Resistivity | 电阻率

📚 Resistivity | 电阻率

Resistivity is a core topic in CIE A Level Physics. It links the microscopic identity of a material to the macroscopic resistance of a wire or component. Understanding resistivity helps explain why copper is chosen for cables, why nichrome is used in heaters, and how temperature sensors and superconductors behave.

电阻率是 CIE A Level 物理中的一个核心主题。它把材料的微观特性与导线或元件的宏观电阻联系起来。理解电阻率有助于解释为什么电缆选用铜、加热器选用镍铬合金，以及温度传感器和超导体的行为。

1. Resistance vs Resistivity | 电阻与电阻率的区别

Resistance R is a property of a particular component. It depends on the material, the length, the cross-sectional area and the temperature of that component.

电阻 R 是某个具体元件的属性。它取决于该元件的材料、长度、横截面积和温度。

Resistivity ρ is an intrinsic property of a material. It does not depend on the size or shape of a sample. Two copper wires of different lengths and thicknesses have different resistances, but the same resistivity if they are at the same temperature.

电阻率 ρ 是材料的内在属性。它与样品的大小或形状无关。两根长度和粗细不同的铜导线电阻不同，但在相同温度下具有相同的电阻率。

2. Definition and Formula | 定义与公式

For a uniform conductor at constant temperature, the resistance is directly proportional to its length L and inversely proportional to its cross-sectional area A. The constant of proportionality is the resistivity ρ.

对于温度恒定的均匀导体，电阻与其长度 L 成正比，与其横截面积 A 成反比。比例常数就是电阻率 ρ。

R = ρL / A or ρ = RA / L

The defining equation ρ = RA / L shows that resistivity is numerically the resistance of a unit cube of the material when current flows perpendicular to one pair of faces.

定义式 ρ = RA / L 表明，电阻率在数值上等于材料单位立方体在电流垂直于其中一对面流动时的电阻。

3. Units of Resistivity | 电阻率的单位

From ρ = RA / L, the SI unit is ohm metre squared divided by metre. This simplifies to ohm metre, written as Ω m.

由 ρ = RA / L 可知，SI 单位是欧姆·米² 除以米，简化为欧姆·米，写作 Ω m。

In calculations, the cross-sectional area must be converted to m² and the length to m. A common error is to use area in mm² or length in cm, which gives an incorrect unit or value.

在计算中，横截面积必须换算为 m²，长度必须换算为 m。常见错误是使用 mm² 表示面积或使用 cm 表示长度，这会导致单位或数值错误。

4. Conductivity | 电导率

Conductivity σ is the reciprocal of resistivity. It describes how easily a material allows electric current to flow.

电导率 σ 是电阻率的倒数。它描述材料允许电流通过的难易程度。

σ = 1 / ρ

Materials with very low resistivity, such as silver and copper, have very high conductivity. Insulators have extremely low conductivity because their resistivity is enormous.

电阻率很低的材料，如银和铜，具有很高的电导率。绝缘体的电导率极低，因为它们的电阻率非常大。

5. Resistivity of Materials: Conductors, Semiconductors and Insulators | 材料的电阻率：导体、半导体和绝缘体

Materials can be grouped by their typical resistivity at room temperature. Conductors have resistivity roughly in the range 10⁻⁸ Ω m to 10⁻⁶ Ω m. Semiconductors have intermediate resistivity, often from about 10⁻³ Ω m to 10³ Ω m. Insulators have very high resistivity, often above 10¹⁰ Ω m.

材料可以按室温下的典型电阻率分组。导体的电阻率大约在 10⁻⁸ Ω m 到 10⁻⁶ Ω m 范围内。半导体的电阻率居中，通常约为 10⁻³ Ω m 到 10³ Ω m。绝缘体的电阻率非常高，通常高于 10¹⁰ Ω m。

Material	Classification	Approximate resistivity at 20 °C / Ω m
Silver	Conductor	1.6 × 10⁻⁸
Copper	Conductor	1.7 × 10⁻⁸
Aluminium	Conductor	2.7 × 10⁻⁸
Nichrome	Alloy conductor	1.1 × 10⁻⁶
Silicon	Semiconductor	about 2.3 × 10³
Glass	Insulator	10¹⁰ to 10¹⁴
PVC	Insulator	about 10¹²

The exact value for a semiconductor depends strongly on temperature, purity and illumination. Semiconductor resistivities are therefore not fixed material constants in the same way as typical metals.

