Ohm's Law and Circuit Analysis | 欧姆定律与电路分析

📚 Ohm’s Law and Circuit Analysis | 欧姆定律与电路分析

Circuit analysis is the foundation of all electrical and electronic systems. At its core lies Ohm’s law, a simple yet powerful relationship between voltage, current, and resistance. This article explains the essential concepts of DC circuit analysis, from individual components to complex networks, with exam-focused clarity.

电路分析是所有电气与电子系统的基础。其核心是欧姆定律——一个简洁却极为重要的电压、电流和电阻关系式。本文从单一元件到复杂网络，系统地讲解直流电路分析的必备知识，紧扣A-level物理考点，帮助你高效复习。

1. Electric Current, Voltage and Resistance | 电流、电压与电阻

Electric current \( I \) is the rate of flow of charge through a conductor. It is measured in amperes (A), where 1 A = 1 C s⁻¹. Conventional current flows from positive to negative, opposite to electron flow.

电流 \( I \) 是电荷通过导体的速率，单位为安培（A），1 A = 1 C s⁻¹。习惯上规定电流方向从正极流向负极，与电子流动方向相反。

Voltage (potential difference) \( V \) between two points is the energy transferred per unit charge. It is measured in volts (V), where 1 V = 1 J C⁻¹. Voltage provides the “driving force” that pushes charge around a circuit.

电压（电势差） \( V \) 是单位电荷在两点间移动时转移的能量，单位为伏特（V），1 V = 1 J C⁻¹。电压为电荷沿电路运动提供“驱动力”。

Resistance \( R \) is the opposition to current flow. It is measured in ohms (Ω), where 1 Ω = 1 V A⁻¹. In a metallic conductor, resistance arises mainly from collisions between conducting electrons and vibrating lattice ions.

电阻 \( R \) 是阻碍电流流动的程度，单位为欧姆（Ω），1 Ω = 1 V A⁻¹。在金属导体中，电阻主要来源于导电电子与晶格离子振动之间的碰撞。

2. Ohm’s Law | 欧姆定律

Ohm’s law states that, for a metallic conductor at constant temperature, the current flowing through it is directly proportional to the potential difference across it. Mathematically:

欧姆定律指出：对于温度恒定的金属导体，通过导体的电流与其两端的电势差成正比。数学表达式为：

V = I × R

Here \( V \) is the voltage across the conductor in volts, \( I \) is the current in amperes, and \( R \) is the resistance in ohms. Rearranging gives \( I = V / R \) and \( R = V / I \).

其中 \( V \) 为导体两端电压（伏特），\( I \) 为电流（安培），\( R \) 为电阻（欧姆）。变形可得 \( I = V / R \) 和 \( R = V / I \)。

It is crucial to remember that Ohm’s law is not a universal rule. It applies only to ohmic conductors in which resistance remains constant over a range of voltages. Non-ohmic devices such as diodes and filament lamps do not obey this simple proportional relationship.

务必注意：欧姆定律并非普遍规律。它只适用于阻值在一定电压范围内保持恒定的欧姆导体。二极管、白炽灯丝等非欧姆元件不满足这种简单正比关系。

3. Resistivity and Conductance | 电阻率与电导

The resistance of a conductor depends on its material and dimensions. For a length \( L \) and uniform cross-sectional area \( A \), resistance is given by:

导体的电阻取决于材料种类与几何尺寸。对于长度为 \( L \)、横截面积为 \( A \) 的均匀导体，电阻为：

R = ρ × L / A

where \( ρ \) (rho) is the resistivity of the material, measured in ohm-metres (Ω m). Resistivity is an intrinsic property of a material and depends on temperature.

其中 \( ρ \)（rho）为材料的电阻率，单位是欧姆·米（Ω m）。电阻率是材料的固有属性，随温度变化。

Conductance \( G \) is the reciprocal of resistance: \( G = 1 / R \), measured in siemens (S). Conductivity \( σ \) is the reciprocal of resistivity: \( σ = 1 / ρ \), measured in S m⁻¹. Good conductors such as copper have low resistivity, while insulators have extremely high resistivity.

电导 \( G \) 是电阻的倒数：\( G = 1 / R \)，单位为西门子（S）。电导率 \( σ \) 是电阻率的倒数：\( σ = 1 / ρ \)，单位为 S m⁻¹。铜等良导体的电阻率很低，而绝缘体的电阻率极高。

For a wire that is stretched to double its length, volume remains constant, so area halves, and resistance becomes four times greater. This type of geometric reasoning appears frequently in exam questions.

