Edexcel物理 电动势 内阻 电阻率 电位分压
电学在 Edexcel A-Level 物理的 Paper 1 里占的分不算最多，题目却最容易扣在细节上。同一张电路图，问的是电流、电压、功率还是内阻，答题路线完全不同。这篇按 Pearson Edexcel 9PH0 的 Topic 3 顺序，把 Electric Circuits 从电荷讲到电位分压器，每个知识点配一个能落到卷面上的算法。
Electric circuits carry a modest share of the marks in Paper 1, yet they punish sloppy wording more than almost any other topic. The same diagram can ask for current, potential difference, power or internal resistance, and each answer needs a different route through the data. This guide follows Topic 3 of the Pearson Edexcel 9PH0 specification in order, so the physics stays in the sequence your teacher will use.
电量、电流与 I = nqvA 的漂移速度
电流的定义是电荷流动的速率，写成 I = ΔQ/Δt，单位安培就是每秒一库仑。金属里真正移动的是自由电子，正电荷方向的约定只是历史遗留，判断电流方向时按约定走即可。把微观量代进去会得到 I = nqvA，n 是单位体积的自由电子数，q 是每个载流子的电荷量，v 是漂移速度，A 是导线截面积。铜的自由电子数密度大约 8 乘 10 的 28 次方每立方米，所以 1 平方毫米的导线通 1 安培时，电子的漂移速度只有 0.1 毫米每秒上下。很多学生以为电子要从电池一路跑到灯泡，实际是电场几乎瞬间建立，慢的只是载流子的平均漂移。常见的错误是算 I = nqvA 时把毫米直接代进公式，面积必须换成平方米。考试里更常考的是同一根导线改变截面积后电流怎么变，这时 n 与 v 不变，A 减半电流也减半的前提是电压重新分配。
Current is the rate of flow of charge, so one ampere means one coulomb passing a point each second. In a metal the moving charges are free electrons, and the conventional positive direction is a sign convention that predates their discovery. The relation I = nqvA links the ammeter reading to the microscopic drift of carriers; because the number density in copper is around 8 x 10^28 per cubic metre, a one square millimetre wire carrying one ampere implies a drift speed near 0.1 millimetres per second, and examiners regularly see millimetres substituted straight into the area term.
电势差、电动势与 V = W/Q
电势差的定义是单位电荷从一点移到另一点时电场力做的功，V = W/Q，所以 1 伏特等于 1 焦耳每库仑。电动势描述的是电源把其他形式的能量转成电能、对单位电荷所做的功，它跟电势差同单位，物理来源却不同。这个区别在文字题里经常被考，答成「电源两端的电压」通常拿不到定义分。理想电源的内阻为零，端电压恒等于电动势；真实电池一通电，端电压就掉下来。很多学生以为电动势和端电压是同一个量，看到题目里两个数不一样就以为抄错了。阅卷时更愿意看到「电动势是电源提供的每库仑能量，端电压是外电路实际得到的每库仑能量」这样的表述。单位换算是另一个常丢分点，毫伏与千伏要先化成伏特再代入。
Potential difference is the work done per unit charge in moving charge between two points, which is why one volt can be read as one joule per coulomb. Electromotive force describes the energy converted per unit charge inside the source, and although it shares the volt with potential difference, the two quantities are not interchangeable in a definition question. A real cell has internal resistance, so the terminal potential difference falls below the emf as soon as current flows, and many candidates lose an easy mark by treating the two readings as a printing error.
电阻、定义式与欧姆定律的适用条件
电阻定义为电压与电流的比值 R = V/I，这个式子在所有元件上都成立，因为它是定义而不是规律。欧姆定律说的是另一件事：在温度不变的前提下，通过金属导体的电流与两端电压成正比，I 与 V 的关系是过原点的直线。把定义当规律是这部分最常见的概念错误，遇到灯丝灯泡的曲线还会答成「不遵守欧姆定律」。灯丝的温度随电流上升，电阻跟着变大，曲线向电流轴弯曲，比值 I/V 越来越小。辨析类题目通常要求写出「非欧姆元件」，而不是「违反欧姆定律」。另一种失分把电阻写成 R = I/V，代数上颠倒了定义式。计算题里如果给了额定电压与功率，先求正常工作时的电阻，再讨论变温后的变化。
Resistance is defined as the ratio of potential difference to current, and that definition holds for every component, ohmic or not. Ohm's law is a separate statement about a specific class of conductor: at constant temperature the current through a metal wire is proportional to the potential difference across it, which is why its graph is a straight line through the origin. A filament lamp fails that condition because its temperature rises with current, so the curve bends towards the current axis, and a candidate who calls the lamp a law-breaker instead of a non-ohmic component is throwing away a definition mark.
