📚 IB Physics B.5 Current and Circuits SL: Experimental Investigation & Answers | IB 物理 B.5 电流与电路 SL：实验探究与解答
Experimental work lies at the heart of Topic B.5 Current and Circuits in the IB Physics SL syllabus. Through hands-on investigations you will not only verify the fundamental laws of electric circuits but also learn to design procedures, collect and process data, and critically evaluate uncertainties. This article guides you through the core experiments—Ohm’s law, resistivity, series and parallel networks, internal resistance, potential dividers and non-ohmic devices—and provides model answers, typical data analyses and examiner advice to help you master the practical skills required for internal assessment and exam questions.
实验探究是 IB 物理 SL 课程 B.5 电流与电路的核心。通过动手实验，你不仅能验证电路的基本定律，还能学会设计步骤、收集处理数据，并批判性地评估不确定度。本文带你梳理欧姆定律、电阻率、串并联网络、内阻、分压器和非欧姆器件等核心实验，并提供模型答案、典型数据分析与考官建议，帮助你掌握内部评估和考试题目要求的实践技能。
1. The Role of Experiments in B.5 | 实验在 B.5 中的作用
Experiments in B.5 transform abstract definitions of current, potential difference and resistance into measurable quantities. They train you to recognise the limitations of idealised models, such as the temperature dependence of a filament or the internal resistance of a real battery. Moreover, investigations give you the opportunity to practise controlled variables, range of measurements, graphical analysis and uncertainty propagation—all of which are assessed in IB physics.
B.5 的实验将电流、电势差和电阻等抽象定义转化为可测量的量。它们训练你认识理想模型的局限性，例如灯丝的温度依赖性，或是真实电池的内阻。此外，探究活动让你有机会练习控制变量、测量范围、图像分析以及不确定度传递——这些都是 IB 物理的评估内容。
2. Key Experimental Setups and Safety | 关键实验装置与安全
A standard circuit investigation kit usually includes a DC power supply (or battery), ammeter, voltmeter, rheostat (variable resistor), fixed resistors, connecting wires and a switch. When building circuits, always check that the ammeter is connected in series and the voltmeter in parallel. Start with the power supply set to a low voltage to prevent components overheating. Never exceed the current rating of the resistor; use a protective series resistor if needed. Keep the circuit switched off while altering connections.
标准的电路探究工具通常包括直流电源（或电池）、安培表、伏特表、变阻器、定值电阻、连接导线和开关。搭建电路时，务必检查安培表串联、伏特表并联。先将电源电压设置在较低值以防元件过热。切勿超过电阻的额定电流；必要时串联保护电阻。改动接线时保持电路断开。
3. Investigating Ohm’s Law | 探究欧姆定律
This classic experiment verifies that for an ohmic conductor at constant temperature, the current I through the resistor is directly proportional to the potential difference V across it.
这个经典实验验证了在恒定温度下，对欧姆导体而言，通过电阻的电流 I 与它两端的电势差 V 成正比。
Typical procedure: Connect a fixed resistor (e.g. 100 Ω) in series with an ammeter and a variable power supply. A voltmeter is placed in parallel with the resistor. Vary the supply voltage in small steps, recording pairs of V and I. Plot a graph of V on the y‑axis against I on the x‑axis.
典型步骤：将一个定值电阻（如 100 Ω）与安培表、可调电源串联。伏特表并联在电阻两端。以较小的步进改变电源电压，记录 V 和 I 的数据对。绘制以 V 为纵轴、I 为横轴的图像。
V = RI
Model answer insight: The graph is a straight line through the origin; its gradient equals the resistance R. If the line curves at high currents, the resistor is heating up and Ohm’s law no longer holds. The uncertainty in R can be found by drawing maximum and minimum gradient lines. A typical result: R = 98.5 Ω ± 1.2 Ω.
模型答案要点：图像是一条过原点的直线，其斜率等于电阻 R。如果图线在大电流处弯曲，说明电阻器温度升高，欧姆定律不再适用。R 的不确定度可以通过绘制最大和最小斜率线获得。典型结果：R = 98.5 Ω ± 1.2 Ω。
4. Measuring Resistivity of a Wire | 测量导线的电阻率
Resistivity ρ (rho) is a material property. For a wire of uniform cross‑sectional area A and length L, the resistance R is given by
电阻率 ρ 是材料的固有属性。对于横截面积 A 恒定、长度为 L 的导线，电阻 R 满足
R = ρL / A
where A = πd²/4, d being the diameter of the wire. The experiment measures R for different lengths L using an ohmmeter or a V‑I method. The diameter is measured with a micrometre screw gauge at several positions and averaged.