若将金属丝拉长至原来的两倍，由于体积不变，横截面积减半，电阻将变为原来的四倍。这类几何推理在考试中经常出现。

4. Series and Parallel Circuits | 串联与并联电路

In a series circuit, components are connected end-to-end, so the same current flows through each component. The total voltage is the sum of individual voltages:

串联电路中，各元件首尾相连，通过每个元件的电流相同。总电压等于各元件电压之和：

R_total = R₁ + R₂ + R₃ + …

In a parallel circuit, components are connected across the same two points, so the voltage across each branch is equal. The total current is the sum of branch currents. The total resistance is given by:

并联电路中，各元件并排连接在相同两点之间，各支路两端的电压相等。总电流等于各支路电流之和。总电阻为：

1 / R_total = 1 / R₁ + 1 / R₂ + 1 / R₃ + …

For two resistors in parallel, the total resistance simplifies to \( R_total = (R₁ × R₂) / (R₁ + R₂) \). Always verify that the parallel combination has a resistance smaller than the smallest individual resistor.

两个电阻并联时，总电阻可化简为 \( R_total = (R₁ × R₂) / (R₁ + R₂) \)。请务必检查：并联总电阻一定小于其中任意一个电阻的阻值。

Common exam traps include adding parallel resistors directly, forgetting to treat series and parallel groups step by step, or confusing current flow in series (same current) with voltage division in series (voltage splits).

常见考试陷阱包括：直接相加并联电阻、忘记按组逐步化简串并联结构、混淆串联中电流处处相同而电压按电阻分配这一原则。

5. Kirchhoff’s Laws | 基尔霍夫定律

Kirchhoff’s first law (current law, KCL): The total current entering a junction equals the total current leaving the junction. This is a statement of conservation of charge. In equation form:

基尔霍夫第一定律（电流定律，KCL）：流入节点的总电流等于流出节点的总电流。这是电荷守恒的体现，公式为：

Σ I_in = Σ I_out

Kirchhoff’s second law (voltage law, KVL): In any closed loop, the sum of the electromotive forces (emfs) equals the sum of the potential differences across all load components. This follows from conservation of energy.

基尔霍夫第二定律（电压定律，KVL）：在任意闭合回路中，电动势之和等于所有负载元件上电势差之和。这由能量守恒导出。

Σ EMF = Σ I × R (around a loop)

When applying KVL, assign a direction for the loop. If current flows through a resistor in the chosen direction, the voltage drop is \( I R \); if opposite, it is \( -I R \). For a battery, add \( +E \) if traversed from negative to positive, and \( -E \) if traversed from positive to negative.

应用KVL时需先规定绕行方向。若电流方向与绕行方向相同，电阻上的电压降为 \( +I R \)；相反则为 \( -I R \)。对于电池，若从负极到正极经过则取 \( +E \)，从正极到负极则取 \( -E \)。

Kirchhoff’s laws are essential for analysing circuits that cannot be reduced to simple series or parallel combinations, such as multi-loop networks.

对于无法简化为简单串并联的多回路网络，基尔霍夫定律是必不可少的分析工具。

6. Internal Resistance and EMF | 内阻与电动势

A real battery is modelled as an ideal emf \( E \) in series with an internal resistance \( r \). When the battery delivers current \( I \) to an external circuit, the terminal voltage \( V \) is lower than the emf because of the voltage drop across the internal resistance:

实际电池可建模为理想电动势 \( E \) 与内阻 \( r \) 的串联。当电池向外部电路提供电流 \( I \) 时，由于内阻上有电压降，端电压 \( V \) 小于电动势：

V = E – I × r

The terminal voltage equals the emf only when no current flows (open circuit). As current increases, the terminal voltage decreases. In the extreme case of a short circuit, \( V = 0 \) and the current reaches \( E / r \).

只有开路（无电流）时，端电压才等于电动势。电流增大时，端电压会下降。在极端短路情况下，\( V = 0 \)，电流达到 \( E / r \)。

When a resistor \( R \) is connected to the battery, the current is:

当外部连接一个电阻 \( R \) 时，电流为：

I = E / (R + r)

Maximum power is delivered to the external load when the load resistance equals the internal resistance (\( R = r \)). This result is known as the maximum power transfer theorem.

当负载电阻等于内阻（\( R = r \)）时，外部负载获得最大功率。这一结论称为最大功率传输定理。

7. I-V Characteristics | 电流-电压特性曲线

The I-V characteristic of a component shows how current changes with applied voltage. For an ohmic resistor at constant temperature, the graph is a straight line through the origin, and the gradient is \( 1 / R \).

元件的I-V特性曲线描述电流随外加电压的变化关系。在恒定温度下，欧姆电阻的I-V图是过原点的直线，斜率为 \( 1 / R \)。

For a filament lamp, as current increases, the metal filament heats up, increasing its resistance. The I-V graph curves upwards at high voltages, showing a decreasing slope (higher resistance).