电阻率 ρ = RA/L 与温度效应
电阻率把材料本身的导电能力与几何尺寸分开，公式是 ρ = RA/L，单位是欧姆米。铜的电阻率大约 1.7 乘 10 的负 8 次方欧姆米，镍铬合金大约 1.1 乘 10 的负 6 次方欧姆米，差了近两个数量级，这就是电热丝用合金、导线用铜的原因。把导线拉长一倍、体积不变时，长度加倍而截面积减半，电阻变成原来的四倍。这个结论考试反复出现，推导时两条关系都要写出来。金属的电阻率随温度升高而增大，因为晶格振动加剧、电子碰撞更频繁。热敏电阻（NTC）反过来，温度升高时电阻明显下降。常见的错误是只记住「电阻随温度变化」，却分不清金属与热敏电阻的方向。
Resistivity separates the material from the geometry, which is why the same copper can make a thick busbar and a hair-thin wire with wildly different resistances. Because ρ = RA/L, stretching a wire to twice its length at constant volume doubles the length and halves the cross-sectional area, so the resistance rises by a factor of four. Metals become more resistive as temperature rises, whereas an NTC thermistor becomes less resistive, and confusion between the two directions is one of the most common slips in this section.
三种元件的 I-V 特性曲线
I-V 曲线的斜率不是电阻，而是 1/R，读图时要先取点再算 V/I，不能拿切线斜率当电阻。欧姆导体的曲线是过原点的直线；灯丝灯泡随电压增大而弯向电流轴；二极管在正向电压超过门槛之后才明显导通，反向几乎不导通。Edexcel 考纲要求能定性地解释这三条曲线，所以答「因为温度升高」之后还要补一句「载流子与晶格的碰撞增多，电阻变大」。很多学生以为二极管的正向曲线也是直的，其实它的电阻随电流迅速减小。作图题要标清坐标轴与单位，阅卷时坐标轴漏写单位是常见扣分点，曲线形状正确也只给一半分。测量时要保证电压从小到大单向调节，来回调会因温度滞后读出重复性很差的数据。
The gradient of an I-V graph is not resistance but its reciprocal, so the safe method is to read a point and divide the potential difference by the current there. An ohmic conductor gives a straight line through the origin, a filament lamp bends towards the current axis as the filament heats up, and a diode conducts appreciably only once the forward potential difference passes its threshold. Marks in this section come from explaining the shape, which means naming the rise in lattice vibrations and the extra collisions rather than simply stating that the temperature went up.
电功率与能量转移 P = VI
功率的三个常用形式是 P = VI、P = I²R 与 P = V²/R，选哪一个取决于题目给了什么量。串联电路里电流相同，用 I²R 最省事；并联电路里电压相同，用 V²/R 更直接。能量写成 W = VIt，单位是焦耳，跟功率乘时间一致。Edexcel 官方公式表里这几条都在同一栏，考场上不必背错符号，前提是分清 I 与 V 各自指哪一段。计算电热器或输电损耗时，先确认所求的是整段电路还是某个元件。常见的错误是把额定功率直接用到电压不足的情境里，功率会随实际电压变化。很多学生以为电阻固定，功率就一定随电压线性增加，实际上 P 与 V 的平方成正比。
Power in a circuit can be written as P = VI, P = I squared R or P = V squared over R, and the choice should follow the quantity the question supplies rather than habit. In a series circuit the current is common, so the squared-current form is fastest, while in parallel branches the potential difference is common and the squared-voltage form saves a step. Energy transferred is simply power multiplied by time, and a frequent error is applying a lamp's rated power to a circuit that supplies a lower potential difference, which overstates both the power and the current.
电动势、内阻与 ε = I(R + r)
把电源看成一个理想电动势串联一个内阻，整个回路就变成 ε = I(R + r)，这是闭合电路的核心方程。端电压写成 V = ε - Ir，所以电流越大，端电压掉得越多。这个模型能解释为什么旧电池接上负载后电压明显下降，而空载时用万用表量着还接近 1.5 伏。测量内阻可以改变外电阻、记录端电压与电流，画 V 对 I 的图像，截距是 ε，斜率是负的内阻。很多学生以为内阻只存在于电池内部，其实电源线、接触点、电流表的内阻都会进入这个 r。计算时先把总电阻写清楚，再求电流，最后求端电压，顺序反了容易把 ε 当成外电路电压。解答题里要先画等效电路，再标出电流方向。
Treating a cell as an ideal emf in series with an internal resistance turns the whole loop into one equation, emf equals the current multiplied by the sum of external and internal resistance. Terminal potential difference is then the emf minus the lost volts, which explains why an old battery reads close to its nominal voltage on an open circuit but sags badly under load. Plotting terminal potential difference against current gives a straight line whose intercept is the emf and whose gradient is minus the internal resistance, and candidates who forget that leads and contacts also contribute usually misread that gradient.
串联与并联的电流、电压与等效电阻
串联电路只有一个支路，电流处处相同，电源电压按电阻比例分配到各元件，等效电阻是各电阻之和。并联电路各支路电压相同，总电流是各支路电流之和，等效电阻小于任何一个支路电阻。两个电阻并联时可以用 R = R1R2/(R1+R2) 快速求解，更多支路则先算电导再加。很多学生以为并联越多电阻越大，实际是每多一条支路就多一条电流通道。常见的错误是把串联的分压规律套到并联上，写出「电压按电阻比例分配」。这一类错误在选择题里几乎每年都出现，判断依据只有一条：先看哪些元件串在同一条支路上。混合电路要拆成几段算，先算并联那一组，再串回主路。
A series circuit offers one path, so the current is identical everywhere while the supply potential difference divides in proportion to resistance, and the equivalent resistance is the simple sum. In parallel the potential difference across each branch is the same, the supply current splits between branches, and the equivalent resistance is smaller than the smallest branch resistance. Many candidates assume that adding branches must raise the total resistance, when each extra branch is really another route for charge, and the usual slip is to apply the series division rule to a parallel network.