其中 A = πd²/4，d 是导线直径。实验用欧姆表或伏安法测量不同长度 L 对应的电阻 R。用螺旋测微器在不同位置多次测量直径并取平均。
Data analysis: Plot R against L. According to the equation, the graph is a straight line through the origin with gradient ρ/A. Hence ρ = gradient × A. Calculate A with its uncertainty, then propagate uncertainties. A common result for nichrome wire is ρ ≈ 1.10 × 10⁻⁶ Ω m.
数据分析：绘制 R 随 L 变化的图像。根据公式，图线为过原点的直线，斜率等于 ρ/A。因此 ρ = 斜率 × A。计算 A 及其不确定度，然后传递不确定度。镍铬合金线的常见结果约为 ρ ≈ 1.10 × 10⁻⁶ Ω m。
Model answer snippet: Better accuracy is achieved by taking at least six different lengths, keeping the wire taut, and measuring the diameter in two perpendicular directions at five points. The major source of uncertainty usually comes from the diameter measurement because it is squared in the area calculation.
模型答案片段：提高精度的方法是至少取六个不同长度，保持导线拉直，并在五个位置的两个垂直方向测量直径。不确定度的主要来源通常是直径的测量，因为它在面积计算中被平方处理。
5. Verifying Series and Parallel Rules | 验证串并联规律
You can experimentally confirm the formulas for equivalent resistance. For resistors R₁ and R₂ in series:
你可以通过实验验证等效电阻的公式。对串联的电阻 R₁ 和 R₂：
Rseries = R₁ + R₂
For parallel combination:
对并联组合：
1 / Rparallel = 1 / R₁ + 1 / R₂
Procedure: Use a multimeter to measure each individual resistor. Then build the series and parallel networks on a breadboard or with crocodile clips. Measure the total resistance of each network directly with the multimeter (circuit disconnected from power). Compare the measured total with the calculated value.
步骤：用万用表测量每个单独的电阻。然后在面包板上或用鳄鱼夹搭建串联和并联网络。用万用表直接测量每个网络的总电阻（电路不与电源连接）。比较实测值与计算值。
Typical results and answer: For R₁ = 220 Ω, R₂ = 330 Ω, the calculated Rseries = 550 Ω and Rparallel ≈ 132 Ω. Measured values might be 548 Ω and 133 Ω respectively, well within tolerance. Discrepancies arise from contact resistance and the tolerance of the resistors themselves. A conclusion should state that the experimental values agree with theory within experimental uncertainty.
典型结果与答案：对于 R₁ = 220 Ω、R₂ = 330 Ω，计算的 Rseries = 550 Ω，Rparallel ≈ 132 Ω。实测值可能分别为 548 Ω 和 133 Ω，在允差范围内。差异来源于接触电阻和电阻器本身的允差。结论应表述为在实验不确定度范围内，实验值与理论值一致。
6. Determining Internal Resistance and EMF | 测定内阻和电动势
A real cell or battery can be modelled as an ideal EMF ε (electromotive force) in series with an internal resistance r. The terminal voltage V across the cell when it delivers a current I is
真实的电池可模拟为理想电动势 ε 与内阻 r 串联。电池输出电流 I 时的端电压 V 为
V = ε − I r
This is a linear relationship with gradient −r and vertical intercept ε.
这是一个线性关系，斜率为 −r，纵截距为 ε。
Experiment: Connect a cell, a variable resistor (or resistance box), an ammeter in series, and a voltmeter across the cell. Vary the resistance and record pairs of I and V. Plot V against I. The graph is a straight line with negative slope. Read r from the magnitude of the gradient and ε from the y‑intercept. A sample data set for a D‑cell might yield ε = 1.52 V and r = 0.85 Ω. Uncertainties are derived from the best‑fit lines.
实验：将电池、可变电阻（或电阻箱）、安培表串联，并将伏特表跨接在电池两端。改变电阻并记录 I 和 V 的数据对。绘制 V 对 I 的图像。图形是一条斜率为负的直线。由斜率的大小读出 r，由纵截距读出 ε。一节 D 型电池的示例数据可能给出 ε = 1.52 V、r = 0.85 Ω。不确定度由最适线导出。
Answer hint: The graph should not be forced through the origin. Taking readings quickly prevents the cell from running down, which would change ε and r during the experiment.
