📚 Resistance and Temperature | 电阻与温度
Resistance is not a fixed property of a material: it changes with temperature. In metals, resistance increases as temperature rises, while in most semiconductors and thermistors it decreases. Understanding this behaviour is essential for circuit design and is a core topic in CIE A-Level Physics.
电阻并不是材料固定不变的属性：它会随温度变化。金属的电阻随温度升高而增大，而大多数半导体和热敏电阻的电阻却随温度升高而减小。理解这一行为对电路设计至关重要，也是 CIE A-Level 物理的核心考点。
This article explains the microscopic reason for the change, the equations you must be able to quote and use, the difference between metals, semiconductors and thermistors, and the experimental and exam skills that carry the marks.
本文将解释这一变化的微观原因、你必须能够引用和使用的公式、金属与半导体及热敏电阻之间的差异，以及能够拿分的实验与应试技巧。
1. Why Resistance Changes with Temperature | 为什么电阻会随温度变化
When a potential difference is applied across a conductor, free electrons drift through the lattice. They collide with vibrating positive ions, transferring energy and limiting the current. These collisions are the main source of resistance.
当导体两端存在电势差时，自由电子在晶格中漂移。它们会与振动的正离子碰撞，传递能量并限制电流。这些碰撞就是电阻的主要来源。
The direction in which resistance changes depends on what happens to the number density of charge carriers and to the frequency of collisions. In a metal the number density hardly changes, so collisions dominate. In a semiconductor the number density changes enormously, and that effect dominates instead.
电阻朝哪个方向变化，取决于载流子数密度和碰撞频率各自发生了什么变化。在金属中，载流子数密度几乎不变，因此碰撞起主导作用；而在半导体中，载流子数密度变化极大，这一效应反而占据了主导地位。
Examiners almost always ask you to 'explain in terms of the behaviour of electrons and ions'. A purely descriptive answer such as 'it just gets hotter so resistance increases' rarely earns full marks.
考官几乎总是要求你"用电子和离子的行为来解释"。仅仅描述性的回答，例如"它变热了，所以电阻变大"，通常拿不到满分。
2. Microscopic Origin of Resistance | 电阻的微观起源
The positive ions in a metal lattice are not stationary. They vibrate about fixed equilibrium positions, and the amplitude of this vibration is a direct measure of the temperature of the metal.
金属晶格中的正离子并非静止不动。它们在固定的平衡位置附近振动，而振动的振幅正是金属温度的直接量度。
At higher temperatures the lattice ions vibrate more violently, so electrons have a shorter mean free path between collisions. Collisions become more frequent, energy transfer to the lattice increases, and this is why the resistance of a metal rises when it is heated.
温度更高时，晶格离子振动更剧烈，电子在两次碰撞之间的平均自由程更短。碰撞更加频繁，向晶格传递的能量增加，这就是金属受热时电阻升高的原因。
It is important to separate the random thermal motion of electrons (typically of the order of 10⁵ m s⁻¹) from the much slower net drift velocity (often only a few millimetres per second). Only the drift velocity produces the current, and it is the drift velocity that is reduced when collisions become more frequent.
必须把电子的随机热运动（通常约为 10⁵ m s⁻¹ 量级）与慢得多的净漂移速度（常常只有几毫米每秒）区分开来。只有漂移速度产生电流，而当碰撞变得更加频繁时，被削弱的正是这个漂移速度。
3. The Drift Velocity Equation I = nAve | 漂移速度方程 I = nAve
The current in a conductor is related to the charge carriers by the equation below, which you should be able to recall and rearrange:
导体中的电流与载流子之间由下面的方程联系起来，你应当能够默写并变形使用它：
I = n A v e
m², v is the mean drift velocity in m s⁻¹ and e is the elementary charge, 1.60 × 10⁻¹⁹ C.
其中 I 是电流（安培），n 是每立方米中载流子的数密度，A 是横截面积（m²），v 是平均漂移速度（m s⁻¹），e 是基本电荷，为 1.60 × 10⁻¹⁹ C。
ixed current, if the number density or the cross-sectional area increases, the drift velocity must fall. This rearrangement is frequently examined in questions about wires of different diameters.
变形后得到 v = I ÷ (n A e)。在电流恒定时，若数密度或横截面积增大，漂移速度必然减小。这一变形常出现在关于不同直径导线的考题中。
In a metal n is almost constant, because heating does not free significantly more electrons. Temperature therefore affects mainly v, through scattering. This single sentence is worth remembering: metals change because of scattering, semiconductors change because of n.