半导体的具体数值在很大程度上取决于温度、纯度和光照。因此，半导体电阻率并不像典型金属那样是固定的材料常数。

6. Temperature Dependence | 温度依赖性

For most metals, resistivity increases with temperature. As temperature rises, the positive metal ions vibrate more vigorously, so conduction electrons collide more frequently and drift more slowly.

对于大多数金属，电阻率随温度升高而增大。随着温度升高，金属正离子振动更剧烈，传导电子碰撞更频繁，漂移更慢。

For many metals over a moderate temperature range, the variation is approximately linear:

在许多金属中，在中等温度范围内，变化近似为线性：

ρ = ρ₀ (1 + α ΔT)

Here ρ₀ is the resistivity at a reference temperature, α is the temperature coefficient of resistivity, and ΔT is the temperature change.

其中 ρ₀ 是参考温度下的电阻率，α 是电阻率温度系数，ΔT 是温度变化。

In semiconductors, resistivity typically decreases as temperature increases because more electrons gain enough energy to become free charge carriers. This effect usually outweighs the increased lattice scattering.

在半导体中，电阻率通常随温度升高而减小，因为更多电子获得足够能量成为自由载流子。这种效应通常超过了晶格散射增加的影响。

7. Measurement of Resistivity | 电阻率的测量

A standard A Level experiment uses a uniform metal wire, a metre rule, a micrometer screw gauge, an ammeter, a voltmeter and a variable power supply.

标准的 A Level 实验使用一根均匀金属丝、米尺、千分尺、电流表、电压表和可变电源。

Measure the diameter of the wire at several points and calculate the mean cross-sectional area A = πd² / 4.
在金属丝的多个位置测量直径，并计算平均横截面积 A = πd² / 4。
Vary the length L of the wire in the circuit and measure the corresponding current I and potential difference V. Calculate R = V / I for each length.
改变电路中金属丝的长度 L，测量相应的电流 I 和电势差 V。对每个长度计算 R = V / I。
Plot a graph of R against L. The graph should be a straight line through the origin, with gradient equal to ρ / A.
绘制 R 对 L 的图像。图像应为过原点的直线，斜率等于 ρ / A。
Use ρ = gradient × A to find the resistivity.
使用 ρ = 斜率 × A 求出电阻率。

The current should be kept small to avoid significant heating, because resistance changes with temperature.

电流应保持较小，以避免明显发热，因为电阻会随温度变化。

8. Microscopic Origin and Drift Velocity | 微观起源与漂移速度

In the free electron model, conduction electrons move randomly and collide with vibrating lattice ions. When an electric field is applied, the electrons gain a small average drift velocity in the direction opposite to the field.

在自由电子模型中，传导电子随机运动并与振动的晶格离子碰撞。当施加电场时，电子在电场相反方向上获得一个较小的平均漂移速度。

Resistivity arises from these collisions. A longer path or a smaller cross-sectional area increases the difficulty for charge carriers to move, which is why R = ρL / A.

电阻率源于这些碰撞。路径越长或横截面积越小，载流子移动越困难，这就是 R = ρL / A 的原因。

The current in a conductor can also be written as I = nAvq, where n is the number density of charge carriers, A is the cross-sectional area, v is the drift velocity and q is the charge on each carrier.

导体中的电流也可写作 I = nAvq，其中 n 是载流子数密度，A 是横截面积，v 是漂移速度，q 是每个载流子的电荷。

Combining I = nAvq with V = IR links resistivity to the scattering of charge carriers inside the material.

将 I = nAvq 与 V = IR 结合，可以将电阻率与材料内部载流子的散射联系起来。

9. Superconductivity | 超导性

A superconductor is a material whose resistivity falls to zero below a critical temperature Tc. Once a current is set up in a superconducting loop, it can flow without a potential difference and without energy loss.