对于灯丝灯泡，电流增大时金属灯丝温度升高，电阻增大。I-V曲线在高压区域向上弯曲，斜率变小（电阻变大）。

A semiconductor diode conducts strongly in the forward direction beyond a small threshold voltage (about 0.7 V for silicon), but only a tiny leakage current flows in reverse until breakdown. The diode is therefore a non-ohmic device.

半导体二极管在正向电压超过阈值（硅管约0.7 V）后开始明显导通，而反向只有极小的漏电流，直到击穿。因此二极管是非欧姆器件。

When sketching I-V graphs, label the axes clearly and note whether the graph is symmetric. A thermistor has a negative temperature coefficient: its resistance decreases as temperature rises, producing a non-linear I-V curve.

绘制I-V图时，务必标注坐标轴，并注意图形是否对称。热敏电阻具有负温度系数：温度升高时电阻下降，因此I-V曲线呈非线性。

8. Power in Circuits | 电路中的功率

The electrical power dissipated by a component is the product of voltage and current:

元件耗散的电动率等于电压与电流的乘积：

P = V × I

Using Ohm’s law, power can also be expressed as \( P = I² R = V² / R \). These equivalent forms are useful in different contexts. For example, the heating effect in a resistor is usually calculated as \( P = I² R \), since current is often the same through a series resistor.

借助欧姆定律，功率也可表示为 \( P = I² R = V² / R \)。这些等价形式在不同场景中各有用途。例如，电阻的发热效应常用 \( P = I² R \) 计算，因为串联电阻中电流往往相同。

Total energy transferred in time \( t \) is \( W = P t = V I t \). In a circuit with a battery of emf \( E \) and internal resistance \( r \), the total power supplied by the battery is \( P_source = E I \), while the useful external power is \( P_external = V I \). The lost power in the internal resistance is \( I² r \).

在时间 \( t \) 内转移的总能量为 \( W = P t = V I t \)。对于含有电动势 \( E \) 和内阻 \( r \) 的电池，电池提供的总功率为 \( P源 = E I \)，外部有用功率为 \( P外 = V I \)，内阻上损耗的功率为 \( I² r \)。

Efficiency of a power supply is defined as \( η = P_external / P_total × 100% = V / E × 100% \) (for a battery delivering current). Note that efficiency increases as current decreases, because less power is wasted on internal resistance.

电源效率定义为 \( η = P外 / P总 × 100% = V / E × 100% \)（电池供电时）。注意：电流越小，效率越高，因为内阻上浪费的功率更少。

9. Wheatstone Bridge and Potentiometer | 惠斯通电桥与电位差计

The Wheatstone bridge is a circuit used to measure unknown resistance accurately. It consists of two ratio resistors \( R₁ \) and \( R₂ \) in one branch, and the unknown \( R_x \) with a known variable resistor \( R₃ \) in the other branch. A sensitive galvanometer is placed between the midpoints.

惠斯通电桥是一种用于精确测量未知电阻的电路。它由两个比例电阻 \( R₁ \) 和 \( R₂ \) 组成一支路，未知电阻 \( R_x \) 和已知可变电阻 \( R₃ \) 组成另一支路，中点之间接入灵敏电流计。

When the bridge is balanced, no current flows through the galvanometer, and the potential at the midpoints is equal. The balance condition is:

当电桥平衡时，电流计中没有电流，两个中点电势相等。平衡条件为：

R₁ / R₂ = R₃ / R_x

Thus \( R_x = R₃ × R₂ / R₁ \). Because the balance condition does not depend on supply voltage, the Wheatstone bridge can achieve very high accuracy.

因此 \( R_x = R₃ × R₂ / R₁ \)。由于平衡条件不依赖电源电压，惠斯通电桥可实现非常高的测量精度。

The potentiometer is a device that compares electromotive forces or measures a potential difference without drawing current from the source when balanced. A uniform resistance wire is connected to a supply; a sliding contact is adjusted until the galvanometer reads zero. At balance, the known emf is proportional to the length of wire.

电位差计用于比较电动势或测量电势差，平衡时不从被测电源吸取电流。将均匀电阻丝接在电源两端，移动滑动触头直到电流计读数为零。平衡时，被测电动势与电阻丝长度成正比。

E₁ / E₂ = L₁ / L₂

Potentiometers are particularly useful for measuring small emfs precisely, such as a thermocouple output, because they do not alter the circuit under test.