在金属中 n 几乎不变，因为加热并不会释放出明显更多的电子。因此温度主要通过散射影响 v。这一句话值得牢记：金属的变化源于散射，半导体的变化源于 n。
4. Metals and Positive Temperature Coefficient | 金属与正温度系数
For a pure metal, resistance increases almost linearly with temperature over a moderate range. We say metals have a positive temperature coefficient of resistance.
对于纯金属，在中等温度范围内，电阻几乎随温度线性增大。我们说金属具有正的电阻温度系数。
ting electrons → shorter mean free path. For a given potential difference the drift velocity is therefore lower, the current is smaller, and since R = V ÷ I the resistance is larger.
完整的推理链条如下：温度升高 → 晶格离子振动幅度增大 → 与漂移电子的碰撞更加频繁 → 平均自由程变短。在给定电势差下，漂移速度因此降低，电流变小；又因为 R = V ÷ I，所以电阻变大。
A common trap is to claim that 'electrons move faster when the metal is hot, so the current increases'. The random speeds do increase, but they are randomly directed and produce no net current. The relevant quantity, the drift velocity, actually decreases at constant applied voltage.
一个常见陷阱是声称"金属变热时电子运动更快，所以电流增大"。随机速率确实增大，但方向杂乱无章，并不产生净电流。真正相关的量——漂移速度——在电压恒定时反而减小。
5. The Linear Approximation R = R₀(1 + αΔθ) | 线性近似公式
Over a limited temperature range, the resistance of a metallic conductor can be modelled by a simple linear relationship:
在有限的温度范围内，金属导体的电阻可以用一个简单的线性关系来建模：
R = R₀ (1 + α Δθ)
ange from the reference value. For copper, α ≈ 3.9 × 10⁻³ K⁻¹; for tungsten it is about 4.5 × 10⁻³ K⁻¹; for nichrome it is only about 0.4 × 10⁻³ K⁻¹.
R₀ 是参考温度（通常取 0 °C）下的电阻，α 是电阻温度系数，单位为 K⁻¹，Δθ 是相对参考温度的温差。铜的 α ≈ 3.9 × 10⁻³ K⁻¹；钨约为 4.5 × 10⁻³ K⁻¹；而镍铬合金仅约 0.4 × 10⁻³ K⁻¹。
o depends on geometry, since ρ = R A ÷ L. When a metal is heated, both ρ and R increase because thermal expansion changes A and L only very slightly.
同一关系也可写成电阻率形式 ρ = ρ₀ (1 + α Δθ)。请记住，电阻率是材料的属性，而电阻还取决于几何尺寸，因为 ρ = R A ÷ L。金属受热时，ρ 和 R 都会增大，因为热膨胀对 A 和 L 的改变非常小。
Do not apply this formula to a thermistor or a filament lamp over a wide temperature range. The graph is not linear for those components, and the mark scheme will penalise unjustified linear extrapolation.
不要把这一公式套用到热敏电阻或大温度范围内的灯丝上。这些元件的图像并非直线，评分标准会扣掉那些未经论证就作线性外推的答案。
6. Comparing Different Materials | 不同材料的比较
The table below summarises how the resistance of common conductors, semiconductors and special components responds to a rise in temperature.
下表总结了常见导体、半导体和特殊元件的电阻如何随温度升高而响应。
| Material | 材料 | Effect of heating | 升温效应 | Dominant reason | 主要原因 |
|---|---|---|
| Pure metal (copper, tungsten) | 纯金属 | R rises nearly linearly; positive α | 电阻近似线性上升 | n constant; more frequent scattering | 载流子数密度不变；散射更频繁 |
| Alloy (nichrome, manganin) | 合金 | R rises very slowly; α close to zero | 电阻上升极慢 | Disorder scattering dominates | 无序散射占主导 |
| Semiconductor (silicon, germanium) | 半导体 | R falls as temperature rises | 电阻随温度下降 | n increases rapidly as pairs are freed | 载流子数密度迅速增大 |
| NTC thermistor (metal oxides) | NTC 热敏电阻 | R falls steeply and non-linearly | 电阻急剧非线性下降 | Thermally generated carriers | 热激发载流子 |
| PTC thermistor (barium titanate) | PTC 热敏电阻 | R almost constant, then rises sharply | 超过临界温度后急剧上升 | Structural phase transition | 结构相变 |
| Superconductor (mercury below 4.2 K) | 超导体 | R drops abruptly to zero | 电阻骤降为零 | Electron pairing below critical temperature | 临界温度以下电子配对 |
| Electrolyte | 电解液 | R usually falls on heating | 升温时电阻通常下降 | Greater ion mobility | 离子迁移率增大 |
Notice that the sign of the temperature coefficient is not a property of 'conductors in general' but of the specific conduction mechanism involved.