超导体是一种在临界温度 Tc 以下电阻率降为零的材料。一旦在超导回路中建立电流，它就可以在没有电势差和没有能量损失的情况下持续流动。

Superconducting magnets are used in MRI scanners, particle accelerators and magnetic levitation trains. The need for very low temperatures can make practical use difficult and expensive.

超导磁体用于 MRI 扫描仪、粒子加速器和磁悬浮列车。需要极低温度可能使实际应用变得困难和昂贵。

10. Worked Example | 例题

A nichrome wire has length 2.00 m and diameter 0.50 mm. Its resistance is measured as 11.2 Ω. Calculate the resistivity of nichrome.

一根镍铬合金丝长度为 2.00 m，直径为 0.50 mm。测得电阻为 11.2 Ω。计算镍铬合金的电阻率。

Step 1: Convert diameter to metres: d = 0.50 mm = 0.50 × 10⁻³ m.

步骤 1：将直径换算为米：d = 0.50 mm = 0.50 × 10⁻³ m。

Step 2: Calculate cross-sectional area A = πd² / 4 = π(0.50 × 10⁻³)² / 4 = 1.96 × 10⁻⁷ m².

步骤 2：计算横截面积 A = πd² / 4 = π(0.50 × 10⁻³)² / 4 = 1.96 × 10⁻⁷ m²。

Step 3: Use ρ = RA / L = (11.2 × 1.96 × 10⁻⁷) / 2.00 = 1.10 × 10⁻⁶ Ω m.

步骤 3：使用 ρ = RA / L = (11.2 × 1.96 × 10⁻⁷) / 2.00 = 1.10 × 10⁻⁶ Ω m。

The result agrees with the expected order of magnitude for nichrome.

结果与镍铬合金电阻率的预期数量级一致。

11. Common Mistakes | 常见错误

Do not confuse resistance with resistivity. Resistance changes with length and area; resistivity is a material property.

不要混淆电阻和电阻率。电阻随长度和面积变化；电阻率是材料属性。

Always convert the wire diameter to metres before calculating area. Using millimetres directly gives an area that is too large by a factor of 10⁶.

在计算面积之前，始终将导线直径换算为米。直接使用毫米会使面积增大 10⁶ 倍。

Remember that area is πd² / 4, not πd². Forgetting the factor of 1/4 is a very common error.

记住面积是 πd² / 4，而不是 πd²。忘记 1/4 因子是非常常见的错误。

Do not assume resistivity always increases with temperature. That is true for most metals, but not for semiconductors and many insulators.

不要假设电阻率总是随温度升高而增大。这对大多数金属成立，但对半导体和许多绝缘体并不成立。

12. Applications | 应用

Copper and aluminium have low resistivity, so they are used for electrical power cables and household wiring to minimise energy loss and voltage drop.

铜和铝的电阻率低，因此被用于电力电缆和家庭布线，以尽量减少能量损失和电压降。

Nichrome and other resistive alloys have higher resistivity and tolerate high temperatures, so they are used in heating elements such as kettles, toasters and electric heaters.

镍铬合金和其他电阻合金具有较高电阻率并能承受高温，因此用于电热水壶、烤面包机和电暖器等加热元件。

Insulators such as PVC and glass have very high resistivity, so they are used to cover wires and support high-voltage equipment safely.

PVC 和玻璃等绝缘体具有非常高的电阻率，因此用于包裹导线和安全支撑高压设备。

Thermistors and light-dependent resistors exploit the change in resistivity with temperature and light intensity. They are used in temperature sensing, circuit protection and automatic lighting.

热敏电阻和光敏电阻利用电阻率随温度和光照强度变化的特性。它们用于温度传感、电路保护和自动照明。

Superconductors, with zero resistivity below their critical temperature, are used where strong magnetic fields or lossless current flow are needed.

超导体在其临界温度以下电阻率为零，用于需要强磁场或无损耗电流流动的场合。

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Resistivity | 电阻率