电位差计特别适合精确测量微小电动势（如热电偶输出），因为它不会改变被测电路的工作状态。

10. Practical Circuit Analysis | 实际电路分析

In practical problems, you should first redraw the circuit clearly, replacing each real battery with an ideal emf and internal resistance. Then identify all nodes and loops, and decide which junction rule or loop rule to apply.

在实际解题中，应首先重新清晰作图，将每个实际电池替换为理想电动势与内阻的串联。然后标出所有节点和回路，决定应用节点电流定律还是回路电压定律。

For complex circuits, a systematic method is to choose a current direction in each branch, label the unknown currents, and write KCL for all but one junction and KVL for independent loops. Solve the resulting linear equations for the unknown currents.

对于复杂电路，系统化方法是：设定每条支路的电流方向，标出未知电流，写出除一个节点外的所有KCL方程，以及独立回路的KVL方程，然后求解线性方程组得到各未知电流。

When a circuit contains a capacitor, remember that in steady state (DC), the current through the capacitor is zero, but the voltage across it can be non-zero. The capacitor then acts as an open circuit, while the rest of the circuit reaches a steady configuration.

当电路中含有电容时，记住在直流稳态下电容支路的电流为零，但电容两端电压可不为零。此时电容相当于开路，其余电路达到稳定工作状态。

Circuit diagrams in exams may omit the return path or use ground symbols. Always check for hidden series/parallel simplifications before applying Kirchhoff’s laws.

考试电路图有时会省略返回路径或使用接地符号。在应用基尔霍夫定律之前，务必检查是否存在隐藏的串并联简化机会。

11. Common Mistakes and Exam Tips | 常见错误与考试要点

One frequent mistake is forgetting that the current in a series circuit is the same everywhere. Always calculate the total resistance first, then the total current, then the voltage across each resistor using \( V = I R \).

常见错误之一是忘记串联电路中电流处处相同。正确的解题顺序是：先求总电阻，再求总电流，最后用 \( V = I R \) 求各电阻上的电压。

In parallel circuits, students often confuse voltage and current. The voltage is the same across each branch, but the current splits inversely to resistance. A larger resistance carries a smaller current if the voltage is fixed.

在并联电路中，学生常混淆电压与电流。各支路电压相等，但电流按电阻反比例分配。电压固定时，电阻越大，电流越小。

When using \( R = V / I \), do not take the gradient of an I-V graph as \( R \) unless the graph is linear and through the origin. For a non-linear device, the resistance at a point is \( V / I \) at that point, not the tangent slope.

使用 \( R = V / I \) 计算电阻时，不要直接将I-V图的斜率视为 \( R \)，除非该图是过原点的直线。对于非线性元件，某点的电阻是该点的 \( V / I \) 值，而非切线斜率。

Always state units in calculations and use the correct SI prefixes (kΩ, MΩ, mA, μA). Show your working clearly, especially when applying Kirchhoff’s laws. Drawing current directions and labelling loops saves marks and reduces errors.

计算中务必注明单位，正确使用SI词头（kΩ、MΩ、mA、μA）。应用基尔霍夫定律时，过程要写清楚。画出电流方向和回路标记，这不仅有助于得分，也能减少错误。

Finally, practise interpreting circuit diagrams with ammeters and voltmeters. An ideal ammeter has zero resistance and is placed in series; an ideal voltmeter has infinite resistance and is placed in parallel.

最后，要练习解读包含电流表和电压表的电路图。理想电流表电阻为零，串联接入；理想电压表电阻无穷大，并联接入。

12. Summary | 总结

Ohm’s law and circuit analysis are indispensable skills in physics. Mastering series/parallel rules, Kirchhoff’s laws, internal resistance, and I-V characteristics enables you to solve a wide range of problems confidently.

欧姆定律与电路分析是物理学习中不可或缺的技能。掌握串并联规律、基尔霍夫定律、内阻和I-V特性，能够帮助你自信地解决各类电路问题。

Key equations to memorise: \( V = I R \), \( R_{series} = R_1 + R_2 + … \), \( 1 / R_{parallel} = 1 / R_1 + 1 / R_2 + … \), \( P = V I = I² R = V² / R \), and \( V = E – I r \).

需要牢记的核心公式：\( V = I R \)、\( R_{串联} = R_1 + R_2 + … \)、\( 1 / R_{并联} = 1 / R_1 + 1 / R_2 + … \)、\( P = V I = I² R = V² / R \)、\( V = E – I r \)。

Always analyse circuits systematically: recognise known combinations, choose an appropriate law, and verify your answers by checking units and physical reasonableness. Good luck with your revision!

分析电路时始终要保持系统性：识别已知组合、选择合适的定律，并通过单位检查和物理合理性验证答案。祝你复习顺利！

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Ohm’s Law and Circuit Analysis | 欧姆定律与电路分析