📚 B.5 Current and Circuits (SL & HL) | 电流与电路（标准与高阶）
In this unit we examine how electric charge flows through conductors to form current, and how circuits harness this flow to do useful work. Understanding current, potential difference, resistance and power is essential for analysing both simple and complex networks. The IB Physics B.5 topic covers standard level ideas such as Ohm's law, series and parallel combinations and potential dividers, and extends into higher level (HL) territory with Kirchhoff's laws, multi-loop circuits, drift velocity, conductance and electrical sensors.
在本单元中，我们研究电荷如何在导体中流动形成电流，以及电路如何利用这种流动做功。理解电流、电势差、电阻和功率，对于分析简单和复杂网络至关重要。IB 物理 B.5 主题涵盖标准水平的概念，如欧姆定律、串并联组合和分压器，并扩展到高阶水平（HL），包括基尔霍夫定律、多回路电路、漂移速度、电导以及电传感器。
1. Electric Current and Charge Flow | 电流与电荷流动
ng a time interval Δt, the average current is I = Δq / Δt. The SI unit of current is the ampere (A), and since 1 A = 1 C s⁻¹, the ampere is a coulomb per second. Current is a scalar quantity: although we refer to the direction of current, it does not add like a vector.
电流定义为电荷流动的速率。若在时间间隔 Δt 内，有电量 Δq 通过导体的某一横截面，则平均电流为 I = Δq / Δt。电流的国际单位是安培（A），由于 1 A = 1 C s⁻¹，1 安培即每秒通过 1 库仑电荷。电流是标量：虽然我们讨论电流的方向，但它并不像矢量那样相加。
I = Δq / Δt and 1 A = 1 C s⁻¹
By convention, the direction of conventional current is the direction in which positive charge carriers would move. In metallic wires the actual charge carriers are free electrons, so electrons drift in the opposite direction to the conventional current. This convention is a historical choice and must be respected when labelling circuit diagrams and applying Kirchhoff's laws.
按照惯例，电流的方向是正电荷载流子移动的方向。在金属导线中，实际的载流子是自由电子，因此电子漂移的方向与常规电流方向相反。这一约定是历史形成的人为规定，在标注电路图和运用基尔霍夫定律时必须遵守。
In most circuits we consider steady direct current (dc), in which the magnitude and direction of the current do not change with time. For current to flow continuously, there must be a complete conducting loop and a source of energy, such as a cell or a power supply. The number of charge carriers, their charge and their drift velocity all influence the magnitude of the current.
在大多数电路中，我们考虑稳定的直流电（dc），即电流的大小和方向不随时间改变。要使电流持续流动，必须有一个完整的导电回路和一个能量源，如电池或电源。载流子的数目、它们所带的电荷以及漂移速度，都影响电流的大小。
2. Potential Difference and EMF | 电势差与电动势
e passes between those points. If work W is done by a charge q moving between two points, then V = W / q. The volt is therefore one joule per coulomb, 1 V = 1 J C⁻¹. A potential difference of 1 V means that 1 C of charge transfers 1 J of energy.
电路中两点之间的电势差（pd）是指电荷在这两点之间通过时，单位电荷从电能形式转移为其他形式的能量。若有电荷 q 从一点移动到另一点时做功 W，则 V = W / q。因此伏特等于焦耳每库仑，1 V = 1 J C⁻¹。1 V 的电势差意味着 1 C 电荷转移 1 J 的能量。
V = W / q and 1 V = 1 J C⁻¹
ther forms (chemical, mechanical, solar) to electrical form per unit charge driven around the complete circuit. Although it is called a "force" 1.5 V gives 1.5 J of electrical energy to every coulomb of charge that passes through it.
电源的电动势（emf）是指电源使其驱动电荷绕完整回路一周时，单位电荷从其他形式（化学能、机械能、太阳能）转移为电能的数量。尽管名称中含有"力"，电动势实际上是一个以伏特为单位的电势差。电动势 ε = 1.5 V 的电池，对通过它的每库仑电荷提供 1.5 J 的电能。
ross a component such as a resistor or lamp, charge loses electrical energy, so the potential difference is the energy lost per unit charge. In symbols, ε = Wsource / q and V = W_component / q.