答题提示：图像不应强制通过原点。快速读取数据可防止电池电量损耗，否则 ε 和 r 会在实验过程中变化。
7. Investigating Potential Dividers | 探究分压器
A potential divider uses two resistors (or one variable resistor) to supply a fraction of the input voltage. The output voltage Vout across R₂ is
分压器利用两个电阻（或一个可变电阻）提供输入电压的一部分。R₂ 两端的输出电压 Vout 为
Vout = Vin × R₂ / (R₁ + R₂)
Typical investigation: Use a fixed 10 kΩ resistor as R₁ and a 10 kΩ potentiometer as R₂. Connect a 6 V DC supply. Vary R₂ and measure Vout with a voltmeter. Record data in a table. Plot Vout against R₂/(R₁+R₂). You should obtain a straight line through the origin with gradient Vin. From the gradient you can check the supply voltage. An alternative is to replace R₂ with a thermistor or LDR and observe how Vout changes with temperature or light, linking to sensor applications.
典型探究：使用一个固定的 10 kΩ 电阻作为 R₁，一个 10 kΩ 电位器作为 R₂。接入 6 V 直流电源。改变 R₂ 并用伏特表测量 Vout。将数据记录在表格中。绘制 Vout 对 R₂/(R₁+R₂) 的图像。你应该得到一条过原点且斜率为 Vin 的直线。由斜率可检验电源电压。另一种方案是将 R₂ 换成热敏电阻或光敏电阻，观察 Vout 随温度或光照的变化，并与传感器应用相联系。
Answers and evaluation: The main sources of error are the resistance tolerances and the accuracy of the voltmeter. The linear fit verifies the formula. If a thermistor is used, the relationship is non‑linear, which leads to a discussion of calibration curves.
答案与评估：主要的误差来源于电阻的允差和伏特表的准确度。线性拟合验证了公式。如果使用热敏电阻，关系是非线性的，由此可引出校准曲线的讨论。
8. Non-Ohmic Devices: Lamp and Diode | 非欧姆器件：灯泡与二极管
Not all components obey Ohm’s law. Two important examples are the filament lamp and the semiconductor diode.
并非所有元件都遵循欧姆定律。两个重要的例子是白炽灯和半导体二极管。
Filament lamp: Set up a circuit to vary the voltage across a small 6 V lamp. Measure I and V. Plot the I‑V or V‑I characteristic. The graph is a curve that passes through the origin. At low voltages the resistance is roughly constant, but as the voltage increases, the filament heats up, the lattice ions vibrate more, increasing the resistance, so the graph bends towards the voltage axis. For an exam‑style answer: The lamp is non-ohmic because its V‑I graph is not a straight line; its resistance increases with temperature.
白炽灯：搭建电路改变小 6 V 灯泡两端的电压。测量 I 和 V。绘制 I‑V 或 V‑I 特性曲线。图像为一条过原点的曲线。低电压时电阻大致恒定，但随着电压升高，灯丝变热，晶格离子振动加剧，电阻增大，因而图线向电压轴弯曲。考试式答案：灯泡是非欧姆的，因为其 V‑I 图像不是直线；其电阻随温度升高而增大。
Diode: Use a silicon diode and a protective series resistor. Sweep the voltage from about −5 V to +1 V. The diode allows very little current (≈0) for reverse bias and even small forward bias until the threshold voltage (~0.7 V for silicon) is reached, after which current rises sharply. The characteristic is highly non‑linear, showing rectifying behaviour.
二极管：使用硅二极管并串联保护电阻。将电压从约 −5 V 扫到 +1 V。在反向偏压下，甚至在到达阈值电压（硅管约 0.7 V）之前的小正向偏压下，二极管几乎不通电流（≈0），此后电流急剧上升。特性曲线高度非线性，展现出整流行为。
9. Data Analysis Techniques and Uncertainties | 数据分析技巧与不确定度
All IB experimental write‑ups require a thorough treatment of uncertainties. After recording raw data, you should calculate absolute and percentage uncertainties for each quantity. For quantities obtained from a graph gradient or intercept, use the max‑min gradient method.
所有 IB 实验报告都需要完整的不确定度处理。记录原始数据后，应计算每个量的绝对和百分不确定度。对于从图像斜率或截距获得的量，应使用最大-最小斜率法。
Example for resistivity experiment: If the gradient of R vs L is m = ΔR/ΔL, then ρ = m × A. The uncertainty in m (Δm) is (mmax − mmin)/2. The uncertainty in A comes from the diameter uncertainty: ΔA = 2 × (Δd/d) × A. The combined relative uncertainty in ρ is the square root of the sum of squares of relative uncertainties, though at SL a simpler addition of relative uncertainties is often accepted. Provide the final result as ρ ± Δρ with the appropriate unit and rounded to the correct number of significant figures.
