Ohm’s Law applies to metals under normal operating conditions, but not to semiconductors, electrolytes, gases, or superconductors. In these materials, the relationship between V and I is either non-linear or the resistance is zero altogether.

值得注意的是，欧姆定律适用于正常工况下的金属，但不适用于半导体、电解质、气体或超导体。在这些材料中，V 与 I 的关系是非线性的，或者电阻根本为零。

4. Ohmic vs Non-Ohmic Conductors | 欧姆导体与非欧姆导体

Conductors that obey Ohm’s Law are classified as ohmic. A fixed-value metallic resistor at constant temperature is the classic example. Its I-V graph is a straight line passing through the origin, and its gradient (1/R) is constant.

遵守欧姆定律的导体被归类为欧姆导体。恒定温度下的定值金属电阻是典型例子。其 I-V 图像是过原点的一条直线，斜率（1/R）保持恒定。

Non-ohmic conductors do not maintain a constant resistance as the voltage changes. Examples include:

非欧姆导体不随电压变化保持恒定电阻。典型例子包括：

Filament lamp (tungsten filament): As the current increases, the filament heats up, increasing its resistance. The I-V curve bends downwards (current grows less than proportionally with voltage).
钨丝灯泡：随着电流增大，灯丝升温，电阻增大。I-V 曲线向下弯曲（电流随电压增长的幅度小于正比关系）。
Diode: In forward bias, resistance drops rapidly; in reverse bias, almost no current flows until breakdown voltage.
二极管：正向偏置时电阻急剧下降；反向偏置时几乎无电流，直至达到击穿电压。
Thermistor: Its resistance decreases sharply as temperature rises, a property exploited in temperature sensing circuits.
热敏电阻：其电阻随温度升高而急剧减小，这一特性被利用于温度传感电路中。
Light-dependent resistor (LDR): Its resistance decreases as light intensity increases.
光敏电阻（LDR）：其电阻随光照强度增大而减小。

For non-ohmic conductors, one distinguishes between static resistance (V/I at a point) and dynamic resistance (ΔV/ΔI, the gradient of the V-I curve at a point). These two values are generally not equal.

对于非欧姆导体，需要区分静态电阻（某点的 V/I）和动态电阻（ΔV/ΔI，即 V-I 曲线上某点的斜率）。这两个值通常不相等。

5. I-V Characteristic Curves | I-V 特性曲线

The I-V characteristic curve is a graphical representation of the relationship between current and voltage for a component. It is obtained by varying the potential difference across the component and measuring the resulting current, then plotting the data.

I-V 特性曲线是元件电流与电压关系的图形表示。它通过改变元件两端的电势差并测量相应电流，然后绘制数据得到。

For an ohmic conductor, the I-V graph is a straight line through the origin with constant gradient. The gradient equals 1/R, so a steeper line indicates a smaller resistance. Under reverse polarity, the line continues symmetrically into the third quadrant.

对于欧姆导体，I-V 图像是过原点的直线，斜率恒定。斜率等于 1/R，因此直线越陡，表示电阻越小。在反向下，直线对称延伸至第三象限。

For a filament lamp, the curve deviates from linearity at higher currents. The gradient decreases as voltage increases, reflecting the increase in resistance due to heating. This is why the lamp’s I-V curve is characteristic of a positive temperature coefficient device.

对于白炽灯，曲线在较高电流处偏离线性。随电压增大，斜率减小，反映了加热引起的电阻增大。这就是灯泡 I-V 曲线表现为正温度系数器件特征的缘故。

For a semiconductor diode in forward bias, the I-V curve rises steeply after a threshold voltage (approximately 0.7 V for silicon). In reverse bias, the current remains essentially zero until the breakdown voltage is reached, at which point a very large current can flow.

对于正向偏置的半导体二极管，I-V 曲线在阈值电压（硅管约为0.7 V）之后急剧上升。在反向偏置时，电流基本为零，直至达到击穿电压，此时会流过非常大的电流。

When drawing I-V characteristic graphs in examinations, always plot current on the y-axis and voltage on the x-axis (I vs V). The gradient of the I-V graph is 1/R, not R. A common error is to confuse the gradient of an I-V graph with that of a V-I graph, which are reciprocals of each other.