关键区别在于能量转移的方向：在电源处，电荷获得电能，因此电动势是单位电荷获得的能量；在电阻或灯泡等元件两端，电荷失去电能，因此电势差是单位电荷失去的能量。用符号表示即 ε = W_电源 / q，V = W_元件 / q。
In practice, a voltmeter measures the potential difference between two points and is therefore connected in parallel with the component, with a very high resistance so that it draws negligible current. An ideal ammeter has zero resistance and is connected in series to measure the current through a component without changing it.
在实际测量中，电压表测量两点之间的电势差，因此与元件并联，并且具有很高的电阻，使其通过的分流可忽略。理想电流表电阻为零，串联接入以测量通过元件的电流，而不会改变该电流。
3. Resistance, Resistivity and Ohm's Law | 电阻、电阻率与欧姆定律
to the current through it, R = V / I. The SI unit is the ohm (Ω), where 1 Ω = 1 V A⁻¹. A resistor of 10 Ω carries 0.5 A when the potential difference across it is 5.0 V.
电阻衡量元件阻碍电流流动的强弱程度，定义为元件两端的电势差与通过它的电流之比，R = V / I。国际单位是欧姆（Ω），1 Ω = 1 V A⁻¹。当 10 Ω 的电阻两端电势差为 5.0 V 时，通过它的电流为 0.5 A。
R = V / I with 1 Ω = 1 V A⁻¹
Ohm's law states that the current through a metallic conductor is directly proportional to the potential difference across it, provided the physical conditions (especially temperature) remain constant. In symbols, I ∝ V, so the I-V graph is a straight line through the origin and the resistance is constant. A component that obeys this relation is called ohmic.
欧姆定律指出：在物理条件（尤其是温度）保持恒定的前提下，通过金属导体的电流与其两端电势差成正比。用符号表示即 I ∝ V，因此其 I-V 图像是一条过原点的直线，电阻为常量。符合这一关系的元件称为欧姆元件。
R = ρL / A, where ρ is the resistivity of the material in Ω m. Doubling the length doubles the resistance; doubling the diameter reduces the resistance to one quarter, because area depends on the square of the radius.
电阻取决于导体的几何形状和材料。对于长度为 L、横截面积为 A 的均匀导线，其电阻满足 R = ρL / A，其中 ρ 为材料的电阻率，单位为 Ω m。长度加倍则电阻加倍；直径加倍则电阻变为原来的四分之一，因为面积与半径的平方成正比。
ρ = R A / L (unit: Ω m)
o be found experimentally by measuring V, I, L and A. For most metals, resistivity increases with temperature because lattice ions vibrate more strongly and scatter the drifting electrons more frequently; for semiconductors such as thermistors, resistivity typically decreases as temperature rises.
把两个关系式合并可得 R = ρL / A = V / I，因此可通过测量 V、I、L 和 A 由实验求出电阻率。对于大多数金属，电阻率随温度升高而增大，因为晶格离子振动加剧，对漂移电子的散射更频繁；对于热敏电阻等半导体，电阻率通常随温度升高而减小。
4. Drift Velocity and Current Density (HL) | 漂移速度与电流密度（HL）
nductor has n charge carriers per cubic metre, each of charge q, cross-sectional area A, and mean drift velocity v, then the current is I = n A v q. This relation links the macroscopic quantity I to microscopic behaviour.
在微观层面上，导体中的电流取决于载流子的数目、每个载流子的电荷以及它们的平均漂移速度。若导体每立方米有 n 个载流子，每个电荷为 q，横截面积为 A，平均漂移速度为 v，则电流为 I = n A v q。这一关系把宏观量 I 与微观行为联系起来。
I = n A v q (for electrons, q = e = 1.60 × 10⁻¹⁹ C)
e around 8 × 10²⁸ m⁻³, which means that for a current of 1 A in a wire of area 10⁻⁶ m², the drift velocity is only about 10⁻⁴ m s⁻¹, roughly 0.1 mm per second.