请注意，温度系数的正负并不是"导体"这一类物质的普遍属性，而是由具体的导电机理决定的。
7. Semiconductors and Negative Temperature Coefficient | 半导体与负温度系数
In a semiconductor such as silicon or germanium, the energy gap between the valence band and the conduction band is small. Thermal energy can therefore break covalent bonds and release electron–hole pairs.
在硅或锗这样的半导体中，价带与导带之间的能隙很小。因此热运动能量可以打断共价键，释放出电子—空穴对。
As temperature rises, the number density n of charge carriers increases roughly exponentially, so the number of carriers available to carry current increases dramatically. Although scattering also becomes more frequent, the increase in n outweighs it by a very large margin.
随着温度升高，载流子数密度 n 大致呈指数增长，可用来导电的载流子数量急剧增加。虽然散射也变得更频繁，但 n 的增大以极大的幅度压倒了这一效应。
The result is that the resistance of a semiconductor falls as temperature rises: a negative temperature coefficient. For silicon the gap is about 1.1 eV and for germanium about 0.67 eV, which is why germanium devices are more temperature-sensitive.
结果是半导体的电阻随温度升高而下降，即负温度系数。硅的能隙约为 1.1 eV，锗约为 0.67 eV，这就是锗器件对温度更敏感的原因。
In an exam, the key marking point is the phrase 'the number density of charge carriers increases' or 'more electron–hole pairs are generated'. Statements about ions vibrating faster alone will not explain a negative coefficient.
在考试中，关键的得分点是"载流子数密度增大"或"产生更多的电子—空穴对"。仅仅说"离子振动更快"是无法解释负温度系数的。
8. Thermistors: NTC and PTC | 热敏电阻：NTC 与 PTC
A thermistor is a semiconductor device whose resistance changes markedly with temperature. The common NTC (negative temperature coefficient) thermistor is made from sintered metal oxides such as manganese, nickel and cobalt oxides.
热敏电阻是一种电阻随温度显著变化的半导体器件。常见的 NTC（负温度系数）热敏电阻由锰、镍、钴等金属氧化物烧结而成。
kΩ at 0 °C to under 500 Ω at 100 °C. The relationship is often written in the form R = a eb ÷ θ where a and b are constants for the device.
NTC 热敏电阻在温度升高时表现出大幅度的非线性电阻下降。典型元件可能从 0 °C 时的约 10 kΩ 降到 100 °C 时的不足 500 Ω。其关系常写成 R = a eb ÷ θ 的形式，其中 a 与 b 是该器件的常数。
A PTC thermistor behaves differently: its resistance stays low up to a critical temperature, then rises very sharply. This makes it useful as a self-resetting overcurrent protection device and as a heating element that limits its own current.
PTC 热敏电阻的行为不同：其电阻在临界温度之前保持较低，超过后急剧上升。这使它可用作自复位过流保护器件，以及能够限制自身电流的加热元件。
Compared with a pure metal, a thermistor gives a much larger and much more non-linear change, which is exactly why it is preferred in temperature sensing circuits and in potential dividers used as thermostats.
与纯金属相比，热敏电阻的变化幅度大得多、非线性也强得多，这正是它在温度传感电路和用作恒温器的分压电路中更受青睐的原因。
9. Superconductivity at Low Temperatures | 低温下的超导现象
If certain materials are cooled below a critical temperature, their resistance drops abruptly to zero. Mercury does this at about 4.2 K, and many modern ceramic materials do so at much higher temperatures.
若将某些材料冷却到临界温度以下，其电阻会骤然降为零。汞大约在 4.2 K 时出现这一现象，而许多现代陶瓷材料则在更高温度下就能实现。
This is superconductivity, and it cannot be explained by the classical collision model used for metals. Below the critical temperature, electrons form pairs that move through the lattice without losing energy to the vibrating ions.
这就是超导现象，它无法用解释金属的经典碰撞模型来说明。在临界温度以下，电子形成配对，可以穿过晶格而不把能量损失给振动的离子。
Applications include powerful electromagnets for MRI scanners, maglev trains and low-loss power transmission. The practical limitation is the cost and complexity of maintaining the very low temperatures required.