在考试中绘制 I-V 特性曲线时，务必以电流为纵轴、电压为横轴（I 对 V）。I-V 图像的斜率是 1/R，而不是 R。一个常见错误是把 I-V 图的斜率与 V-I 图的斜率混为一谈，两者互为倒数。

6. Temperature Dependence of Resistance | 电阻的温度依赖性

For metallic conductors, resistance increases with temperature. This is because higher temperatures cause the lattice ions to vibrate with greater amplitude, increasing the frequency of collisions with conduction electrons. This reduces the mean drift velocity of electrons for a given electric field, effectively increasing resistance.

对于金属导体，电阻随温度升高而增大。这是因为温度升高使晶格离子振幅增大，与传导电子的碰撞频率增加。在给定电场下，电子的平均漂移速度减小，有效电阻增大。

The relationship for metals near room temperature can be approximated as linear over a limited temperature range:

金属在室温附近的电阻-温度关系在有限温度范围内可近似为线性：

R = R₀(1 + αΔT)

where R₀ is the resistance at a reference temperature, α is the temperature coefficient of resistance (in K⁻¹), and ΔT is the temperature change in kelvin.

其中 R₀ 是参考温度下的电阻，α 是电阻温度系数（单位 K⁻¹），ΔT 是温度变化量（单位开尔文）。

For semiconductors (such as thermistors), the opposite behaviour is observed: resistance decreases sharply with rising temperature. This is because higher temperatures excite more electrons into the conduction band, dramatically increasing the charge carrier density. Although lattice vibrations also increase, the carrier density effect dominates, resulting in a net decrease in resistance.

对于半导体（如热敏电阻），观察到相反的行为：电阻随温度升高而急剧下降。这是因为温度升高将更多电子激发到导带，电荷载流子密度急剧增加。虽然晶格振动也增强，但载流子密度效应占主导，导致电阻净减小。

7. Resistivity | 电阻率

The resistance of a conductor depends on its dimensions as well as its material. For a wire of uniform cross-sectional area, the resistance is given by:

导体的电阻不仅取决于材料，还取决于其尺寸。对于横截面积均匀的导线，电阻由下式给出：

R = ρL/A

where ρ (rho) is the resistivity of the material (in Ω·m), L is the length (in m), and A is the cross-sectional area (in m²).

其中 ρ（rho）是材料的电阻率（单位 Ω·m），L 是长度（单位 m），A 是横截面积（单位 m²）。

Resistivity is an intrinsic property of the material and is independent of the conductor’s shape or size. Metals such as copper and silver have very low resistivity (about 1.7 × 10⁻⁸ Ω·m and 1.6 × 10⁻⁸ Ω·m respectively), while insulators like rubber have resistivity values exceeding 10¹³ Ω·m.

电阻率是材料的内在属性，与导体的形状和尺寸无关。铜和银等金属的电阻率非常低（分别约为 1.7 × 10⁻⁸ Ω·m 和 1.6 × 10⁻⁸ Ω·m），而橡胶等绝缘体的电阻率可超过 10¹³ Ω·m。

Resistivity varies with temperature exactly as resistance does: for metals, ρ increases with T; for semiconductors, ρ decreases with T. In A-Level problems, you may be asked to find the resistivity of a wire given its resistance, length, and diameter — remember that A = πd²/4 when using the diameter.

电阻率随温度的变化与电阻一致：金属的 ρ 随温度升高而增大；半导体的 ρ 随温度升高而减小。在A-Level题目中，可能会要求你根据电阻、长度和直径求导线的电阻率——注意使用直径时 A = πd²/4。

8. Application of Ohm’s Law in Circuits | 欧姆定律在电路中的应用

Ohm’s Law is the fundamental tool for analysing circuits. In a series circuit, the same current flows through every component, while the total potential difference across the battery is the sum of the individual voltage drops:

欧姆定律是分析电路的基本工具。在串联电路中，每个元件流过的电流相同，而电池两端的电势差等于各元件电压降之和：

V_total = V₁ + V₂ + V₃ = I(R₁ + R₂ + R₃)

In a parallel circuit, the voltage across each branch is identical, while the total current is the sum of the branch currents:

在并联电路中，每个支路两端的电压相同，而总电流等于各支路电流之和：

I_total = I₁ + I₂ + I₃ = V(1/R₁ + 1/R₂ + 1/R₃)

Combining resistors in series: R_total = R₁ + R₂ + R₃. Combining in parallel: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃. For just two parallel resistors, this simplifies to R_total = R₁R₂/(R₁ + R₂).