对于金属中的电子，q 即基本电荷 e，因此方程写为 I = n A v e。整理得 v = I / (n A e)。铜中 n 的典型值约为 8 × 10²⁸ m⁻³，这意味着在横截面积为 10⁻⁶ m² 的导线中通过 1 A 电流时，漂移速度仅为约 10⁻⁴ m s⁻¹，大约是每秒 0.1 毫米。
This very small drift velocity may seem surprising, but it explains why a lamp lights up almost instantly when the switch is closed: the electric field is established along the whole circuit at close to the speed of light, so all the electrons in the wire begin to drift at essentially the same moment. The electrons themselves move slowly, but the signal travels very fast.
如此小的漂移速度看似令人惊讶，但它解释了为什么开关一闭合灯泡几乎立刻发光：电场以接近光速在整个电路中建立，所以导线中所有电子几乎同时开始漂移。电子本身移动很慢，但信号传播极快。
ul when comparing wires of different thicknesses, because a thin wire carrying the same current has a much larger current density and therefore a larger drift velocity and a greater heating effect.
电流密度 J 定义为单位横截面积上的电流，J = I / A = n v q，单位为 A m⁻²。在比较粗细不同的导线时它很有用，因为同样电流下细导线的电流密度大得多，因此漂移速度更大，发热效应也更明显。
5. Series and Parallel Combinations | 串联与并联组合
ss the components add up to the supply pd, and the total resistance is the sum of the individual resistances: Rtotal = R₁ + R₂ + R₃ + ... . A break anywhere in a series loop stops the current everywhere.
在串联电路中，各元件依次相连，因此通过每个元件的电流相同。各元件两端的电势差之和等于电源电势差，总电阻等于各电阻之和：Rtotal = R₁ + R₂ + R₃ + ……。串联回路中任何一处断开，整个回路的电流都会停止。
Series: I = I₁ = I₂ | V = V₁ + V₂ | R = R₁ + R₂ + …
nts in the branches add up to the total current from the supply, and the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances: 1 / Rtotal = 1 / R₁ + 1 / R₂ + ... .
在并联电路中，各元件接在相同的两点之间，因此每条支路两端的电势差相同。各支路电流之和等于电源提供的总电流，总电阻的倒数等于各电阻倒数之和：1 / Rtotal = 1 / R₁ + 1 / R₂ + ……。
Parallel: V = V₁ = V₂ | I = I₁ + I₂ | 1/R = 1/R₁ + 1/R₂ + …
rallel combinations the total resistance is always smaller than the smallest individual resistance, because adding a branch provides an extra path for the current.
若只有两个电阻并联，可用"积除以和"的公式快速求合电阻：R = R₁R₂ / (R₁ + R₂)。注意并联组合的总电阻总是小于其中最小的单个电阻，因为增加一条支路就为电流提供了额外的通路。
| Quantity | 物理量 | Series | 串联 | Parallel | 并联 |
| Current 电流 | Same everywhere 处处相同 | Splits between branches 在支路间分流 |
| Potential difference 电势差 | Divides between components 在各元件间分配 | Same across each branch 各支路相同 |
| Resistance 电阻 | R = ΣRᵢ | 1/R = Σ(1/Rᵢ) |
| Failure of one component 一个元件断路 | Whole circuit stops 整个电路停止工作 | Other branches still work 其他支路仍工作 |
capacity, whereas identical cells in parallel give a larger total current capability and a smaller effective internal resistance but the same emf. This is why a torch uses cells in series to obtain a higher voltage.