电位分压器与传感器信号的转换
电位分压器由两个电阻串联组成，输出取自其中一个电阻两端，公式是 Vout = Vin × R2/(R1 + R2)。它的用途是把固定电源变成一个可调的电压源，或者把传感器的电阻变化转成可读的电压变化。热敏电阻放在分压器的下臂时，温度升高电阻下降，输出电压随之下降；放在上臂则相反。光敏电阻的逻辑相同，光照增强时电阻减小。判断方向时只看一件事：所求电压落在哪个电阻上，那个电阻变大，输出就变大。很多学生以为传感器一定要接电流表读数，其实分压器给出的电压信号更容易被数据记录仪采集。公式里的 R2 必须是与输出端并联的那一只。
A potential divider is two resistors in series with the output taken across one of them, giving an output potential difference equal to the supply multiplied by that resistance divided by the total. Its practical value is turning a fixed supply into a variable voltage, or converting a change in a sensor's resistance into a readable voltage. With a thermistor in the lower arm the output falls as temperature rises, and reversing the arms reverses the response, so the question can be settled by asking which resistor the meter is connected across.
分压器接上负载后的输出变化
分压器的公式只在输出端没有电流时严格成立，一旦接上负载，负载与下臂电阻并联，等效阻值下降，输出电压就低于按公式算出的值。题目里出现「接一个灯泡或电压表」时，先判断这个元件是否消耗明显电流。理想电压表的内阻视为无穷大，所以接上它不改变分压比，这是很多计算题的默认前提。真实电压表内阻有限，测量值偏低，误差大小取决于它与下臂电阻的比值。常见的错误是直接用分压公式算带负载的输出，忽略了并联这一步。阅卷时看到先算并联等效电阻、再算分压的写法，通常会给满步骤分。判断题目是否考虑负载，看它有没有给负载电阻或电表内阻。
The divider formula is exact only when nothing draws current from the output, because a load in parallel with the lower resistor lowers the effective resistance and pulls the output voltage down. When a question mentions a lamp or a voltmeter across the output, the first step is to judge whether that component takes a significant current, which is why an ideal voltmeter with infinite resistance leaves the ratio unchanged. Real meters do load the circuit, and candidates who apply the formula directly to a loaded divider tend to lose the method marks that come from combining the parallel pair first.
Topic 3 两项核心实验的误差来源
Topic 3 的两项核心实验分别是测量材料的电阻率、以及测量电池的电动势与内阻。测电阻率要用米尺量长度、用千分尺量直径，直径误差被平方放大，所以同一个位置要换方向量三次取平均。测内阻要用可变电阻改变电流，同时读电流表与电压表，用图像求截距与斜率比单点计算可靠。接触电阻与导线电阻会让测得的电阻率偏大，这是最常见的系统误差。电流表外接还是内接，取决于待测电阻与两只电表内阻的相对大小。很多学生以为多做几次读数就能消除系统误差，其实重复测量只压随机误差。表格设计题要写清物理量与单位，缺少单位通常扣分。
The two core practicals for this topic determine the resistivity of a material and the emf and internal resistance of a cell. The resistivity experiment is dominated by the diameter measurement, because that quantity is squared in the area term, so readings in two perpendicular directions at several points matter more than extra repeats of the length. Systematic effects such as contact resistance and lead resistance push the answer high, and repeated readings cannot remove them, which is why the graphical method is preferred for the emf and internal resistance practical.
Paper 1 与 Paper 3 的答题落点
Topic 3 出现在 Paper 1，也会在 Paper 3 的综合题里与实验设计合并考查。Paper 1 的题目偏重公式的直接运用，两到三步就能出结果；Paper 3 更常给一段情境或一组数据，要求设计测量步骤、说明控制变量。计算题的分通常分给公式、代入与单位三处，中间步骤写清楚比直接给答案稳。单位不统一是最高频的扣分点，毫安要换安培、平方毫米要换平方米、分钟要换秒。很多学生以为写出正确数值就能拿满分，其实缺单位或有效数字过多同样会扣。答题时先用符号推到终点，最后一步再代数，出错时容易回头检查。Edexcel 的评分标准对「显示推理过程」写得明确，纯答案的容错空间很小。
Topic 3 is examined in Paper 1 and reappears in Paper 3, where circuits are often combined with experimental design. Paper 1 questions tend to test direct use of an equation in two or three steps, while Paper 3 supplies a context or a data set and asks for a method, a control of variables and a judgement about uncertainty. Marks are normally awarded for the equation, the substitution and the unit, so a neat line of working protects more credit than a bare final number, and conversions from milliamps, square millimetres or minutes are where most of the avoidable losses occur.
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