串联电阻合并：R_total = R₁ + R₂ + R₃。并联电阻合并：1/R_total = 1/R₁ + 1/R₂ + 1/R₃。若仅有两个并联电阻，可简化为 R_total = R₁R₂/(R₁ + R₂)。

Ohm’s Law also enables the calculation of power dissipated in a resistor. Combining P = IV with V = IR gives three equivalent forms:

欧姆定律还可用于计算电阻中消耗的功率。将 P = IV 与 V = IR 结合，可得到三种等价形式：

P = IV = I²R = V²/R

In examination problems, choose the most convenient form based on the known variables. If I and R are known, use I²R; if V and R are known, use V²/R.

在解题时，根据已知量选择最方便的形式。若已知 I 和 R，用 I²R；若已知 V 和 R，用 V²/R。

9. Worked Example | 例题精讲

Problem: A 12 V battery is connected to a circuit consisting of a 4 Ω resistor in series with a parallel combination of a 6 Ω resistor and a 12 Ω resistor. Calculate (a) the total resistance of the circuit, (b) the current drawn from the battery, (c) the voltage across the 4 Ω resistor, and (d) the current through the 6 Ω resistor.

例题：一个12 V电池接入由4 Ω电阻与6 Ω和12 Ω并联组合串联而成的电路。求：(a) 电路总电阻；(b) 电池输出的电流；(c) 4 Ω电阻两端的电压；(d) 通过6 Ω电阻的电流。

Solution (a): First, find the parallel resistance. Using the product-over-sum formula: R_parallel = (6 × 12)/(6 + 12) = 72/18 = 4 Ω. Total resistance: R_total = 4 + 4 = 8 Ω.

解 (a)：首先求并联电阻。用积化和公式：R_并联 = (6 × 12)/(6 + 12) = 72/18 = 4 Ω。总电阻：R_total = 4 + 4 = 8 Ω。

Solution (b): Using Ohm’s Law: I = V/R_total = 12/8 = 1.5 A.

解 (b)：由欧姆定律：I = V/R_total = 12/8 = 1.5 A。

Solution (c): Voltage across the 4 Ω resistor: V = IR = 1.5 × 4 = 6 V. The remaining 6 V appears across the parallel combination.

解 (c)：4 Ω电阻两端电压：V = IR = 1.5 × 4 = 6 V。剩余的6 V加在并联组合两端。

Solution (d): Since the parallel branches have the same voltage (6 V), current through the 6 Ω resistor: I₆ = V/R = 6/6 = 1 A. Checking: current through the 12 Ω resistor is 6/12 = 0.5 A, and 1 + 0.5 = 1.5 A, which matches the total current.

解 (d)：由于并联支路两端电压相同（6 V），通过6 Ω电阻的电流：I₆ = V/R = 6/6 = 1 A。验证：通过12 Ω电阻的电流为 6/12 = 0.5 A，且 1 + 0.5 = 1.5 A，与总电流一致。

10. Experimental Verification | 实验验证

The classic experiment to verify Ohm’s Law uses a circuit with a power supply, an ammeter in series, a voltmeter in parallel, and a variable resistor (rheostat) to adjust the current. A fixed-value resistor or a length of wire serves as the test conductor.

验证欧姆定律的经典实验使用包含电源、串联电流表、并联电压表和可变电阻（变阻器）的电路。以定值电阻或一段导线作为被测导体。

Procedure: Connect the circuit, set the variable resistor to give a small current, record the ammeter and voltmeter readings. Repeat for at least 6–8 different current values by adjusting the rheostat. Plot current (y-axis) against voltage (x-axis). A straight line through the origin confirms Ohm’s Law, and its gradient gives 1/R.

操作步骤：连接电路，将变阻器调至较小电流，记录电流表和电压表读数。通过调节变阻器，至少重复6–8组不同电流值。以电流（纵轴）对电压（横轴）作图。过原点的直线确认欧姆定律成立，其斜率即为 1/R。

Practical considerations: The test wire should be kept at constant temperature—use a low current to minimise Joule heating. The ammeter must be in series (so all current passes through it), and the voltmeter in parallel (so it measures the voltage across the conductor only). An ammeter has negligible resistance; a voltmeter has very high resistance.