其应用包括 MRI 扫描仪用的强电磁铁、磁悬浮列车和低损耗输电。实际限制在于维持所需极低温的成本与复杂性。
10. Experimental Investigation | 实验探究
A standard practical is to determine how the resistance of a metal wire and of an NTC thermistor varies with temperature. The wire or thermistor is placed in a water bath together with a thermometer, and the bath is heated slowly.
一个标准的实验是测定金属丝和 NTC 热敏电阻的电阻如何随温度变化。把金属丝或热敏电阻与温度计一起放入水浴中，然后缓慢加热。
R = V ÷ I. Take at least six readings spread over the available range, and record them in a table with the temperature in °C.
电压用并联的电压表测量，电流用串联的电流表测量，再由 R = V ÷ I 计算电阻。在可用温度范围内至少取六组读数，并把温度（°C）记入数据表格。
Precautions include using a small current so that the heating effect of the current itself is negligible, stirring the water continuously so the temperature is uniform, and waiting for thermal equilibrium before each reading. Plot resistance on the y-axis against temperature on the x-axis.
注意事项包括：使用小电流，使电流自身的热效应可以忽略；持续搅拌水以保证温度均匀；每次读数前等待热平衡。作图时以电阻为纵轴、温度为横轴。
A metal gives a straight line of positive gradient, whereas an NTC thermistor gives a curve whose gradient becomes steadily less negative as temperature rises.
金属得到的是正斜率的直线，而 NTC 热敏电阻给出的是斜率随温度升高而逐渐变得不那么负的曲线。
11. Worked Examples | 典型例题
Example 1. ing R = R₀(1 + αΔθ), R = 5.0 × (1 + 3.9 × 10⁻³ × 60) = 5.0 × 1.234 = 6.17 Ω, so R ≈ 6.2 Ω.
例 1. 一个铜线圈在 0 °C 时电阻为 5.0 Ω。取 α = 3.9 × 10⁻³ K⁻¹，求 60 °C 时的电阻。由 R = R₀(1 + αΔθ)，R = 5.0 × (1 + 3.9 × 10⁻³ × 60) = 5.0 × 1.234 = 6.17 Ω，即 R ≈ 6.2 Ω。
Example 2. brate more, electrons collide more frequently, drift velocity falls for the same applied voltage, so R = V ÷ I increases.
例 2. 一盏白炽灯接通后亮度逐渐增大。解释其电阻为何上升。灯丝温度升高，晶格离子振动加剧，电子碰撞更频繁，在相同电压下漂移速度下降，因此 R = V ÷ I 增大。
Example 3. An NTC thermistor is connected as the top resistor of a potential divider. As the temperature rises its resistance falls, so it takes a smaller share of the supply voltage and the output voltage across the bottom resistor increases.
例 3. 将 NTC 热敏电阻作为分压电路的上方电阻接成电路。温度升高时其电阻下降，因而分得的电压减小，下方电阻两端的输出电压随之增大。
12. Exam Tips and Common Mistakes | 考试要点与常见错误
reases at constant pd → current falls → resistance rises. Marking schemes reward each link in the chain.
第一，永远先讲机理再下结论：离子振动加剧 → 碰撞更频繁 → 平均自由程变短 → 恒定电压下漂移速度减小 → 电流下降 → 电阻上升。评分标准会给链条中的每一个环节计分。
Second, do not confuse resistance with resistivity, and do not say that n changes in a metal. Third, remember that a thermistor is not ohmic in the ordinary sense: its resistance changes with temperature as well as with current, so its I–V graph is not a straight line.
第二，不要把电阻与电阻率混为一谈，也不要说金属中的 n 会变化。第三，记住热敏电阻在通常意义上并非欧姆元件：它的电阻既随温度变化也随电流变化，因此其 I–V 图像不是直线。
Finally, be precise about direction. Metals: positive coefficient, resistance rises. Semiconductors and NTC thermistors: negative coefficient, resistance falls. PTC thermistors and superconductors are special cases with their own distinctive behaviour, and questions often use them to test whether you really understand the mechanism rather than a memorised rule.
最后，要精确说明变化方向。金属：正温度系数，电阻上升。半导体与 NTC 热敏电阻：负温度系数，电阻下降。PTC 热敏电阻与超导体属于特例，各有其独特行为，考题常用它们来检验你是否真正理解机理，而不是死记结论。
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