电池也可以组合使用：相同电池串联给出更大的总电动势（ε_total = ε₁ + ε₂ + ……），但容量不变；相同电池并联则能提供更大的总电流能力、更小的等效内阻，但电动势不变。这正是手电筒采用电池串联以获得更高电压的原因。
6. Kirchhoff's Laws and Multi-loop Circuits (HL) | 基尔霍夫定律与多回路电路（HL）
Kirchhoff' the sum of the currents entering a junction equals the sum of the currents leaving it: ΣIin = ΣIout vation of electric charge, since charge cannot accumulate at a junction in a steady state.
基尔霍夫第一定律（节点定律）指出：流入节点的电流之和等于流出节点的电流之和，即 ΣIin = ΣIout。这实质上就是电荷守恒的表述，因为在稳定状态下电荷不可能在节点处堆积。
Σ Iin = Σ Iout (junction rule, charge conservation)
Kirchhoff' lently, the algebraic sum of all potential changes around a closed loop is zero. This expresses the conservation of energy, because a charge returning to its starting point must have the same energy as when it left.
基尔霍夫第二定律（回路定律）指出：沿电路中任意闭合回路，电动势之和等于电势降落之和，即 Σε = ΣIR。等价地说，沿闭合回路所有电势变化的代数和为零。这体现了能量守恒，因为电荷回到出发点时必须具有与出发时相同的能量。
ε = Σ I R or Σ ΔV = 0 (loop rule, energy conservation)
When solving multi-loop problems, a consistent sign convention is essential. A common approach is to choose an arbitrary direction for each unknown current, then travel around each loop in a chosen sense: an emf is positive if it drives current in the direction of travel, and a resistor contributes −IR if the assumed current flows with the travelling direction and +IR if it flows against it. A negative answer simply means the real current flows opposite to the assumed direction.
求解多回路问题时，始终一致的正负号约定至关重要。常用方法是：为每个未知电流任意设定方向，然后沿选定方向绕行回路：若电源电动势驱动电流沿绕行方向则该电动势取正；若设定电流与绕行方向相同，该电阻项为 −IR，相反则为 +IR。若解出负值，只表示实际电流方向与设定方向相反。
A typical HL problem involves two cells and several resistors forming two or three loops. Apply the junction rule to reduce the number of unknown currents, apply the loop rule to each independent loop to obtain simultaneous equations, then solve them. Always finish by checking that the currents satisfy the junction rule and that the energy balance around each loop is consistent.
典型的高阶题目涉及两个电源和若干电阻，构成两到三个回路。先应用节点定律减少未知电流的个数，再对每个独立回路应用回路定律得到联立方程，然后求解。最后一定要检验电流是否满足节点定律，以及每个回路的能量收支是否自洽。
7. Potential Dividers and Their Applications | 分压器及其应用
n a smaller, controlled output pd. For two resistors R₁ and R₂ in series across a supply Vin, the potential difference across R₂ is Vout = Vin × R₂ / (R₁ + R₂). The output is taken across R₂.
分压器由跨接在电源上的两个或多个串联电阻组成，用于获得较小的、可控的输出电势差。若 R₁ 与 R₂ 串联后接在电源 Vin 上，则 R₂ 两端的电势差为 Vout = Vin × R₂ / (R₁ + R₂)，输出电压即取自 R₂ 两端。
Vout = Vin × R₂ / (R₁ + R₂)
Because the output depends only on the ratio of the resistances, a potential divider is ideal for use with sensors. If R₂ is replaced by an NTC thermistor, the output voltage rises as the temperature falls, giving a temperature-sensing circuit. If R₂ is replaced by a light-dependent resistor (LDR), the output voltage increases when the light intensity decreases, which can be used to switch on a lamp at dusk.
由于输出电压仅取决于电阻之比，分压器非常适合同传感器配合使用。若把 R₂ 换成负温度系数（NTC）热敏电阻，则温度降低时输出电压升高，构成温度传感电路。若把 R₂ 换成光敏电阻（LDR），则光照强度降低时输出电压升高，可用于在黄昏时自动开启路灯。
One important practical point is the loading effect. When a load of resistance RL is connected across the output, RL is effectively in parallel with R₂, so the output voltage falls below the
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