实验注意事项：被测导线应保持恒温——使用较小电流以尽量减少焦耳热。电流表必须串联（保证全部电流通过），电压表必须并联（仅测量导体两端电压）。电流表电阻可忽略不计，电压表电阻非常大。

A common experimental error is ignoring the internal resistance of the ammeter or the resistance of connecting wires, which can introduce small systematic errors. In a well-designed experiment, the wire should also be straight and of uniform cross-section to eliminate geometric variations.

一个常见的实验误差是忽略电流表内阻或连接导线的电阻，这会引入小的系统误差。在精心设计的实验中，导线应为笔直且横截面积均匀，以消除几何差异。

11. Common Exam Pitfalls | 常见考试误区

Below are some of the most frequent mistakes students make with Ohm’s Law questions in CIE A-Level Physics, along with strategies to avoid them:

以下是学生在 CIE A-Level 物理欧姆定律题目中最常犯的错误及规避策略：

Pitfall \| 误区	Correction \| 纠正
Applying V = IR to semiconductors or diodes at all voltages	Ohm’s Law applies only to ohmic conductors; use V = IR for non-ohmic elements only at a specific operating point
将 V = IR 任意用于半导体或二极管的所有电压	欧姆定律仅适用于欧姆导体；对非欧姆元件只能在特定工作点使用 V = IR
Forgetting that the gradient of an I-V graph is 1/R, not R	I-V曲线斜率为 1/R；若绘制 V-I 图，斜率才是 R
忽略 I-V 图斜率为 1/R 而非 R	For an I-V graph the gradient is 1/R; only in a V-I graph is the gradient R
Ignoring internal resistance of the battery when using V = IR	The terminal voltage V = E − Ir, where E is e.m.f. and r is internal resistance
使用 V = IR 时忽略电池内阻	端电压 V = E − Ir，其中 E 为电动势，r 为内阻
Saying a filament lamp obeys Ohm’s Law above the threshold	A filament lamp never obeys Ohm’s Law—its resistance changes with temperature
认为白炽灯在阈值以上遵守欧姆定律	白炽灯从不遵守欧姆定律——其电阻随温度改变

Another frequent error is in unit conversion: remember that 1 kΩ = 10³ Ω, 1 MΩ = 10⁶ Ω, and 1 mA = 10⁻³ A. Always convert to base units before substituting into V = IR.

另一个常见错误是单位换算：记住 1 kΩ = 10³ Ω，1 MΩ = 10⁶ Ω，1 mA = 10⁻³ A。代入 V = IR 前务必先转换为基本单位。

12. Summary | 总结

Ohm’s Law, V = IR, is a cornerstone of circuit theory in A-Level Physics. It applies strictly to metallic conductors at constant temperature, where the I-V graph is linear through the origin. The resistance of a conductor is determined by its resistivity, length, and cross-sectional area through R = ρL/A, and it varies with temperature—increasing for metals and decreasing for semiconductors.

欧姆定律 V = IR 是 A-Level 物理电路理论的基石。它严格适用于恒定温度下的金属导体，此时 I-V 图像是过原点的直线。导体的电阻由其电阻率、长度和横截面积通过 R = ρL/A 决定，并随温度变化——金属增大而半导体减小。

To excel in examinations, practise sketching I-V characteristic curves for different components, understand the difference between ohmic and non-ohmic behaviour, and always verify that conditions for Ohm’s Law are satisfied before applying V = IR in a given problem.

要在考试中取得优异成绩，请多加练习绘制不同元件的 I-V 特性曲线，深刻理解欧姆导体与非欧姆导体之间的区别，并在应用 V = IR 解题前始终确认条件是否满足欧姆定律的适用前提。

Mastering Ohm’s Law not only earns marks in circuit questions but also provides the intuitive foundation needed for more advanced topics such as Kirchhoff’s laws, capacitance, and alternating current analysis in the remainder of the A-Level syllabus.

掌握欧姆定律不仅能在电路题中获得分数，还为后续A-Level课程中的基尔霍夫定律、电容和交变电流分析等高级课题奠定直观基础。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Ohm’s Law | 欧姆定律