电阻率实验示例：如果 R 与 L 图像的斜率为 m = ΔR/ΔL，则 ρ = m × A。m 的不确定度 Δm = (mmax − mmin)/2。A 的不确定度来源于直径的不确定度：ΔA = 2 × (Δd/d) × A。ρ 的合成相对不确定度是各相对不确定度平方和的平方根，不过在 SL 水平通常接受较简单的相对不确定度相加。最终结果应以 ρ ± Δρ 给出，带有适当单位并按正确有效数字位数修约。
10. Common Mistakes and Pitfalls | 常见错误与陷阱
Many students lose marks by overlooking simple details. Here are the most frequent errors together with fixes:
许多学生因忽略简单细节而失分。以下是最常见的错误及修正方法：
- Polarity and connection: Connecting the ammeter in parallel or voltmeter in series. Fix: Always redraw the circuit before wiring and check with a teacher.
- 极性及连接：将安培表并联或伏特表串联。修正：接线前务必重绘电路，并请老师检查。
- Ignoring wire resistance: In low‑resistance measurements, the resistance of connecting leads and contact resistance can be significant. Fix: Use short, thick wires; null the multimeter or use a four‑wire (Kelvin) method.
- 忽略导线电阻：在低电阻测量中，连接线和接触电阻的影响可能很大。修正：使用短而粗的导线；将万用表归零或采用四线（开尔文）法。
- Letting components heat: In Ohm’s law or lamp experiments, holding the voltage too high for too long changes the resistance. Fix: Take readings quickly and allow cooling between measurements.
- 使元件发热：在欧姆定律或灯泡实验中，电压过高并保持过久会改变电阻。修正：快速读取数据，并在两次测量之间让元件冷却。
- Plotting errors: Forcing a best‑fit line through the origin when theory does not require it, or using unequal scales that distort the slope. Fix: Only force through the origin if there is a valid theoretical reason; use as much of the graph paper as possible with sensible scales.
- 绘图错误：在理论不要求的情况下强制最适线过原点，或使用不等分刻度扭曲斜率。修正：仅当有正当理论依据时才强制过原点；尽可能充分利用坐标纸并使用合理刻度。
- Inadequate repeats: Not taking enough data points or not repeating measurements. Fix: Take at least six distinct settings and, where feasible, measure each value twice and average.
- 重复不足：数据点不够或未重复测量。修正：至少设定六个不同条件，并在可行时每个值测量两次后取平均。
11. Exam-Style Questions and Model Answers | 考试风格问题与模型答案
Below is a typical Section B question that integrates experiment design and data analysis.
以下是一道整合实验设计与数据分析的典型 Section B 题目。
Question: A student wishes to determine the internal resistance r and EMF ε of a battery. She uses a variable resistor, an ammeter, a voltmeter and connecting wires. Describe how she should set up the circuit, the measurements she must take, how she should analyse the data, and explain how the values of ε and r are obtained. Include any precautions.
问题：某学生欲测定一节电池的内阻 r 和电动势 ε。她使用了可变电阻、安培表、伏特表和连接导线。请描述她应如何搭建电路、必须进行的测量、应如何分析数据，并解释如何获得 ε 和 r 的值。包括注意事项。
Model answer outline: Connect the battery, ammeter, variable resistor in series, and voltmeter across the battery terminals. Vary the resistance to obtain at least six pairs of I (current) and V (terminal voltage). Record data in a table. Plot a graph of V on the y‑axis against I on the x‑axis. The equation V = ε − Ir shows that the graph is a straight line with gradient −r and intercept ε. Draw the best‑fit straight line; its y‑intercept equals ε (in volts) and the magnitude of the gradient equals r (in ohms). For uncertainty, draw maximum and minimum reasonable lines to find rmax and rmin, then calculate Δr. Precautions: use a switch to avoid draining the battery, take readings quickly, start with the highest resistance to limit current, and check that the ammeter and voltmeter are zeroed correctly. This method yields reliable values because the linear relationship is directly tested.
模型答案概要：将电池、安培表、可变电阻串联，伏特表跨接在电池两端。改变电阻，获得至少六对 I（电流）和 V（端电压）数据。在表格中记录数据。以 V 为纵轴、I 为横轴绘图。方程 V = ε − Ir 表明图像为一条直线，斜率为 −r、截距为 ε。绘制最佳拟合直线；其纵截距等于 ε（伏特），斜率的大小等于 r（欧姆）。为求不确定度，绘制最大和最小合理直线得出 rmax 和 rmin，然后计算 Δr。注意事项：使用开关避免电池放电，快速读数，从最高电阻开始以限制电流，并检查安培表和伏特表是否正确调零。该方法得到的结果可靠，因为它直接检验了线性关系。
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